# Spaces of polynomials with constrained divisors as Grassmanians for
traversing flows

Gabriel Katz  MIT, Department of Mathematics, 77 Massachusetts Ave.,
Cambridge, MA 02139, U.S.A<EMAIL_ADDRESS>
(Date: 03/22/2022)

###### Abstract.

We study traversing vector flows $v$ on smooth compact manifolds $X$ with
boundary. For a given compact manifold $\hat{X}$, equipped with a traversing
vector field $\hat{v}$ which is convex with respect to $\partial\hat{X}$, we
consider submersions/embeddings $\alpha:X\to\hat{X}$ such that $\dim
X=\dim\hat{X}$ and $\alpha(\partial X)$ avoids a priory chosen tangency
patterns $\Theta$ to the $\hat{v}$-trajectories. In particular, for each
$\hat{v}$-trajectory $\hat{\gamma}$, we restrict the cardinality of
$\hat{\gamma}\cap\alpha(\partial X)$ by an even number $d$. We call
$(\hat{X},\hat{v})$ a convex pseudo-envelop/envelop of the pair $(X,v)$. Here
the vector field $v=\alpha^{\dagger}(\hat{v})$ is the $\alpha$-transfer of
$\hat{v}$ to $X$.

For a fixed $(\hat{X},\hat{v})$, we introduce an equivalence relation among
convex pseudo-envelops/ envelops $\alpha:(X,v)\to(\hat{X},\hat{v})$, which we
call a quasitopy. The notion of quasitopy is a crossover between bordisms of
pseudo-envelops and their pseudo-isotopies. In the study of quasitopies
$\mathcal{QT}_{d}(Y,\mathbf{c}\Theta)$, the spaces
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ of real univariate polynomials of degree
$d$ with real divisors whose combinatorial types avoid the closed poset
$\Theta$ play the classical role of Grassmanians.

We compute, in the homotopy-theoretical terms that involve $(\hat{X},\hat{v})$
and $\mathcal{P}_{d}^{\mathbf{c}\Theta}$, the quasitopies of convex envelops
which avoid the $\Theta$-tangency patterns. We introduce characteristic
classes of pseudo-envelops and show that they are invariants of their
quasitopy classes. Then we prove that the quasitopies
$\mathcal{QT}_{d}(Y,\mathbf{c}\Theta)$ often stabilize, as $d\to\infty$.

##  1\. Introduction

This paper is the second in a series, which is inspired by the works of Arnold
[Ar], [Ar1], and Vassiliev [V]. It is a direct continuation of [K9], where
many ideas of this article are present. Here we apply these ideas to
traversing flows on manifolds with boundary. As [K9], this article relies
heavily on computations from [KSW1], [KSW2]. While [K9] is studying immersions
of compact $n$-dimensional manifolds into products $\mathbb{R}\times Y$, where
$Y$ is a compact $n$-dimensional manifold, the present paper deals with
special traversing flows on compact $(n+1)$-dimensional manifolds $X$, the
flows that admit a “virtual global section”. The existence of such sections
allows to establish a transparent correspondence between the universe of
immersions into the products and the universe of traversing flows with virtual
sections. In fact, with this correspondence in place, many results of the
present article are just reformulations of the corresponding results from
[K9], thus motivating and justifying them. However, we prove also many
propositions with no analogues in [K9] (for example, Proposition 4.1, Theorem
4.1, Corollary 4.4, Proposition 4.2, and Theorem 4.8).

Let us describe here our results informally, in a manner that clarifies their
nature, but does not involve their most general forms, which carry the burden
of combinatorial decorations.

As we have mentioned above, we study traversing vector flows $v$ (see
Definition 4.1 and [K1], [K2]) on smooth compact $(n+1)$-dimensional manifolds
$X$ with boundary. Every traversing flow admits a Lyapunov function, and this
fact may serve as a working definition of such flows. We are interested in the
combinatorial patterns $\omega$ that describe how the $v$-trajectories are
tangent to the boundary $\partial X$. Specifically, we are concerned with the
traversing vector flows whose tangency patterns $\omega$ do not belong to a
given closed poset $\Theta$. Depending on $\Theta$, the very existence of such
flows puts strong restrictions on the topology of $X$ (see [K1]-[K8], [K10]).
For example, in Fig.1, $\Theta$ includes the cubic and the double tangencies
of $\partial X$ to the family of $u$-directed lines. In other words, in this
figure, the tangency patterns $\omega=(3)$ and $\omega=(22)$ are forbidden.

In order to simplify the investigation of such flows, we “envelop” them into,
so called, convex pseudo-envelops $(\hat{X},\hat{v})$. A convex pseudo-envelop
consists of: (i) a compact smooth $(n+1)$-manifold $\hat{X}$, (ii) a
traversing vector field $\hat{v}$ on it such that the boundary
$\partial\hat{X}$ is convex (see Definition 4.5) with respect to the
$\hat{v}$-flow, (iii) a submersion $\alpha:X\to\hat{X}$ such that
$\alpha(\partial X)$ avoids a forbidden tangency patterns $\Theta$ to the
$\hat{v}$-trajectories, and (iv) the pull-back $\alpha^{\dagger}(\hat{v})$ of
$\hat{v}$, coinciding with the given vector field $v$ (see Fig.1, in which
$\hat{X}$ is a cylinder).

Figure 1. Convex pseudo-envelops
$\alpha:(X,\alpha^{\dagger}(\partial_{u}))\to(\hat{X},\partial_{u})$ of a
punctured torus $X$ (on the top), and of a punctured surface $X$ of genus $2$
(on the bottom). The envelop $\hat{X}$ is the cylinder $[0,1]\times S^{1}$,
equipped with the constant vector field $\partial_{u}$. Both submersions
$\alpha$ are generic relative to the vertical vector field $\partial_{u}$ on
the cylinder. The dots mark the multiplicity $2$ tangencies of
$\alpha(\partial X)$ to the $\theta$-fibers; the self-intersections of
$\alpha(\partial X)$ are unmarked. In both examples, the cardinality of the
fibers $\theta\circ\alpha^{\partial}:\partial X\to S^{1}$ does not exceed $6$.

The existence of a convex pseudo-envelop provides the $v$-flow on $X$ with a
“virtual global section”, a significantly effective tool. Its existence allows
us to transfer the results from [K9] about immersions/embeddings
$\\{\beta:M\to\mathbb{R}\times Y\\}$, where $M$ and $Y$ are compact smooth
$n$-manifolds, to the parallel results about convex envelops of traversing
flows.

Next, we organize convex pseudo-envelops
$\\{\alpha:(X,v)\to(\hat{X},\hat{v})\\}$ with the fixed target
$(\hat{X},\hat{v})$ into the quasitopy equivalence classes
$\mathcal{QT}_{d}(\hat{X},\hat{v};\mathbf{c}\Theta)$, where $d$ is a given
even natural number. For a typical $\hat{v}$-trajectory
$\hat{\gamma}\subset\hat{X}$, $d$ gives an upper bound of the cardinality of
the set $\hat{\gamma}\cap\alpha(\partial X)$. Fig. 3 may give some impression
of the nature of quasitopy; it depicts a pair of submersions
$\alpha_{0},\alpha_{1}$, linked by a kind of cobordism with a strict
combinatorial control of its tangent patterns.

The quasitopies come in two flavors:
$\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, formed by
regular embeddings $\alpha$, and
$\mathcal{QT}_{d}^{\mathsf{sub}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, formed by
more general submersions $\alpha$.

Consider the locus $\partial_{1}^{+}\hat{X}(\hat{v})\subset\partial\hat{X}$,
where $\hat{v}$ points inside of $\hat{X}$, and the locus
$\partial_{2}^{-}\hat{X}(\hat{v})\subset\partial_{1}^{+}\hat{X}(\hat{v})$,
where $\hat{v}$ is tangent to $\partial\hat{X}$. Then we construct two
classifying maps

(1.1)
$\displaystyle\Phi^{\mathsf{emb}}:\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)\to[(\partial_{1}^{+}\hat{X}(\hat{v}),\partial_{2}^{-}\hat{X}(\hat{v})),\,(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)],$
$\displaystyle\Phi^{\mathsf{sub}}:\mathcal{QT}_{d}^{\mathsf{sub}}(\hat{X},\hat{v};\mathbf{c}\Theta)\to[(\partial_{1}^{+}\hat{X}(\hat{v}),\partial_{2}^{-}\hat{X}(\hat{v})),\,(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)],$

whose targets are sets of homotopy classes from the quotient space
$\partial_{1}^{+}\hat{X}(\hat{v})\big{/}\partial_{2}^{-}\hat{X}(\hat{v})$ to
the universal space $\mathcal{P}_{d}^{\mathbf{c}\Theta}$—the “Grassmanian”—,
formed by real polynomials of degree $d$ whose real divisors avoid a given
poset $\Theta$.

Then we prove (see Theorem 4.2) that $\Phi^{\mathsf{emb}}$ is a bijection and
$\Phi^{\mathsf{sub}}$ is a split surjection.

For a constant vector field $\hat{v}$ on the standard ball $D^{n+1}$, the sets
$\mathcal{QT}_{d}^{\mathsf{sub/emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ are
groups; the group operation is induced by the boundary connected sum of
pseudo-envelops/envelops. These groups are abelian for $n>1$. The group
$\mathcal{QT}_{d}^{\mathsf{sub}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ is a split
extension of
$\mathcal{QT}_{d}^{\mathsf{emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ (see
Theorem 4.4). We prove that

$\mathcal{QT}_{d}^{\mathsf{emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)\approx\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt).$

In the process, in the Subsection 4.2, we notice an interesting relation
between the connected components of the of the space of convex traversing
flows on the standard $(n+1)$-ball and the monoid $\mathbf{HS}_{n-1}$ of
smooth types of integral homology $(n-1)$-spheres, being considered up to
connected sums with smooth homotopy $(n-1)$-spheres (see Proposition 4.1).

The groups
$\mathcal{QT}_{d}^{\mathsf{sub/emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ act on
the quasitopies of convex pseudo-envelops
$\mathcal{QT}_{d}^{\mathsf{sub/emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, and
the classifying maps $\Phi^{\mathsf{sub/emb}}$ in (1.1) are equivariant.

These classifying maps deliver a variety of (co)homotopy and (co)homology
invariants of traversing flows which admit convex pseudo-envelops (see Theorem
4.7, Proposition 4.6, and Theorem 4.8).

We also establish several of results (Theorem 4.3, Corollary 4.6) about the
stabilization of the sets
$\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$ as
$d\to\infty$.

Finally, we compute
$\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$ for may
special cases of $\Theta$ and $(\hat{X},\hat{v})$.

##  2\. Spaces of real polynomials with constrained real divisors

For our reader convenience, we state here a number of results from [KSW1] and
[KSW2] about the topology of spaces of real monic univariate polynomials with
constrained real divisors. These results are crucial for the applications to
follow. To make the present paper even partially self-contained, we are
basically recycling Section 2 from [K9].

Let $\mathcal{P}_{d}$ denote the space of real monic univariate polynomials of
degree $d$. Given a polynomial $P(u)=u^{d}+a_{d-1}u^{d-1}+\cdots+a_{0}$ with
real coefficients, we consider its _real divisor_ $D_{\mathbb{R}}(P)$. Let
$\omega_{i}$ denotes the multiplicity of the $i$-th real root of $P$, the real
roots being ordered by their magnitude. We call the ordered $\ell$-tuple of
natural numbers $\omega=(\omega_{1},\ldots,\omega_{\ell})$ the real root
multiplicity pattern of $P(u)$, or the multiplicity pattern for short.

Such sequences $\omega$ form a universal poset $(\mathbf{\Omega},\succ)$. The
partial order “$\succ$” in $\mathbf{\Omega}$ is defined in terms of two types
of elementary operations: merges $\\{\mathsf{M}_{i}\\}_{i}$ and inserts
$\\{\mathsf{I}_{i}\\}_{i}$ The operation $\mathsf{M}_{i}$ merges a pair of
adjacent entries $\omega_{i},\omega_{i+1}$ of
$\omega=(\omega_{1},\dots,\omega_{i},\omega_{i+1},\dots,\omega_{q})$ into a
single component $\tilde{\omega}_{i}=\omega_{i}+\omega_{i+1}$, thus forming a
new shorter sequence
$\mathsf{M}_{i}(\omega)=(\omega_{1},\dots,\tilde{\omega}_{i},\dots,\omega_{q})$.
The insert operation $\mathsf{I}_{i}$ either insert $2$ in-between
$\omega_{i}$ and $\omega_{i+1}$, thus forming a new longer sequence
$\mathsf{I}_{i}(\omega)=(\dots,\omega_{i},2,\omega_{i+1},\dots)$, or, in the
case of $\mathsf{I}_{0}$, appends $2$ before the sequence $\omega$, or, in the
case $\mathsf{I}_{q}$, appends $2$ after the sequence $\omega$.

We define the order $\omega\succ\omega^{\prime}$, where
$\omega,\omega^{\prime}\in\mathbf{\Omega}$, if one can produce
$\omega^{\prime}$ from $\omega$ by applying a sequence of these elementary
operations.

For a sequence
$\omega=(\omega_{1},\omega_{2},\,\dots\,,\omega_{q})\in\mathbf{\Omega}$, we
introduce the norm and the reduced norm of $\omega$ by the formulas:

(2.1)
$\displaystyle|\omega|=_{\mathsf{def}}\sum_{i}\;\omega_{i}\quad\text{and}\quad|\omega|^{\prime}=_{\mathsf{def}}\sum_{i}\;(\omega_{i}-1).$

Note that $q$, the cardinality of the support of $\omega$, is equal to
$|\omega|-|\omega|^{\prime}$.

Let $\mathring{{\mathsf{R}}}^{\omega}_{d}$ be of the set of all polynomials
with the real root multiplicity pattern $\omega$, and let
${\mathsf{R}}_{d}^{\omega}$ be its closure in $\mathcal{P}_{d}$.

For a given collection $\Theta$ of multiplicity patterns $\\{\omega\\}$, which
share the pairity of their norms $\\{|\omega|\\}$ and is closed under the
merge and insert operations, we consider the union $\mathcal{P}_{d}^{\Theta}$
of the subspaces $\mathring{{\mathsf{R}}}_{d}^{\omega}$, taken over all
$\omega\in\Theta$ such that $|\omega|\leq d$ and $|\omega|\equiv d\mod 2$. We
denote by $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ its complement
$\mathcal{P}_{d}\setminus\mathcal{P}^{\Theta}_{d}$.

Since $\mathcal{P}^{\Theta}_{d}$ is contractible, it makes more sense to
consider its one-point compactification $\bar{\mathcal{P}}_{d}^{\Theta}$. If
the set $\bar{\mathcal{P}}_{d}^{\Theta}$ is closed in $\bar{\mathcal{P}}_{d}$,
by the Alexander duality on the sphere $\bar{\mathcal{P}}_{d}\cong S^{d}$, we
get

$H^{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\approx
H_{d-j-1}(\bar{\mathcal{P}}_{d}^{\Theta};\mathbb{Z}).$

This implies that the spaces $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and
$\bar{\mathcal{P}}_{d}^{\Theta}$ carry the same (co)homological information.
Let us describe it in _pure combinatorial terms_.

Let us consider the following domain in $\mathbb{R}^{d+1}$:

$\displaystyle\mathcal{E}_{d}$ $\displaystyle=_{\mathsf{def}}$
$\displaystyle\big{\\{}(u,P)\in\mathbb{R}\times\mathcal{P}_{d}|\;P(u)\leq
0\big{\\}}\text{\; and its boundary}$ (2.2)
$\displaystyle\partial\mathcal{E}_{d}$ $\displaystyle=_{\mathsf{def}}$
$\displaystyle\big{\\{}(u,P)\in\mathbb{R}\times\mathcal{P}_{d}|\;P(u)=0\big{\\}}.$

The pair $(\mathcal{E}_{d},\partial\mathcal{E}_{d})$ is of a fundamental
importance for us.

For a subposet $\Theta\subset{\mathbf{\Omega}}$ and natural numbers $d,k$, we
introduce the following notations:

(2.3) $\displaystyle\Theta_{[d]}=_{\mathsf{def}}$
$\displaystyle\\{\omega\in\Theta:\;|\omega|=d\\},\quad\Theta_{\langle
d]}=_{\mathsf{def}}\\{\omega\in\Theta:\;|\omega|\leq d\\},$
$\displaystyle\Theta_{|\sim|^{\prime}=k}=_{\mathsf{def}}$
$\displaystyle\\{\omega\in\Theta:\;|\omega|^{\prime}=k\\},\quad\Theta_{|\sim|^{\prime}\geq
k}=_{\mathsf{def}}\\{\omega\in\Theta:\;|\omega|^{\prime}\geq k\\}.$

Assuming that $\Theta\subset{\mathbf{\Omega}}$ is a _closed_ sub-poset, let

$\mathbf{c}\Theta=_{\mathsf{def}}\mathbf{\Omega}\setminus\Theta\text{\; and
\;}\mathbf{c}\Theta_{\langle d]}=_{\mathsf{def}}{\mathbf{\Omega}}_{\langle
d]}\setminus\Theta_{\langle d]}.$

We denote by $\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$ the free
$\mathbb{Z}$-module, generated by the elements of ${\mathbf{\Omega}}_{\langle
d]}$. Using the merge operators ${\mathsf{M}}_{k}$ and the insert operators
${\mathsf{I}}_{k}$ from [KSW2] on ${\mathbf{\Omega}}$, we define homomorphisms
$\partial_{{\mathsf{M}}}:\mathbb{Z}[{\mathbf{\Omega}}_{\langle
d]}]\to\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$,
$\partial_{{\mathsf{I}}}:\mathbb{Z}[{\mathbf{\Omega}}_{\langle
d]}]\to\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$ by

$\partial_{{\mathsf{M}}}(\omega)=_{\mathsf{def}}-\sum_{k=1}^{s_{\omega}-1}(-1)^{k}{\mathsf{M}}_{k}(\omega)\,\text{
and
}\,\partial_{{\mathsf{I}}}(\omega)=_{\mathsf{def}}\sum_{k=0}^{s_{\omega}}(-1)^{k}{\mathsf{I}}_{k}(\omega),$

where $s_{\omega}=_{\mathsf{def}}|\omega|-|\omega|^{\prime}$. In fact,
$\partial_{{\mathsf{M}}}$ and $\partial_{{\mathsf{I}}}$ are anti-commuting
differentials [KSW2]. Therefore, the sum

(2.4)
$\displaystyle\partial=_{\mathsf{def}}\partial_{{\mathsf{M}}}+\partial_{{\mathsf{I}}}:\mathbb{Z}[{\mathbf{\Omega}}_{\langle
d]}]\to\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$

is a differential.

For a closed poset $\Theta$, the restrictions of the operators
$\partial,\partial_{{\mathsf{M}}}$, and $\partial_{{\mathsf{I}}}$ to the free
$\mathbb{Z}$-module $\mathbb{Z}[\Theta]$ are well-defined. Thus, for any
closed subposet $\Theta_{\langle d]}\subset\mathbf{\Omega}_{\langle d]}$, we
may consider the differential complex $\partial:\mathbb{Z}[\Theta_{\langle
d]}]\to\mathbb{Z}[\Theta_{\langle d]}]$, whose $(d-j)$-grading is defined by
the module $\mathbb{Z}[\Theta_{\langle d]}\cap\Theta_{|\sim|^{\prime}=j}]$. We
denote by $\partial^{\ast}:\mathbb{Z}[\Theta_{\langle
d]}]^{\ast}\to\mathbb{Z}[\Theta_{\langle d]}]^{\ast}$ its dual operator, where
$\mathbb{Z}[\Theta_{\langle
d]}]^{\ast}=_{\mathsf{def}}\mathsf{Hom}(\mathbb{Z}[\Theta_{\langle
d]}],\mathbb{Z})$.

Then we consider the quotient set $\mathbf{\Theta}^{\\#}_{\langle
d]}:=\mathbf{\Omega}_{\langle d]}\big{/}\Theta_{\langle d]}$. For the closed
subposet $\Theta_{\langle d]}$, the partial order in $\mathbf{\Omega}_{\langle
d]}$ induces a partial order in the quotient $\mathbf{\Theta}^{\\#}_{\langle
d]}$.

Finally, we introduce a new differential complex
$(\mathbb{Z}[\mathbf{\Theta}^{\\#}_{\langle d]}],\partial^{\\#})$ by including
it in the short exact sequence of differential complexes:

(2.5) $\displaystyle 0\to(\mathbb{Z}[\Theta_{\langle
d]}],\partial)\to(\mathbb{Z}[\mathbf{\Omega}_{\langle
d]}],\partial)\to(\mathbb{Z}[\mathbf{\Theta}^{\\#}_{\langle
d]}],\partial^{\\#})\to 0.$

We will rely on the following result from [KSW2], which reduces the
computation of the reduced cohomology
$\tilde{H}^{\ast}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})$ to Algebra
and Combinatorics.

######  Theorem 2.1.

([KSW2]) Let $\Theta\subset{\mathbf{\Omega}}_{\langle d]}$ be a closed
subposet. Then, for any $j\in[0,d]$, we get group isomorphisms

(2.6)
$\displaystyle\tilde{H}^{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\;\stackrel{{\scriptstyle\mathcal{D}}}{{\approx}}\;H_{d-j}(\bar{\mathcal{P}}_{d},\bar{\mathcal{P}}_{d}^{\Theta};\mathbb{Z})\;\approx
H_{d-j}\big{(}\partial^{\\#}:\mathbb{Z}[\mathbf{\Theta}^{\\#}]\to\mathbb{Z}[\mathbf{\Theta}^{\\#}]\big{)},$
$\displaystyle\quad\tilde{H}_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\;\stackrel{{\scriptstyle\mathcal{D}}}{{\approx}}\;H^{d-j}(\bar{\mathcal{P}}_{d},\bar{\mathcal{P}}_{d}^{\Theta};\mathbb{Z})\;\approx
H_{d-j}\big{(}(\partial^{\\#})^{\ast}:(\mathbb{Z}[\mathbf{\Theta}^{\\#}])^{\ast}\to(\mathbb{Z}[\mathbf{\Theta}^{\\#}])^{\ast}\big{)},$

where $\mathcal{D}$ is the Poincaré duality isomorphism. $\diamondsuit$

Consider the embedding $\epsilon_{d,d+2}:\mathcal{P}_{d}\to\mathcal{P}_{d+2}$,
defined by the formula

(2.7) $\displaystyle\epsilon_{d,d+2}(P)(u)=_{\mathsf{def}}(u^{2}+1)\cdot
P(u).$

It preserves the ${\mathbf{\Omega}}$-stratifications of the two spaces by the
combinatorial types $\omega$ of real divisors. The embedding
$\epsilon_{d,d+2}$ makes it possible to talk about _stabilization of the
homology/cohomology_ of the spaces $\bar{\mathcal{P}}_{d}^{\Theta}$ and
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$, as $d\to\infty$.

For a closed poset $\Theta\subset{\mathbf{\Omega}}$, consider the closed
finite poset $\Theta_{\langle d]}={\mathbf{\Omega}}_{\langle d]}\cap\Theta$.
It is generated by some _maximal_ elements
$\omega_{\star}^{(1)},\dots,\omega_{\star}^{(\ell)}$, where $\ell=\ell(d)$. We
introduce two useful quantities:

(2.8) $\displaystyle\eta_{\Theta}(d)$ $\displaystyle=_{\mathsf{def}}$
$\displaystyle\max_{i\,\in\,[1,\,\ell(d)]}\big{\\{}\big{(}|\omega_{\star}^{(i)}|-2|\omega_{\star}^{(i)}|^{\prime}\big{)}\big{\\}},$
(2.9) $\displaystyle\psi_{\Theta}(d)$ $\displaystyle=_{\mathsf{def}}$
$\displaystyle\frac{1}{2}\big{(}d+\eta_{\Theta}(d)\big{)}.$

Note that

$\psi_{\Theta}(d)=\frac{1}{2}\;\min_{i\,\in\,[1,\ell(d)]}\big{\\{}(d-|\omega_{\star}^{(i)}|^{\prime})+(|\omega_{\star}^{(i)}|-|\omega_{\star}^{(i)}|^{\prime})\big{\\}}<d,$

where both summands, the “codimension” $(d-|\omega_{\star}^{(i)}|^{\prime})$
and the “support” $(|\omega_{\star}^{(i)}|-|\omega_{\star}^{(i)}|^{\prime})$,
are positive and each does not exceed $d$. At the same time,
$\eta_{\Theta}(d)$ may be negative.

In the stabilization results about quasitopies, the quantity

(2.10) $\displaystyle\xi_{\Theta}(d+2)=_{\mathsf{def}}d+2-\psi_{\Theta}(d+2).$

plays the key role.

Now we are in position to state the main stabilization result from [KSW2]:

######  Theorem 2.2.

(short stabilization: $\mathbf{\\{d\Rightarrow d+2\\}}$) Let $\Theta$ be a
closed subposet of $\mathbf{\Omega}$. Let the embedding
$\epsilon_{d,d+2}:\,\mathcal{P}_{d}\subset\mathcal{P}_{d+2}$ be as in (2.7).

  * •

Then, for all $j\geq\psi_{\Theta}(d+2)-1$, we get a homological isomorphism

$(\epsilon_{d,d+2})_{\ast}:H_{j}(\bar{\mathcal{P}}_{d}^{\Theta};\mathbb{Z})\approx
H_{j+2}(\bar{\mathcal{P}}_{d+2}^{\Theta};\mathbb{Z}),$

  * •

and, for all $j\leq d+2-\psi_{\Theta}(d+2)$, a homological isomorphism

$(\epsilon_{d,d+2})_{\ast}:H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\approx
H_{j}(\mathcal{P}_{d+2}^{\mathbf{c}\Theta};\mathbb{Z}).\quad\quad\diamondsuit$

######  Definition 2.1.

A closed poset $\Theta\subseteq\mathbf{\Omega}$ is called profinite if, for
all integers $q\geq 0$, there exist only finitely many elements
$\omega\in\Theta$ such that $|\omega|^{\prime}\leq q$. $\diamondsuit$

######  Corollary 2.1.

(long stabilization: $\mathbf{\\{d\Rightarrow\infty\\}}$) For any closed
profinite poset $\Theta\subset\mathbf{\Omega}$, and for each $j$, the
homomorphism

$(\epsilon_{d,d^{\prime}})_{\ast}:\;H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\approx
H_{j}(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta};\mathbb{Z})$

is an isomorphism for all sufficiently big $d\leq d^{\prime}$, $d\equiv
d^{\prime}\mod 2$. $\diamondsuit$

As a result, we may talk about the stable homology
$H_{j}(\mathcal{P}_{\infty}^{\mathbf{c}\Theta};\mathbb{Z})$, the direct limit
$\lim_{d\to\infty}H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})$.

Let us describe a few special cases of stabilization from [KSW2]. For $k\geq
1$, $q\in[0,d]$, and $q\equiv d\mod 2$, let us consider the closed poset

(2.11) $\displaystyle{\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq
k}^{(q)}=_{\mathsf{def}}\,\big{\\{}\omega\in{\mathbf{\Omega}}_{\langle
d]}\text{ such that }|\omega|^{\prime}\geq k\text{ and }|\omega|\geq
q\big{\\}}.$

Note that, for $\Theta={\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq
k}^{(0)}=_{\mathsf{def}}{\mathbf{\Omega}}_{|\sim|^{\prime}\geq k}$, the space
$\bar{\mathcal{P}}_{d}^{\Theta}$ is the entire $(d-k)$-skeleton of
$\bar{\mathcal{P}}_{d}$.

(2.12) $\displaystyle\text{ Let
\;}A(d,k,q)=_{\mathsf{def}}\;\big{|}\chi\big{(}(\mathbb{Z}\big{[}{\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq
k}^{(q)}\big{]},\partial)\big{)}\big{|}=\big{|}\chi\big{(}\bar{\mathcal{P}}_{d}^{{\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq
k}^{(q)}}\big{)}-1\big{|},$

the absolute value of the Euler number of the differential complex
$\big{(}\mathbb{Z}\big{[}{\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq
k}^{(q)}\big{]},\partial\big{)}$.

######  Proposition 2.1.

([KSW2]) Fix $k\in[1,d)$ and $q\geq 0$ such that $q\equiv d\mod 2$. Let
$\Theta={\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq k}^{(q)}$.

Then the one-point compactification $\bar{\mathcal{P}}^{\Theta}_{d}$ has the
homotopy type of a bouquet of $(d-k)$-dimensional spheres. The number of
spheres in the bouquet equals $A(d,k,q)$. $\diamondsuit$

######  Proposition 2.2.

([KSW1]) Let $\Theta\subseteq{\mathbf{\Omega}}_{\langle
d],\,|\sim|^{\prime}\geq 2}$ be a closed poset. For $d^{\prime}\geq d+2$ such
that $d^{\prime}\equiv d\mod 2$, let $\hat{\Theta}_{\\{d^{\prime}\\}}$ be the
smallest closed poset in ${\mathbf{\Omega}}_{\langle{d^{\prime}}]}$ containing
$\Theta$.

Then for $d^{\prime}\geq d+2$, we have an isomorphism
$\pi_{1}\big{(}\mathcal{P}_{d^{\prime}}^{\mathbf{c}\hat{\Theta}_{\\{d^{\prime}\\}}}\big{)}\cong\pi_{1}\big{(}\mathcal{P}_{d+2}^{\mathbf{c}\Theta_{\\{d+2\\}}}\big{)}$
of the fundamental groups. $\diamondsuit$

##  3\. Submersions & embeddings of manifolds whose boundary has constrained
tangency patterns to the product $1$-foliations

This section forms a bridge between the results from [K9] and our main results
from Section 4. The reader may choose to surf Section 3 or to proceed directly
to Section 4. Since all the results of this section are instant derivations of
the similar results from [K9], we provide just an outline of their
validations. Although we will not use directly the results of Section 3 in
Section 4, this section may induce the “right mindset” for the reader.

Let $Y$ be a smooth compact $n$-manifold. Having in mind applications to
traversing vector flows, we move away from immersions and embeddings
$\beta:M\to\mathbb{R}\times Y$ of $n$-manifolds $M$, the topic of [K9], to
submersions and regular embeddings $\alpha:X\to\mathbb{R}\times Y$ of compact
smooth $(n+1)$-manifolds $X$ with boundary into the product $\mathbb{R}\times
Y$. When $\dim(X)=\dim(Y)+1$, the submersions and immersions are the same
notion. Moreover, the restriction of a submersion $\alpha$ to $\partial X$ is
an immersion. Of cause, if $\alpha:X\to\mathbb{R}\times Y$ is an embedding, so
is $\alpha^{\partial}:=\alpha|:\partial X\to\mathbb{R}\times Y$.

Therefore many constructions and notions from [K9], with the help of the
correspondence $\alpha\leadsto\alpha^{\partial}$, apply instantly to
submersions $\alpha:X\to\mathbb{R}\times Y$ such that $\dim X=\dim Y+1$.

######  Remark 3.1.

_From the viewpoint of this paper, the main difference between immersions
$\beta:M\to\mathbb{R}\times Y$ and submersions $\alpha:X\to\mathbb{R}\times
Y$, where $\dim X=\dim M+1$, is that not any $\beta$ is a boundary
$\alpha^{\partial}$ of some $\alpha$. For example, the figure $\infty$ in the
plane does not bound a submersion of a $2$-manifold. See [Pa] for the
comprehensive theory of possible extensions of a given immersion $\beta$ to a
submersion $\alpha$._ $\diamondsuit$

######  Example 3.1.

_The following simple construction provides _models of submersions_ that
animate our treatment. Let $W$ be a codimension zero compact submanifold of a
given manifold $V$. Consider a covering map $\pi:\tilde{W}\to W$ with a finite
fiber. Let $X\subset\tilde{W}$ be a compact codimension zero submanifold. It
is possible to isotop the imbedding $X\subset\tilde{W}$ so that $\pi:\partial
X\to W$ will be an immersion with all the multiple crossings of $\pi(\partial
X)$ being in general position. Of course, each crossing has the multiplicity
that does not exceed the cardinality of the $\pi$-fiber. Then $\pi:X\to V$ is
the model example of a submersion to keep in mind._ $\diamondsuit$

Let us introduce the central to this paper notion of quasitopy for submersions
$\alpha:X\to\mathbb{R}\times Y$, an analogue of Definition 3.7 from [K9]. We
fix a natural number $d$ and consider a closed sub-poset
$\Theta\subset\mathbf{\Omega}$ such that $(\emptyset)\notin\Theta$ (the, so
called, $\Lambda$-condition (3.10) from [K9]). For topological reasons, we
will consider only the case $d\equiv 0\mod 2$.

Let $\mathcal{L}$ be the $1$-foliation of $\mathbb{R}\times Y$ by the fibers
of the obvious projection $\mathbb{R}\times Y\to Y$, and
$\mathcal{L}^{\bullet}$ be the $1$-foliation of $\mathbb{R}\times
Y\times[0,1]$ by the fibers of $\mathbb{R}\times Y\times[0,1]\to
Y\times[0,1]$.

Let $X$ be a compact smooth $(n+1)$-dimensional manifold with boundary.
Consider a smooth map $\alpha:X\to\mathbb{R}\times Y$ such that:

  * •

$\alpha$ is a submersion,

  * •

for each $y\in Y$, the total multiplicity $m_{\beta}(y)$ of $\alpha(\partial
X)$ with respect to the foliation $\mathcal{L}$ (see [K9], formula (3.4)) is
less than or equal to $d$ and $m_{\beta}(y)\equiv d\mod 2$,

  * •

for each $y\in Y$, the combinatorial tangency pattern
$\omega^{\alpha}(y)\in\mathbf{\Omega}_{\langle d]}$ of $\alpha(\partial X)$
with respect to $\mathcal{L}$ does not belong to $\Theta$,

  * •

for each $y\in\partial Y$, $\omega^{\alpha}(y)=(\emptyset)$.

Note that the normal bundle $\nu^{\alpha}$ to $\alpha(\partial X)$ in
$\mathbb{R}\times Y$ is trivial.

Let $X_{0},X_{1}$ be two compact smooth (oriented) $(n+1)$-dimensional
manifolds with boundary. We consider a compact smooth (oriented)
$(n+2)$-manifold $W$ with conners $\partial X_{0}\coprod\partial X_{1}$ such
that $\partial W=(X_{0}\coprod X_{1})\bigcup_{\\{\partial X_{0}\coprod\partial
X_{1}\\}}\delta W$, where $\delta W$ is a smooth (oriented) cobordism between
$\partial X_{0}$ and $\partial X_{1}$. Let $Z=_{\mathsf{def}}\mathbb{R}\times
Y$ and $Z^{\partial}=_{\mathsf{def}}\mathbb{R}\times\partial Y$.

Let $A:W\to\mathbb{R}\times Z$, where $\dim(W)=\dim(Z)+1$, be a submersion. In
particular, $A:\delta W\to\mathbb{R}\times Z,\;A|_{\partial
X_{0}}\to\mathbb{R}\times(Y\times\\{0\\})$, and $A|_{\partial
X_{1}}\to\mathbb{R}\times(Y\times\\{1\\})$ are immersions.

The next two definitions lay down the foundation for notions of quasitopy of
traversing vector fields, the main subject of Section 4 (see Fig. 2).

######  Definition 3.1.

Let us fix natural numbers $d^{\prime}\geq d$, $d^{\prime}\equiv d\equiv 0\mod
2$. Consider closed subposets
$\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}$ such that
$(\emptyset)\notin\Theta$.

We say that a two submersions $\alpha_{0}:X_{0}\to\mathbb{R}\times Y$ and
$\alpha_{1}:X_{1}\to\mathbb{R}\times Y$ are
$(d,d^{\prime};\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$-quasitopic, if
there exists a compact smooth $(n+2)$-manifold $W$ as above and a smooth
submersion $A:W\to\mathbb{R}\times Z$ so that:

  * •

$A|_{X_{0}}=\alpha_{0}$ and $A|_{X_{1}}=\alpha_{1}$;

  * •

for each $z\in Z$, the total multiplicity $m_{A}(z)$ of $A(\delta W)$ with
respect to the fiber $\mathcal{L}^{\bullet}_{z}$ is such that $m_{A}(z)\leq
d^{\prime}$, $m_{A}(z)\equiv d^{\prime}\mod 2$, and the combinatorial tangency
pattern $\omega^{A}(z)$ of $A(\delta W)$ with respect to
$\mathcal{L}^{\bullet}_{z}$ belongs to $\mathbf{c}\Theta^{\prime}$;

  * •

for each $z\in Y\times(\\{0\\}\cup\\{1\\})$, the total multiplicity $m_{A}(z)$
of $A(\delta W)$ with respect to the fiber $\mathcal{L}_{z}$ is such that
$m_{A}(z)\leq d$, $m_{A}(z)\equiv d\mod 2$, and the combinatorial tangency
pattern $\omega^{A}(z)$ of $A(\delta W)$ with respect to $\mathcal{L}$ belongs
to $\mathbf{c}\Theta$.

  * •

for each $z\in Z^{\partial}$, $\omega^{A}(z)=(\emptyset)$.

We denote by
$\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
the set of quasitopy classes of such submersions/ embeddings
$\alpha:X\to\mathbb{R}\times Y$. $\diamondsuit$

It is easy to check that the quasitopy of submersions is an equivalence
relation. Recall that in [K9], Definition 3.7, we have introduced a similar
notion of quasitopy for immersions/embeddings $\beta:(M,\partial
M)\to(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$. There, we used the
notation

$\mathsf{QT}^{\mathsf{imm/emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})=_{\mathsf{def}}\mathcal{QT}^{\mathsf{imm/emb}}_{d,d^{\prime}}(Y,\partial
Y;\hfill\break\mathbf{c}\Theta,(\emptyset);\mathbf{c}\Theta^{\prime})$

for the set of equivalence classes of immersions/embeddings $\beta$ under the
quasitopy relation.

As for immersions $\beta:M\to\mathbb{R}\times Y$, for any choice of connected
components $\kappa_{1}\in\pi_{0}(\partial Y_{1}),\kappa_{2}\in\pi_{0}(\partial
Y_{2})$, the connected sum operation (see [K9], formula (3.15))

$\displaystyle\uplus:\;\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y_{1};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\times\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y_{2};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to$
(3.2)
$\displaystyle\to\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y_{1}\\#_{\partial}Y_{2};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime}),$

is well-defined for the quasitopies of submersions.

It converts the set
$\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(D^{n};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
into a group
$\underline{\overline{\mathsf{H}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
(abelian for $n>1$). This group acts, via the connected sum operation
$\uplus$, on the set
$\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$,
provided that a connected component of $\partial Y$ is chosen. To get a better
insight, compare this claim with Proposition 3.2 from [K9].

For two pairs $X_{1}\supset A_{1}$ and $X_{2}\supset A_{2}$ of topological
spaces, we denote by $[(X_{1},A_{1}),\hfill\break(X_{2},A_{2})]$ the set of
homotopy classes of continuous maps $g:X_{1}\to X_{2}$, where $g(A_{1})\subset
A_{2}$.

######  Definition 3.2.

Given three pairs of spaces $X_{1}\supset A_{1}$, $X_{2}\supset A_{2}$,
$X_{3}\supset A_{3}$, and a fixed continuous map
$\epsilon:(X_{2},A_{2})\to(X_{3},A_{3})$, we denote by

(3.3)
$\displaystyle[[(X_{1},A_{1}),\,\epsilon:(X_{2},A_{2})\to(X_{3},A_{3})]]$

the set of homotopy classes $[g]$ of continuous maps
$g:(X_{1},A_{1})\to(X_{2},A_{2})$, modulo the following equivalence relation:
by definition, $[g_{0}]\sim[g_{1}]$, where
$g_{0}:(X_{1},A_{1})\to(X_{2},A_{2})$ and
$g_{1}:(X_{1},A_{1})\to(X_{2},A_{2})$ are continuous maps, if the compositions
$\epsilon\circ g_{0}$ and $\epsilon\circ g_{1}$ are homotopic as maps from
$(X_{1},A_{1})$ to $(X_{3},A_{3})$. $\diamondsuit$

Following the proof of Proposition 3.3 and Theorem 3.2 from [K9], we get
Theorem 3.1, their analogue for submersions. It is crucial that here $d\equiv
0\mod 2$, which implies that any regular embedding $\beta:M\to\mathbb{R}\times
Y$, such that all the multiplicities $\\{m_{\beta}(y)\\}_{y\in Y}$ are even,
bounds a regular embedding $\alpha:X\to\mathbb{R}\times Y$, where $\partial
X=M$ and $\alpha|_{\partial X}=\beta$.

For an $n$-dimensional $Y$, the next lemma reduces the computation of
quasitopies
$\underline{\overline{\mathsf{QT}}}^{\mathsf{\,emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$,
based on regular embeddings $\alpha:X\to\mathbb{R}\times Y$ of
$(n+1)$-dimensional manifolds $X$, to the computation of quasitopies
$\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$,
based on the regular embeddings $\beta:M\to\mathbb{R}\times Y$ of
$n$-dimensional manifolds $M$.

######  Lemma 3.1.

For closed posets $\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}$ such
that $(\emptyset)\notin\Theta$ and $d\leq d^{\prime}$, $d\equiv
d^{\prime}\equiv 0\mod 2$, the map

$\Delta:\,\underline{\overline{\mathsf{QT}}}^{\mathsf{\,emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$

that takes an embedding $\alpha:X\to\mathbb{R}\times Y$ to the embedding
$\alpha^{\partial}:\partial X\to\mathbb{R}\times Y$ is a bijection.

As a result, the obvious map
$\mathcal{A}^{\bullet}:\,\underline{\overline{\mathsf{QT}}}^{\mathsf{\,emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to\underline{\overline{\mathsf{QT}}}^{\mathsf{\,imm}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
is injective; in other words, if two embedding are quasitopic via a
submersion, they are quasitopic via an embedding.

###### Proof.

The main step is contained in the proof of Lemma 3.6 from [K9]. Let us
describe its flavor. Since, for $d\equiv 0\mod 2$ and a closed $n$-manifold
$M$, any $(\partial\mathcal{E}_{d})$-regular (see Definition 3.4 in [K9])
embedding $\beta:M\subset\mathbb{R}\times\text{int}(Y)$ bounds a (orientable
when $Y$ is orientable) $(n+1)$-manifold
$\alpha:X_{\beta}\subset\mathbb{R}\times Y$, the map $\Delta$ is onto.
Evidently, the combinatorial tangency types to $\mathcal{L}$ are determined by
$\beta(M)$. Thus, every embedding
$\alpha\in\underline{\overline{\mathsf{QT}}}^{\mathsf{\,emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
produces an element $\alpha^{\partial}$ in
$\mathsf{QT}^{\mathsf{\,emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$.
By the same token, if a $(\partial\mathcal{E}_{d})$-regular embedding
$\beta:M\subset\mathbb{R}\times\text{int}(Y)$ bounds a
$(\partial\mathcal{E}_{d^{\prime}})$-regular embedding
$B:N\subset\mathbb{R}\times Z$ (where $Z=_{\mathsf{def}}Y\times[0,1]$), whose
tangency to $\mathcal{L}^{\bullet}$ patterns belong to
$\mathbf{c}\Theta^{\prime}$, then $X_{\beta}\cup_{\partial M}N$ bounds (an
orientable when $Y$ is orientable) $(n+2)$-manifold $W\subset\mathbb{R}\times
Z$, provided $d^{\prime}\equiv 0\mod 2$. Here $N\subset\mathbb{R}\times Z$ is
a compact $(n+1)$-manifold such that $\partial N=M$ and $B|_{M}=\beta$. Thus
the map $\Delta$ is injective. By the previous argument, it is bijective.

By [K9], Proposition 3.5, the map
$\mathcal{A}:\mathsf{QT}^{\mathsf{\,emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to\mathsf{QT}^{\mathsf{\,imm}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
(see formula (3.32) in [K9]) is injective. By the argument above, $\Delta$ is
injective. Chasing the obvious square diagram, formed by the sources and
targets of the maps $\mathcal{A},\mathcal{A}^{\bullet}$, we conclude that
$\mathcal{A}^{\bullet}$ is injective as well. ∎

Combining Lemma 3.1 with Theorem 3.2 from [K9], we get the following results.

######  Theorem 3.1.

We fix even natural numbers $d^{\prime}\geq d$, $d^{\prime}\equiv d\mod 2$,
and closed subposets
$\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}_{\langle d]}$ such that
$(\emptyset)\notin\Theta$. Let $Y$ be a smooth compact $n$-manifold.

Then any submersion $\alpha:X\to\mathbb{R}\times Y$ as in
(LABEL:multiplicity_condition) gives rise111not in a canonical fashion to a
map $\Psi(\alpha):(Y,\partial
Y)\to(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{(\emptyset)})$.
Moreover,
$(d,d^{\prime};\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$-quasitopic
submersions/embeddings $\alpha_{0}:X_{0}\to\mathbb{R}\times Y$ and
$\alpha_{1}:X_{1}\to\mathbb{R}\times Y$ produce homotopic maps
$\Psi(\alpha_{0})$ and $\Psi(\alpha_{1})$.

In this way, we get a map

$\Psi_{d,d^{\prime}}^{\mathsf{\,sub/emb}}:\;\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to\big{[}\big{[}(Y,\partial
Y),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{(\emptyset)})\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},\mathcal{P}_{d^{\prime}}^{(\emptyset)})\big{]}\big{]},$

Conversely, the homotopy class of any continuous map $G:(Y,\partial
Y)\to(\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{(\emptyset)})$ is
realized by a smooth regular _embedding_
$\alpha:X\hookrightarrow\mathbb{R}\times Y$ which satisfies
(LABEL:multiplicity_condition); that is,
$G=\Psi_{d,d^{\prime}}^{\mathsf{\,emb}}(\alpha)$.

Moreover, $\Psi_{d,d^{\prime}}^{\mathsf{\,emb}}$ is a bijection, and
$\Psi_{d,d^{\prime}}^{\mathsf{\,sub}}$ is a surjection, admitting a right
inverse. $\diamondsuit$

##  4\. Convex envelops of traversing flows, their quasitopies, &
characteristic classes

###  4.1. Traversing, generic, and convex vector fields. Morse
stratifications. Spaces of convex traversing vector fields

######  Definition 4.1.

A vector field $v\neq 0$ on a compact smooth manifold $X$ is called
traversing, if each $v$-trajectory is homeomorphic either to a closed
interval, or to a point. $\diamondsuit$

Let $v$ be a traversing and boundary generic (see [K1], [K2], and Definition
4.4 below) vector field on a compact smooth $(n+1)$-manifold $X$ with
boundary. As we will see soon, every trajectory $\gamma$ of such a vector
field $v$ generates its tangency divisor $D_{\gamma}$, an ordered sequence of
points in $\gamma$, together with their multiplicities (natural numbers).

We try to “go around” the fundamental discontinuity of the map
$x\to\gamma_{x}\to D_{\gamma_{x}}$, where $\gamma_{x}$ stands for the
$v$-trajectory through $x\in X$. This requires “to envelop” the pair $(X,v)$
in a convex envelop/pseudo-envelop $(\hat{X},\hat{v})$ (see Definition 4.8).
The convex pseudo-envelops, when available, will greatly simplify our analysis
of traversing flows. In the spirit of Section 3, we will apply our results
about immersions and submersions (against the background of product
$1$-foliations) from [K9] to the new environment of convex envelops of
traversing flows.

Following [Mo], for a generic vector field $v$ on a smooth compact
$(n+1)$-dimensional manifold $X$, such that $v\neq 0$ along $\partial X$, let
us describe an important Morse stratification
$\\{\partial_{j}^{\pm}X(v)\\}_{j\in[1,\dim X]}$ of the boundary $\partial X$.
The stratum $\partial_{j}X=_{\mathsf{def}}\partial_{j}X(v)$ has the following
description (see [K1]) in terms of an auxiliary function
$z:\hat{X}\to\mathbb{R}$ that satisfies the three properties:

  * •

$0$ is a regular value of $z$,

  * •

$z^{-1}(0)=\partial X$, and

  * •

$z^{-1}((-\infty,0])=X$.

In terms of $z$, the locus $\partial_{j}X=_{\mathsf{def}}\partial_{j}X(v)$ is
defined by the equations:

$\big{\\{}z=0,\;\mathcal{L}_{v}z=0,\;\ldots,\;\mathcal{L}_{v}^{(j-1)}z=0\big{\\}},$

where $\mathcal{L}_{v}^{(k)}$ stands for the $k$-th iteration of the Lie
derivative operator $\mathcal{L}_{v}$ in the direction of $v$ (see [K2]). The
pure stratum $\partial_{j}X^{\circ}\subset\partial_{j}X$ is defined by the
additional constraint $\mathcal{L}_{v}^{(j)}z\neq 0$. The locus
$\partial_{j}X$ is the union of two loci: (1) $\partial_{j}^{+}X$, defined by
the constraint $\mathcal{L}_{v}^{(j)}z\geq 0$, and (2) $\partial_{j}^{-}X$,
defined by the constraint $\mathcal{L}_{v}^{(j)}z\leq 0$. The two loci,
$\partial_{j}^{+}X$ and $\partial_{j}^{-}X$, share a common boundary
$\partial_{j+1}X$.

For a generic $v$, all the strata $\partial_{j}X$ are smooth
$(n+1-j)$-manifolds. The requirement of $v$ being generic with respect to
$\partial X$ may be expressed as the property of the $j$-form

(4.2) $\displaystyle dz\wedge
d(\mathcal{L}_{v}z)\wedge\;\ldots\;\wedge\;d(\mathcal{L}_{v}^{(j-1)}z)$

being a nonzero section of the bundle $\bigwedge^{j}T_{\ast}X$ along the locus
${\partial_{j}X}$ for all $j\in[1,n+1]$. If $v$ on $X$ is generic to $\partial
X$, then each point $x\in\partial X$ belongs to a unique minimal stratum
$\partial_{j}X\subset\partial X$ with a maximal $j=j(x)\leq n+1$. In the
generic case, at each $b\in\partial X$, a flag

$\mathsf{Flag}_{b}(v)={\mathsf{def}}\\{T_{b}(\partial X)=F^{n}\supset
F^{n-1}\supset\ldots\supset F^{n-j(b)+1}\\}$

is generated by the tangent spaces at $b$ to all the Morse strata
$\\{\partial_{j}X\\}_{j\leq j(b)}$ that contain $b$.

Let $\hat{v}$ be a traversing vector field on a compact smooth
$(n+1)$-dimensional manifold $\hat{X}$ with boundary. Consider a submersion
$\alpha:X\to\mathsf{int}(\hat{X})$, $\dim X=\dim\hat{X}$, such that the self-
intersections of $\alpha(\partial X)$ mutually transversal. Let
$v=\alpha^{\dagger}(\hat{v})$ be the transfer of $\hat{v}$ to $X$.

For general submersions $\alpha$, which are not necessarily embeddings, the
situation is more complex: not only one gets multiple self-intersections
$\\{\Sigma_{k}\\}_{k\in[2,n+1]}$ of various branches of $\alpha(\partial X)$,
but such self-intersections may be tangent to the $\hat{v}$-flow in a variety
of ways that produce similar Morse-type stratifications of the loci
$\Sigma_{k}$, $k\geq 2$, as well. Prior to Theorem 4.1, we will revisit this
complication.

We associate several flags $\\{\mathsf{Flag}_{b}(v)\\}_{b\in\alpha^{-1}(a)}$
with each point $a\in\alpha(\partial X)$. Let $\alpha^{\partial}_{\ast}$
denote the differential of the immersion $\alpha^{\partial}:\partial
X\to\hat{X}$.

######  Definition 4.2.

We say that several vector subspaces $\\{V_{i}\subset W\\}_{i}$ of a given
vector space $W$ are in general position, if the obvious map
$W\to\oplus_{i}(W/V_{i})$ is onto. Note that this definition allows for any
numbers of $V_{i}$’s to coincide with the ambient $W$.

We say that the flags
$\\{(\alpha^{\partial}_{\ast})[\mathsf{Flag}_{b}(v)]\\}_{b\in\alpha^{-1}(a)}$
are in general position in the ambient space $T_{a}\hat{X}$, if the
$\alpha_{\ast}$-images of the minimal strata
$\\{F^{n-j(b)+1}(v)\\}_{b\in\alpha^{-1}(a)}$ of the flags
$\\{\mathsf{Flag}_{b}(v)\\}_{b\in\alpha^{-1}(a)}$ are in general position in
$T_{a}\hat{X}$. $\diamondsuit$

######  Example 4.1.

_Consider the case $n=2$, depicted in Fig.2._

_If $\\#(\alpha^{\partial})^{-1}(a)=1$, then each flag
$(\alpha^{\partial}_{\ast})\\{F^{2}\supset F^{1}\supset F^{0}\\}$, or
$(\alpha^{\partial}_{\ast})\\{F^{2}\supset F^{1}\\}$, or
$(\alpha^{\partial}_{\ast})\\{F^{2}\\}$ is in general position at $a$. _

_If $\\#(\alpha^{\partial})^{-1}(a)=2$, a pair of flags is in general
position, if and only if, it is of the form
$(\alpha^{\partial}_{\ast})\\{F^{2}\supset F^{1}\\}$ and
$(\alpha^{\partial}_{\ast})\\{G^{2}\\}$, or of the form
$(\alpha^{\partial}_{\ast})\\{F^{2}\\}$ and
$(\alpha^{\partial}_{\ast})\\{G^{2}\\}$._

_If $\\#(\alpha^{\partial})^{-1}(a)=3$, a triple of flags is in general
position only if the pair is of the form
$(\alpha^{\partial}_{\ast})\\{F^{2}\\}$,
$(\alpha^{\partial}_{\ast})\\{G^{2}\\}$,
$(\alpha^{\partial}_{\ast})\\{H^{2}\\}$. The rest of combinations fail to be
generic._ $\diamondsuit$

Figure 2. Six locally generic configurations of $\alpha^{\partial}(\partial
X)$ in 3D. The numbers $1,2,3$ reflect the local multiplicity of the marked
point $a$ on the trajectory $\hat{\gamma}_{a}$ through it.

######  Definition 4.3.

Let $X,\hat{X}$ be smooth compact $(n+1)$-manifolds with boundary and
$\hat{v}$ a traversing vector field on $\hat{X}$. We assume that a submersion
$\alpha:X\to\hat{X}$ has the following properties: for each point
$a\in\alpha(\partial X)$, there exist a natural number $k=k(a)\leq n+1$, an
open neighborhood $U_{a}$ of $a$ in $\hat{X}$, and smooth functions
$\\{z_{1},\dots,z_{k}:U_{a}\to\mathbb{R}\\}$ such that:

  1. (1)

$0$ is a regular value for each $z_{i}$,

  2. (2)

in $U_{a}$, the locus $\alpha(\partial X)$ is given by the equation
$\\{z_{1}\cdot\ldots\cdot z_{k}=0\\}$,

  3. (3)

the differential $k$-form $dz_{1}\wedge\ldots\wedge dz_{k}\,|_{U_{a}}\neq
0.$222Thus $\alpha^{\partial}$ is $k$-normal in the sense of Definition 3.3
from [K9].

Let $\hat{\gamma}_{a}$ denote the $\hat{v}$-trajectory through $a$.

We say that a point $a\in\alpha(\partial X)$ has a multiplicity $j=j(a)$ with
respect to $\hat{v}$, if the jet
$\mathsf{jet}^{j-1}_{a}\big{(}z_{1}\cdot\ldots\cdot
z_{k}\big{|}_{\hat{\gamma}_{a}}\big{)}=0$, but
$\mathsf{jet}^{j}_{a}\big{(}z_{1}\cdot\ldots\cdot
z_{k}\big{|}_{\hat{\gamma}_{a}}\big{)}\neq 0$. $\diamondsuit$

######  Definition 4.4.

Let $X,\hat{X}$ be smooth compact $(n+1)$-manifolds with boundary and
$\hat{v}$ a traversing vector field on $\hat{X}$. Let $U$ be an open
$\hat{v}$-flow adjusted neighborhood of $a$ in $\hat{X}$. For each point
$a\in\mathsf{int}(\hat{X})$, consider a smooth transversal section $S$ of the
$\hat{v}$-flow at $a$ and the flow-generated local projections $\pi:U\to S$.

We say that a submersion $\alpha:X\to\mathsf{int}(\hat{X})$ is locally generic
relative to $\hat{v}$ if, for each point $a\in\alpha(\partial X)$,

  * •

the images of the flags333equivalently, of their minimal strata
$\\{\mathsf{Flag}_{b}(v)\\}_{b\in\alpha^{-1}(a)}$, under the differentials
$(\alpha^{\partial})_{\ast}$, are in general position in $T_{a}(\hat{X})$,

  * •

the images of the flags $\\{\mathsf{Flag}_{b}(v)\\}_{b\in\alpha^{-1}(a)}$,
under the differentials $(\pi\circ\alpha^{\partial})_{\ast}$, are in general
position in the tangent space $T_{a}S$. $\diamondsuit$

One may compare the next definition, which utilizes the notion of convexity,
with Definition 4.6, introducing the more general notion of $k$-convexity.

######  Definition 4.5.

Let $\hat{X}$ be a compact connected smooth manifold with boundary, equipped
with a vector field $\hat{v}$. We say that the pair $(\hat{X},\hat{v})$ is
convex if

  * •

$\hat{v}$ admits a Lyapunov function $\hat{f}:\hat{X}\to\mathbb{R}$ (i.e.,
$d\hat{f}(\hat{v})>0$); equivalently, $\hat{v}$ is traversing,

  * •

$\partial\hat{X}$ is locally generic with respect to $\hat{v}$,

  * •

$\partial_{2}^{+}\hat{X}(\hat{v})=\emptyset$ (equivalently, $\hat{v}$ is
convex). $\diamondsuit$

######  Remark 4.1.

_Any compact connected manifold $\hat{X}$ with boundary has a traversing
vector field [K1]. However, not any compact manifold with boundary admits a
traversing_ convex _vector field! For example, consider any surface $\hat{X}$,
obtained from a closed oriented connected surface, different from the
$2$-sphere, by removing an open disk. Such an $\hat{X}$ does not admit a
traversing convex vector field [K5]._

_By[K1], Lemma 4.1, any traversing vector field $\hat{v}$, admits a Lyapunov
function._

_If $Y$ is a _closed_ manifold, then any vector field $\hat{v}\neq 0$ that is
tangent to the fibers of the obvious projection $[0,1]\times Y\to Y$ is
evidently convex with respect to $\partial[0,1]\times Y$. The obvious function
$\hat{f}:[0,1]\times Y\to[0,1]$ has the desired Lyapunov property
$d\hat{f}(\hat{v})>0$._ $\diamondsuit$

######  Example 4.2.

_Any non-trapping Riemannian metric $g$ on a compact smooth manifold $M$ with
a convex boundary $\partial M$ produces the _geodesic vector field_
$\hat{v}^{g}$ on the unit spherical bundle $SM\to M$, which is traversing and
convex with respect to $\partial(SM)$ [K6]. The very existence of such a
metric $g$ puts severe restrictions on the topological nature of $M$._
$\diamondsuit$

Let $\mathsf{conv}(\hat{X})$ denote the space of traversing vector fields
$\hat{v}$ on $\hat{X}$ such that $(\hat{X},\hat{v})$ is a convex pair, as in
Definition 4.5. The space $\mathsf{conv}(\hat{X})$ is considered in the
$C^{\infty}$-topology.

######  Lemma 4.1.

For any $\hat{v}_{0},\hat{v}_{1}\in\mathsf{conv}(\hat{X})$ that belong to the
same path-connected component of the space $\mathsf{conv}(\hat{X})$, there
exists a smooth diffeomorphism $\phi:\hat{X}\to\hat{X}$ such that $\phi$ maps
$\hat{v}_{0}$-trajectories to $\hat{v}_{1}$-trajectories, while preserving
their orientations.

If, for a pair $\hat{v}_{0},\hat{v}_{1}\in\mathsf{conv}(\hat{X})$, there
exists a smooth isotopy
$\\{\psi^{t}:\partial\hat{X}\to\partial\hat{X}\\}_{t\in[0,1]}$ such that
$\psi^{1}(\partial_{1}^{+}\hat{X}(\hat{v}_{0}))=\partial_{1}^{+}\hat{X}(\hat{v}_{1})$,
then exists a smooth diffeomorphism $\phi:\hat{X}\to\hat{X}$, an extension of
$\psi^{1}$, that maps $\hat{v}_{0}$-trajectories to
$\hat{v}_{1}$-trajectories, while preserving their orientations.

###### Proof.

If $\hat{v}\in\mathsf{conv}(\hat{X})$ is a convex boundary generic vector
field, then by Theorem 6.6 from [K7], the stratification
$\hat{X}\supset\partial_{1}^{+}\hat{X}(\hat{v})\supset\partial_{2}^{-}\hat{X}(\hat{v})$
is stable, up to an isotopy of $\hat{X}$, under sufficiently small
perturbations of $\hat{v}$. As a result, within a path connected component of
$\hat{v}$ in $\mathsf{conv}(\hat{X})$, the smooth topological type of this
Morse stratification is stable via an isotopy. In particular,
$\partial_{1}^{+}\hat{X}(\hat{v}_{0})$ and
$\partial_{1}^{+}\hat{X}(\hat{v}_{1})$ are isotopic in $\hat{X}$, provided the
$\hat{v}_{0}$ and $\hat{v}_{1}$ are connected by a path in
$\mathsf{conv}(\hat{X})$.

Let us denote by $\\{\hat{\psi}^{t}:\hat{X}\to\hat{X}\\}_{t\in[0,1]}$ the
isotopy that transforms the $\hat{v}_{0}$-induced Morse stratification of
$\partial\hat{X}$ to the $\hat{v}_{1}$-induced Morse stratification of
$\partial\hat{X}$. Let us compare the vector fields
$\tilde{v}_{0}=_{\mathsf{def}}(\hat{\psi}^{1})_{\ast}(\hat{v}_{0})$ and
$\hat{v}_{1}$. Both vector fields point inside of $\hat{X}$ exactly along
$\partial_{1}^{+}\hat{X}(\hat{v}_{1})$. Since $\hat{v}_{0},\hat{v}_{1}$ are
traversing, they admit some Lyapunov functions
$f_{0},f_{1}:\hat{X}\to\mathbb{R}$. Put
$\tilde{f}_{0}=_{\mathsf{def}}((\hat{\psi}^{1})^{-1})^{\ast}(f_{0})$. It
serves as Lyapunov’s function for $\tilde{v}_{0}$. For
$x\in\partial_{1}^{+}\hat{X}(\hat{v}_{1})$, we denote by
$\gamma_{x}^{\\{0\\}}$ and $\gamma_{x}^{\\{1\\}}$ the $\tilde{v}_{0}$\- and
$\hat{v}_{1}$-trajectories through $x$. Let $var_{0}(x)$ stands for the
variation of the function $\tilde{f}_{0}$ along $\gamma_{x}^{\\{0\\}}$ and
$var_{1}(x)$ for the variation of the function $\hat{f}_{1}$ along
$\gamma_{x}^{\\{1\\}}$.

For $y\in\gamma_{x}^{\\{0\\}}$, consider the unique point
$\tilde{\phi}(y)\in\gamma_{x}^{\\{1\\}}$ such that
$f_{1}(\tilde{\phi}(y))/var_{1}(x)=\tilde{f}_{0}(y)/var_{0}(x)$. Now the
diffeomorphism $\phi:=\tilde{\phi}\circ\hat{\psi}^{1}$ takes
$\hat{v}_{0}$-trajectories to $\hat{v}_{0}$-trajectories, while preserving
their orientations. Hence, we have shown that the isotopy class of
$\partial_{1}^{+}\hat{X}(\hat{v})$ in $\partial X$ determines the smooth
topological type of a convex pair $(\hat{X},\mathcal{L}(\hat{v}))$, where
$\mathcal{L}(\hat{v})$ denotes the oriented $1$-foliation, determined by a
convex traversing $\hat{v}$. ∎

######  Lemma 4.2.

Let $\hat{X}$ is a connected compact smooth manifold with boundary. If
$\\#(\pi_{0}(\partial\hat{X}))\geq 3$, then
$\mathsf{conv}(\hat{X})=\emptyset$.

###### Proof.

For a convex $\hat{v}$, $\partial_{1}^{+}\hat{X}(\hat{v})$ and
$\partial_{1}^{-}\hat{X}(\hat{v})$, each is a deformation retract of
$\hat{X}$. Therefore, each of these two loci must be connected. Thus each of
the loci $\partial_{1}^{+}\hat{X}(\hat{v})$,
$\partial_{1}^{-}\hat{X}(\hat{v})$ must be contained in some connected
component of $\partial\hat{X}$. When $\\#(\pi_{0}(\partial\hat{X}))\geq 3$,
this argument forces at least one component of the boundary to be free from
both loci. However, the union of the two loci is the entire boundary. This
contradiction proves the claim. ∎

###  4.2. Homology spheres and convex flows

Let $\mathbf{HS}_{n-1}$ denote the monoid of smooth types of integral homology
$(n-1)$-spheres, being considered up to connected sums with smooth homotopy
$(n-1)$-spheres.

For $n\neq 4$, let $\mathbf{\Theta}_{n}$ denote the group of $h$-cobordism
classes of smooth homotopy $n$-spheres. The operations in $\mathbf{HS}_{n-1}$
and in $\mathbf{\Theta}_{n}$ are the connected sums of spheres. We denote the
order of the group $\mathbf{\Theta}_{n}$ by $|\mathbf{\Theta}_{n}|$.

######  Proposition 4.1.

Any convex vector field $\hat{v}$ on the standard ball $D^{n+1}$ defines a
smooth involution $\tau_{\hat{v}}$ on $S^{n}=\partial D^{n+1}$, whose fixed
point set $\partial_{2}^{-}D^{n+1}(\hat{v})$ is an integral homology
$(n-1)$-sphere.

For $n\geq 6$, any element of the set $\mathbf{HS}_{n-1}$ arises as the locus
$\partial_{2}^{-}\hat{X}(\hat{v})$ for a convex traversing vector field
$\hat{v}$ on a smooth compact contractible $(n+1)$-dimensional manifold
$\hat{X}$, whose boundary $\partial\hat{X}$ is a smooth homotopy sphere.
Moreover, $\partial\hat{X}$ admits a smooth involution $\tau_{\hat{v}}$ such
that $(\partial\hat{X})^{\tau_{\hat{v}}}=\partial_{2}^{-}\hat{X}(\hat{v})$.

For $n\geq 6$, the multiple $|\mathbf{\Theta}_{n}|\cdot[\Sigma]$ of any given
element $[\Sigma]\in\mathbf{HS}_{n-1}$ arises as the locus
$\partial_{2}^{-}D^{n+1}(\hat{v})$ for a convex traversing vector field
$\hat{v}$ on the ball $D^{n+1}$. The sphere $\partial D^{n+1}$ admits a smooth
involution $\tau_{\hat{v}}$ such that $(\partial
D^{n+1})^{\tau_{\hat{v}}}=\partial_{2}^{-}D^{n+1}(\hat{v})$.

###### Proof.

Using a convex $\hat{v}$-flow, $\partial_{1}^{+}D^{n+1}(\hat{v})$ is a
deformation retract of $D^{n+1}$ and thus a contractible manifold. By Poincaré
duality, $\partial_{2}^{-}D^{n+1}(\hat{v})$, the boundary of
$\partial_{1}^{+}D^{n+1}(\hat{v})$, is a homology sphere.

By [Ke], Theorem 3, for $n-1\geq 5$, any smooth homology sphere
$\Sigma^{n-1}$, after a connected sum $\Sigma^{n-1}\\#\,\Sigma^{n-1}_{H}$ with
a unique smooth homotopy $(n-1)$-sphere $\Sigma^{n-1}_{H}$, bounds a
contractible smooth manifold $W^{n}$.

Consider a smooth metric $g$ on $W^{n}$. We denote by $d_{g}(x,\partial
W^{n})$ the smooth distance function to $\partial W^{n}$ on a collar $U$ of
$\partial W^{n}$ in $W^{n}$. Let $F:W^{n}\to\mathbb{R}_{+}$ be a function that
is strictly positive and smooth in the interior of $W^{n}$ and coincides with
the function $\tilde{F}(x):=\sqrt{d_{g}(x,\partial W^{n})}$ in the collar $U$.

Consider a smooth manifold $\hat{X}^{n+1}\subset\mathbb{R}\times W^{n}$, given
by the inequality $\\{(t,x):|t|\leq F(x)\\}$. It comes with the vector field
$\hat{v}$ that is tangent to the fibers of the obvious projection
$p:\mathbb{R}\times W^{n}\to W^{n}$. Since $W^{n}$ is contractible, the
boundary of $\hat{X}^{n+1}$, the double of $W^{n}$, is a smooth homotopy
$n$-sphere, and $\hat{X}^{n+1}$ is a homotopy ball. The vector field $\hat{v}$
on $\hat{X}^{n+1}$ is convex and defines an involution
$\tau_{\hat{v}}:\partial\hat{X}^{n+1}\to\partial\hat{X}^{n+1}$, whose fixed
point set
$(\partial\hat{X}^{n+1})^{\tau_{\hat{v}}}=\partial_{2}^{-}\hat{X}^{n+1}(\hat{v})=\partial
W=\Sigma^{n-1}\\#\,\Sigma^{n-1}_{H}$. This validates the second claim.

The connected sum of $|\mathbf{\Theta}_{n}|$ copies of $\partial\hat{X}^{n+1}$
is a standard $n$-sphere. Consider the boundary connected sum
$(\hat{Y}^{n+1},\tilde{v})$ of the $|\mathbf{\Theta}_{n}|$ copies of the pair
$(\hat{X}^{n+1},\hat{v})$. Here the $1$-handles $H=D^{n}\times[0,1]$ are
attached at pairs of points that belong to different pairs of $\partial
W^{n}$’s so that $|\mathbf{\Theta}_{n}|$ copies of $W^{n}$ are connected by
the chain of $1$-handles $D^{n-1}_{+}\times[0,1]$, where
$D^{n-1}_{+}\subset\partial D^{n}$ is a hemisphere. The fields $\tilde{v}$ in
the different copies extend concavely across the $1$-handles. Then
$\partial\hat{Y}^{n+1}$ is the standard sphere $S^{n}$ which bounds a
contractible manifold $\hat{Y}^{n+1}$. By the $h$-cobordism theorem (see
[Mi]), applied to $\hat{Y}^{n+1}\setminus D^{n+1}$, we conclude that
$\hat{Y}^{n+1}$ is the standard ball. Thus we managed to build a convex vector
field $v^{\\#}$ on $D^{n+1}$ whose locus $\partial_{2}^{-}D^{n+1}(v^{\\#})$ is
a homology sphere $\tilde{\Sigma}^{n-1}$, the $|\mathbf{\Theta}_{n}|$-multiple
of the given class $\Sigma^{n-1}\in\mathbf{HS}_{n-1}$. In particular,
$\partial D^{n+1}$ admits a smooth involution $\tau_{v^{\\#}}$ whose fixed
point set is $|\mathbf{\Theta}_{n}|\cdot\Sigma^{n-1}$. ∎

######  Example 4.3.

_Consider a free action of the icosahedral group $\mathsf{I}_{120}$ on
$S^{3}$. Then $\mathsf{I}_{120}$ acts freely on
$S^{7}=\mathsf{join}(S^{3},S^{3})$. Hence the orbit space
$\Sigma^{7}=_{\mathsf{def}}S^{7}/\mathsf{I}_{120}$ is a homology sphere. By
Proposition 4.1, there is a convex traversing vector field $\hat{v}$ on the
ball $D^{9}$, such that its locus $\partial_{2}^{-}D^{9}(\hat{v})$ is a
connected sum of $28$ copies of $[\Sigma^{7}]\in\mathbf{HS}_{7}$, and its
locus $\partial_{1}^{+}D^{9}(\hat{v})$ is contractible. Note that
$\pi_{1}(\partial_{2}^{-}D^{9}(\hat{v}))$ is a free product of 28 copies of
$\mathsf{I}_{120}$._ $\diamondsuit$

######  Corollary 4.1.

For $n>6$, $\pi_{0}(\mathsf{conv}(D^{n+1}))$ admits a surjection onto a
subgroup $\mathbf{G}_{n-1}$ of $\mathbf{HS}_{n-1}$ of index
$|\mathbf{\Theta}_{n}|$ at most.

In particular, here are a few “clean” surjections:

$\pi_{0}(\mathsf{conv}(D^{7}))\to\mathbf{HS}_{5},\;\pi_{0}(\mathsf{conv}(D^{13}))\to\mathbf{HS}_{11},\;\pi_{0}(\mathsf{conv}(D^{62}))\to\mathbf{HS}_{60}.$

###### Proof.

For $n-1\geq 5$, let $\Sigma^{n-1}$ be a given smooth homology sphere, and let
$\Sigma^{n-1}_{H}$ be the unique homotopy sphere such that
$\Sigma^{n-1}\\#\,\Sigma^{n-1}_{H}$ bounds a smooth contractible manifold
[Ke]. By Proposition 4.1, any convex traversing vector field $\hat{v}$
produces a homology sphere $\partial_{2}^{-}D^{n+1}(\hat{v})$, and the
$|\mathbf{\Theta}_{n}|$-multiple of any $\Sigma^{n-1}\\#\,\Sigma^{n-1}_{H}$ is
produced this way. On the other hand, deforming $\hat{v}$ within the space of
boundary generic vector fields does not change the smooth isotopy type of the
pair $\partial_{1}^{+}D^{n+1}(\hat{v})\supset\partial_{2}^{+}D^{n+1}(\hat{v})$
(see Lemma 4.1, or [K1], [K2]). In particular, the smooth topological type of
the pair is preserved along a path in the space $\mathsf{conv}(D^{n+1})$.
Therefore, $\pi_{0}(\mathsf{conv}(D^{n+1}))$ admits a surjection onto a
subgroup $\mathbf{G}_{n-1}$ of $\mathbf{HS}_{n-1}$ of index
$|\mathbf{\Theta}_{n}|$ at most.

The three examples of surjections in the corollary are based on the
computations of $|\mathbf{\Theta}_{n}|$ in [KeM], [WaX]; we just use some
$n$’s for which $|\mathbf{\Theta}_{n}|=1$. ∎

###  4.3. Convex pseudo-envelops of traversing flows

The next key lemma incapsulates a given convex traversing flow
$(\hat{X},\hat{v})$ into the obvious traversing flow $\tilde{v}$ in a box
$[0,1]\times Y$ for an appropriate choice of a compact manifold $Y$, $\dim
Y=\dim\partial\hat{X}$. The construction that realizes the embedding
$(\hat{X},\hat{v})\subset([0,1]\times Y,\tilde{v})$ delivers a global “virtual
section” $\\{0\\}\times Y$ of $\hat{v}$. In turn, this section enables us to
apply the key Theorem 3.1 from [K9] to any convex traversing flow
$(\hat{X},\hat{v})$. Therefore, we will be able to transfer many results from
[K9] and from Section 3 to the environment of convex envelops and pseudo-
envelops (see Definition 4.5) of traversing boundary generic vector fields.

######  Lemma 4.3.

Let $\dim(\hat{X})=n+1$. If a pair $(\hat{X},\hat{v})$ is convex, then there
exists a compact smooth $n$-manifold $Y$ such that:

  1. (1)

$\hat{X}\subset[0,1]\times Y$,

  2. (2)

$\hat{v}$ is tangent to the fibers of the projection $p:[0,1]\times Y\to Y$,

  3. (3)

the obvious function $h:[0,1]\times Y\to[0,1]$ has the property
$dh(\hat{v})>0$,

  4. (4)

with the help of $p$, the loci $\partial_{1}^{+}\hat{X}(\hat{v})$ and
$\partial_{1}^{-}\hat{X}(\hat{v})$ each is homeomorphic to $Y$.

###### Proof.

Since $v$ is a traversing field, it admits a Lyapunov function
$\hat{f}:\hat{X}\to\mathbb{R}$ so that $d\hat{f}(v)>0$ in $\hat{X}$ [K1]. We
add a collar $U$ to $\hat{X}$ along its boundary $\partial\hat{X}$ and denote
$\hat{X}\cup_{\partial\hat{X}}U$ by $\tilde{X}$. Then, we smoothly extend
$\hat{v}$ and $\hat{f}$ in $\tilde{X}$ and denote these extensions by
$\tilde{v}$ and $\tilde{f}$. We adjust $U$ so that $\tilde{v}\neq 0$ and
$\tilde{f}(\tilde{v})>0$ there.

Let $F:\tilde{X}\to\mathbb{R}$ be a smooth function such that $0$ is its
regular value and $F^{-1}(0)=\partial\hat{X}$, $F^{-1}((-\infty,0])=\hat{X}$.
Let
$\tilde{X}(F,\epsilon)=_{\mathsf{def}}\\{x\in\tilde{X}|\;F(x)\leq\epsilon\\}$.
For a sufficiently small $\epsilon>0$, $\tilde{X}(F,\epsilon)$ is a smooth
compact manifold, contained in $\tilde{X}$.

By definition,
$\partial_{2}\hat{X}(\tilde{v}):=\\{x\in\partial\hat{X}|\;F(x)=0,\text{ and
}(\mathcal{L}_{\tilde{v}}F)(x)=0\\}$. The convexity of $\hat{v}$ in $\hat{X}$
means that $\partial_{2}^{+}\hat{X}(\hat{v})=\emptyset$, thus
$\partial_{2}^{-}\hat{X}(\hat{v})=\partial_{2}\hat{X}(\hat{v})$. By Morin’s
Theorem [Mor] (see [K2] for details), in the vicinity of each point
$x\in\partial_{2}^{-}\hat{X}(\tilde{v})$ there is a system of smooth
coordinates $(u,w,y_{1},\ldots,y_{n-1})$ in which
$F((u,w,y_{1},\ldots,y_{n-1}))=u^{2}+w$, so that $\partial\hat{X}$ is given by
the equation $u^{2}+w=0$, $\hat{X}$ by the inequality $u^{2}+w\leq 0$, and
each $\tilde{v}$-trajectory is produced by freezing the coordinates $w$ and
$y_{1},\ldots,y_{n-1}$.

Since $d\tilde{f}(\tilde{v})>0$ in $\tilde{X}(F,\epsilon)$, the field
$\tilde{v}$ is traversing in $\tilde{X}(F,\epsilon)$. Hence each
$\tilde{v}$-trajectory $\gamma\subset\tilde{X}(F,\epsilon)$ is either
transversal to $\partial\hat{X}$ at a pair of points, or it is simply tangent
to $\partial\hat{X}$ at a singleton, or does not intersect $\hat{X}$.

Consider the set $Z\subset\tilde{X}(F,\epsilon)$ of $\tilde{v}$-trajectories
through the points of $\hat{X}$ (equivalently, the set of
$\tilde{v}$-trajectories through $\partial_{1}^{+}\hat{X}(\hat{v})$). Every
such trajectory $\gamma$ is a closed oriented segment $[a(\gamma),b(\gamma)]$,
where $a(\gamma)\neq b(\gamma)\in\partial
Z\subset\partial\tilde{X}(F,\epsilon)$. Thus, the variation
$var_{\gamma}(\tilde{f})$ of the function $\tilde{f}$ along $\gamma$ is
strictly positive. Using compactness of $\hat{X}$, we get that
$\inf_{\gamma\subset Z}\;\\{var_{\gamma}(\tilde{f})\\}>0$.

Let $Y=_{\mathsf{def}}\bigcup_{\gamma\subset Z}a(\gamma)\subset\partial Z$.
Using the local model $\\{u^{2}+w\leq\epsilon\\}$, we see that $\tilde{v}$ is
transversal to $Y$ for all sufficiently small $\epsilon>0$.

Now, let us consider a new function on $\gamma$:

$h_{\gamma}(x)=_{\mathsf{def}}(var_{[a(\gamma),\,x]}\tilde{f})\big{/}(var_{[a(\gamma),\,b(\gamma)]}\tilde{f}).$

The function $h:Z\to\mathbb{R}$, defined as a collection of functions
$\\{h_{\gamma}:\gamma\to\mathbb{R}\\}_{\gamma\subset Z}$, is evidently a new
Lyapunov function for $\tilde{v}$ on $Z$. It is smooth thanks to the
transversality of $\tilde{v}$ to $Y$ and the smooth dependence of solutions of
ODEs on initial data. In fact, $h$ gives the product structure $[0,1]\times Y$
to $Z$. Indeed, any point $x\in Z$ is determined by the unique trajectory
$\gamma_{x}$ through $x$ and by the value at $x$ of the Lyapunov function $h$
of the $\tilde{v}$-flow.

Finally, the $(-\tilde{v})$-flow defines a smooth map
$p:\partial_{1}^{+}\hat{X}(\hat{v})\to Y$ which is a homeomorphism. In fact,
$\partial_{1}^{+}\hat{X}(\hat{v})$ is diffeomorphic to $Y$ by a small
perturbation of $p$. ∎

Let $\alpha:X\to\hat{X}$ be a smooth submersion of a compact smooth manifold
$X$ with boundary into the interior of a compact connected smooth manifold
$\hat{X}$ of the same dimension, where $\partial\hat{X}\neq\emptyset$. Assume
that $\hat{X}$ is equipped with a traversing vector field $\hat{v}$ so that
the pair $(\hat{X},\hat{v})$ is convex in the sense of Definition 4.5. As
before, we denote by $v=\alpha^{\dagger}(\hat{v})$ the pull-back of $\hat{v}$
under $\alpha$. We denote by $f$ the pull-back $\alpha^{\ast}(\hat{f})$ of the
Lyapunov function $\hat{f}$. When $\alpha$ is a regular embedding, to simplify
the notations, we identify $X$ and $\alpha(X)$. Under this identification,
$v=\hat{v}|_{X}$ and $f=\hat{f}|_{X}$.

######  Definition 4.6.

Let $(\hat{X},\hat{v})$ be a convex pair and let $\alpha:X\to\hat{X}$ be a
submersion for which $\alpha^{\partial}$ is locally generic (in the sense of
Definition 4.4) with respect to $\hat{v}$.

We say that $\alpha$ is $k$-convex if
$\partial_{k}^{+}X(\alpha^{\dagger}(\hat{v}))=\emptyset$, $k$-concave if
$\partial_{k}^{-}X(\alpha^{\dagger}(\hat{v}))=\emptyset$, and $k$-flat if
$\partial_{k}X(\alpha^{\dagger}(\hat{v}))=\emptyset$. $\diamondsuit$

Let a regular immersion $\alpha^{\partial}:\partial X\to\hat{X}$ be locally
generic relative to $\hat{v}$ in the sense of Definition 4.4. For each
$\hat{v}$-trajectory $\hat{\gamma}\subset\hat{X}$, we pick a point
$a\in\hat{\gamma}\cap\alpha(\partial X)$. Such $a$ belongs to the intersection
of $k=k(a)\in[1,n+1]$ local branches of $\alpha^{\partial}(\partial X)$, where
$k(a)=\\#((\alpha^{\partial})^{-1}(a))$. In particular, as we remarked before,
$a$ belongs to a unique collection of the $\alpha$-images of the Morse strata
$\\{\alpha(\partial_{j(b)}X(v))\\}_{b\,\in\alpha^{-1}(a)}$ with the _maximal
possible_ indexes $j(b)$ (recall that $v:=\alpha^{\dagger}(\hat{v})$ is the
$\alpha$-transfer of $\hat{v}$). The tangent spaces of these strata are in
general position at $a$, thanks to $\alpha^{\partial}$ being locally generic.
Recall that this setting includes the cases where some or all the strata
$\\{\alpha(\partial_{j(b)}X(v))\\}_{b\,\in\alpha^{-1}(a)}$ are
$n$-dimensional, i.e., $j(b)=1$.

$\displaystyle\text{ Let
}\;j(a)=_{\mathsf{def}}\sum_{b\,\in\alpha^{-1}(a)}j(b)\;\geq\;k(a)\text{ \;
and}$ (4.3)
$\displaystyle\partial_{j(a)}\big{(}\alpha(X)\big{)}(\hat{v})=_{\mathsf{def}}\bigcap_{\,b\,\in\alpha^{-1}(a)}\alpha\big{(}\partial_{j(b)}X(v)\big{)},$

the latter equality being understood as an identity of the two germs at $a$ of
the LHS and RHS loci. By Definition 4.4, the germ of
$\partial_{j(a)}(\alpha(X))(\hat{v})$ at $a$ is a smooth submanifold of
$\hat{X}$, transversal to the trajectory $\hat{\gamma}_{a}$.

Let $\mathsf{T}_{a}\subset T_{a}\hat{X}$ be the tangent space to the minimal
stratum $\partial_{j(a)}(\alpha(X))$ at $a$ (see (4.3)). With the help of the
$\hat{v}$-flow, the subspace $\mathsf{T}_{a}\subset T_{a}\hat{X}$ spreads to
form a $(\dim(\hat{X})-j(a))$-dimensional subbundle $\mathsf{T}^{\bullet}_{a}$
of the tangent bundle $T\hat{X}\big{|}_{\hat{\gamma}}$ along the trajectory
$\hat{\gamma}$. We denote by $\mathsf{T}^{\clubsuit}_{a}$ the image of
$\mathsf{T}^{\bullet}_{a}$ under the quotient map $T\hat{X}|_{\hat{\gamma}}\to
T\hat{X}|_{\hat{\gamma}}\big{/}T\hat{\gamma}$, where $T\hat{\gamma}$ stands
for the $1$-bundle, tangent to $\hat{\gamma}$.

We introduce a sightly modified version of Definition 3.2 from [K2], a
modification that applies to submersions $\alpha:X\to\hat{X}$.

######  Definition 4.7.

Let $\hat{v}$ be a traversing vector field on a compact connected smooth
manifold on $\hat{X}$ with boundary. We say that a submersion
$\alpha:X\to\textup{int}(\hat{X})$ is traversally generic relative to
$\hat{v}$, if:

  * •

$\alpha^{\partial}$ is locally generic in the sense of Definition 4.4 with
respect to $\hat{v}$,

  * •

for each $\hat{v}$-trajectory $\hat{\gamma}\subset\hat{X}$, the subbundles
$\\{\mathsf{T}^{\clubsuit}_{a}\\}_{a\,\in\,\hat{\gamma}\,\cap\,\alpha(\partial
X)}$ are in general position in the normal to $\hat{\gamma}$ (trivial)
$n$-bundle $(T\hat{X}|_{\hat{\gamma}})\big{/}T\hat{\gamma}$. $\diamondsuit$

######  Example 4.4.

_The patterns in Fig. 2 may be stacked vertically along a trajectory
$\hat{\gamma}$. To get a traversally generic piles in the vicinity of
$\gamma$, we obey the following rules: (1) to any stack, we may add any number
of configuration of type $a$ from Fig.1, as long as the prescribed parity of
$m(\hat{\gamma})\equiv 0\mod 2$ is not violated; (2) no two configurations of
multiplicity $3$ (of the types $c,e,f$) reside on $\hat{\gamma}$; (3) at most
two configurations of multiplicity $2$ (of the types $b,d$) reside on
$\hat{\gamma}$, moreover, the $(-\hat{v})$-directed projections on the
transversal section $S$ to $\hat{\gamma}$ of the fold loci (as in $d$) or/and
of the simple self-intersections (as in $b$) must be transversal in $S$. _
$\diamondsuit$

Let $\alpha$ be traversally generic relative to $\hat{v}$. Then, for any
$\hat{v}$-trajectory $\hat{\gamma}$, by counting the dimensions of the bundles
$\\{\mathsf{T}^{\clubsuit}_{a}\\}_{a\,\in\,\hat{\gamma}\,\cap\,\alpha(\partial
X)}$ and using that they are in general position in the $n$-dimensional
bundle, normal to $\hat{\gamma}$, we get that the reduced multiplicity

(4.4) $\displaystyle
m^{\prime}(\hat{\gamma})=_{\mathsf{def}}\sum_{a\,\in\,\hat{\gamma}\,\cap\,\alpha(\partial
X)}\big{(}j(a)-1\big{)}\,\leq\,n.$

For a traversally generic $\alpha$ and a trajectory $\hat{\gamma}$, let
$\gamma\subset X$ be any segment in $\alpha^{-1}(\hat{\gamma})\subset X$ which
is bounded by a pair of points in $\partial X$. Then, by Theorem 3.5 from
[K2], the total multiplicity $m(\gamma)\leq 2n+2$. Although, for a traversally
generic $\alpha$, there is no $\alpha$-independent constraint on the total
multiplicity
$m(\hat{\gamma})=_{\mathsf{def}}\sum_{x\,\in\,\hat{\gamma}\,\cap\,\alpha(\partial
X)}j(x)$, there is an universal constraint on the cardinality of the subset
$\hat{\gamma}_{\\{\geq 2\\}}\subset\hat{\gamma}\cap\alpha(\partial X)$,
consisting of points $x$ whose multiplicity $m(x)\geq 2$. Namely,
$\\#(\hat{\gamma}_{\\{\geq 2\\}})\leq n$, since no more than $n$ proper vector
subspaces may be in general position in an ambient $n$-dimensional vector
space.

Now we are ready to introduce the central notion of a convex pseudo-envelop.

######  Definition 4.8.

Let $\hat{v}$ be a traversing vector field on a compact connected smooth
manifold $\hat{X}$ with boundary. We assume that $(\hat{X},\hat{v})$ is convex
in the sense of Definition 4.5.

We call such a pair $(\hat{X},\hat{v})$ a convex pseudo-envelop of a
submersion $\alpha:X\to\textup{int}(\hat{X})$, if $\alpha^{\partial}$ is
locally generic relative to $\hat{v}$. We think of $X$ as being equipped with
the pull-back vector field $v=\alpha^{\dagger}(\hat{v})$, so that $(X,v)$ is
“enveloped” by $(\hat{X},\hat{v})$.

If $\alpha$ is a locally generic _regular embedding_ , then we call
$(\hat{X},\hat{v})$ a convex envelop of $\alpha$.

$\diamondsuit$

######  Remark 4.2.

_Not all traversally generic pairs $(X,v)$ admit convex envelops
$(\hat{X},\hat{v})$. For example, if $X$ has two closed submanifolds (or
singular cycles), $M$ and $N$, of complementary dimensions with a nonzero
algebraic intersection number $M\circ N$, then no convex envelop
$(\hat{X},\hat{v})$ of $(X,v)$ exists. Indeed, with the help of the
$(-\hat{v})$-flow, $M$ is cobordant in $\hat{X}$ to a cycle $M^{\prime}$ which
resides in $\partial_{1}^{+}\hat{X}(\hat{v})$. Similarly, with the help of the
$\hat{v}$-flow, $N$ is cobordant in $\hat{X}$ to a cycle $N^{\prime}$ which
resides in $\partial_{1}^{-}\hat{X}(\hat{v})$. Note that if $M\circ N\neq 0$
in $X$, then evidently the same property holds in any $\hat{X}\supset X$.
Since $M^{\prime}$ and $N^{\prime}$ are disjoint cycles, their intersection
number $M^{\prime}\circ N^{\prime}=0$, which contradicts to the assumption
$M\circ N\neq 0$._ _In particular, the non-triviality of $\mathsf{sign}(X)$,
the Wall relative signature of $X$ [W], obstructs the existence of a convex
envelop for any traversing $v$ on $X$._ _As Fig. 1 testifies, this argument
does not rule out the existence of a convex _pseudo_ -envelop for $(X,v)$. In
fact, Fig.1 shows that any compact oriented surface $X$ can be enveloped._
$\diamondsuit$

We denote by $\mathsf{Sub}(X,\hat{X})$ the space of smooth submersions
$\alpha:X\to\mathsf{int}(\hat{X})$.

When $\alpha$ is an embedding, by Theorem 3.5 from [K2], in the space
$\mathcal{V}_{\mathsf{trav}}(\hat{X},\alpha)$ of traversing vector fields
$\hat{v}$ on $\hat{X}$, there is an open and dense subset
$\mathcal{V}^{\ddagger}(\hat{X},\alpha)$ such that $\alpha$ is traversally
generic (see Definition 4.7) with respect to
$\hat{v}\in\mathcal{V}^{\ddagger}(\hat{X},\alpha)$.

On the other hand, the property of a submersion $\alpha$ to be traversally
generic with respect to a _given_ traversing vector field $\hat{v}$ (see
Definition 4.7) is an “open” property in the $C^{\infty}$-topology on
$\mathsf{Sub}(X,\hat{X})$, since it may be expressed in terms of mutual
transversality of the relevant strata in the appropriate jet spaces.

We conjecture that the transversal generality of submersions $\alpha$ with
respect to a fixed convex pair $(\hat{X},\hat{v})$ is also a “dense” property.
Among other things, the next Theorem 4.1 shows that this conjecture is valid
for the regular _embeddings_ $\alpha$ which admit convex envelops. However,
for general submersions $\alpha$, we are able only to prove that a somewhat
_weaker_ property “is dense”. That property is described in the second claim
of Theorem 4.1. Speaking informally, we can insure by $\alpha$-perturbations
the general positions of the singularities of the maps
$\alpha^{\partial}:\partial X\to\hat{X}$ and
$\pi\circ\alpha^{\partial}:\partial X\to\partial_{1}^{+}\hat{X}(\hat{v})$
separately, but not mutually. Here the map
$\pi:\hat{X}\to\partial_{1}^{+}\hat{X}(\hat{v})$ is defined by the
$(-\hat{v})$-directed convex flow.

######  Theorem 4.1.

Let $(\hat{X},\hat{v})$ be a convex pair, and $X$ a compact smooth manifold
with boundary, $\dim(X)=\dim(\hat{X})=n+1$. Assume that
$\mathsf{Sub}(X,\hat{X})\neq\emptyset$.

  * •

There is an open and dense subset $\mathcal{O}\subset\mathsf{Sub}(X,\hat{X})$
such that, for any $\alpha\in\mathcal{O}$, the local branches of
$\alpha(\partial X)$ are in general position in $T_{a}\hat{X}$ at every point
$a\in\alpha(\partial X)$.

  * •

For any $\alpha\in\mathcal{O}$ and for each $\hat{v}$-trajectory
$\hat{\gamma}\subset\hat{X}$, the $\hat{v}$-invariant subbundles
$\\{\mathsf{T}^{\clubsuit}_{a}\\}_{a\,\in\;\hat{\gamma}\,\cap\,\alpha(\partial
X)}$, generated by the intersections

$\bigcap_{b\in(\alpha^{\partial})^{-1}(a),\;j(b)\geq
2}\alpha_{\ast}\big{[}T_{b}\big{(}\partial_{j(b)}X(v)\big{)}\big{]}$

of the tangent spaces to the Morse strata
$\big{\\{}\partial_{j(b)}X(v)\big{\\}}_{b\in(\alpha^{\partial})^{-1}(a),\;j(b)\geq
2}$, are in general position in the normal to $\hat{\gamma}$ (trivial)
$n$-bundle $(T\hat{X}|_{\hat{\gamma}})\big{/}T\hat{\gamma}$. Here $v$ is the
pull-back of $\hat{v}$ under $\alpha$.

###### Proof.

We assume that the openness of $\mathcal{O}$ in $\mathsf{Sub}(X,\hat{X})$ is
clear, due to the compactness of $X$, and will present the arguments that
validate the density of $\mathcal{O}$ in $\mathsf{Sub}(X,\hat{X})$. We divide
the proof into three steps, marked as (i), (ii), and (iii).

(i) By [LS], the set $\mathcal{N}\subset C^{\infty}(\partial X,\hat{X})$ of
smooth maps $\beta:\partial X\to\hat{X}$, such that $\beta$ is a $k$-normal
immersion in the sense of [LS] (see also Definition 3.3 from [K9]) for all
$k\leq n+1$, is open and dense.

First we aim to show that, for a given submersion
$\alpha\in\mathsf{Sub}(X,\hat{X})$, there is an open set
$\mathcal{O}_{\alpha}\subset\mathsf{Sub}(X,\hat{X})$ such that
$\alpha\in\mathsf{closure}(\mathcal{O}_{\alpha})$ and, for any
$\tilde{\alpha}\in\mathcal{O}_{\alpha}$, the submersion
$\tilde{\alpha}^{\partial}\in\mathcal{N}$.

With this goal in mind, we choose an auxiliary metric $g$ on $\hat{X}$ such
that: $(1)$ the boundary $\partial\hat{X}$ is convex in $g$, and $(2)$ there
is $\epsilon_{0}>0$ such that any two points $x,y\in\hat{X}$ that are less
than $\epsilon_{0}$-apart are connected by a single geodesic arc. Using the
submersion $\alpha$, we pull-back $g$ to a Riemannian metric
$g^{\dagger}_{\alpha}$ on $X$.

For some $\epsilon<\epsilon_{0}$, the $\epsilon$-neighborhood
$C_{\epsilon}\subset X$ of $\partial X$ in the metric $g^{\dagger}_{\alpha}$
has a product structure $\psi:\partial
X\times[0,\epsilon]\stackrel{{\scriptstyle\approx}}{{\to}}C_{\epsilon}$, so
that the curve $\delta_{x}=_{\mathsf{def}}\psi(x\times[0,\epsilon])$ is the
unique geodesic in $g^{\dagger}_{\alpha}$, normal to $\partial X$ at
$x\in\partial X$. Using the diffeomorphism $\psi$, we construct a smooth
diffeomotopy $\\{\psi_{t}:X\to X\\}_{t\in[0,1]}$ (via the flow inward normal
to $\partial X$) so that $\psi_{0}=\mathsf{id}_{X}$ and
$\psi_{1}(X)=X\setminus\mathsf{int}(C_{\epsilon})$.

Using the convexity of $\partial\hat{X}$ in $g$ and the choice of
$\epsilon<\epsilon_{0}$ we get the following claim: for any $\beta$ that
approximates $\alpha^{\partial}$ and each $x\in\partial X$, there exists the
unique geodesic $\hat{\delta}_{x}\subset\hat{X}$ in the metric $g$ that
connects the points $\beta(x)$ and $\alpha\circ\psi_{1}(x)$.

We pick such an approximation $\beta\in\mathcal{N}$ of $\alpha^{\partial}$.
Let a smooth map $\alpha^{\sharp}(\beta,\epsilon):X\to\hat{X}$ be defined by
the two properties: (1) $\alpha^{\sharp}(\beta,\epsilon)|_{X\setminus
C_{\epsilon}}:=\alpha\circ\psi_{1}$, (2) for each $x\in\partial X$, the
diffeomorphism
$\alpha^{\sharp}(\beta,\epsilon)|:\delta_{x}\to\hat{\delta}_{x}$ is an
isometry with respect to $g^{\dagger}_{\alpha}$ and $g$ along the two geodesic
arcs. By picking $\epsilon$ small enough and $\beta\in\mathcal{N}$
sufficiently $C^{\infty}$-close to $\alpha^{\partial}$, we get that
$\alpha^{\sharp}(\beta,\epsilon)$ is $C^{\infty}$-close to $\alpha$.
Therefore, we may assume that
$\alpha^{\sharp}(\beta,\epsilon)\in\mathsf{Sub}(X,\hat{X})$ for an appropriate
choice of $\beta$ that approximates $\alpha^{\partial}$. By its construction,
$(\alpha^{\sharp}(\beta,\epsilon))^{\partial}\in\mathcal{N}$.

(ii) For a given smooth map $\beta:\partial
X\to\mathsf{int}(\partial_{1}^{+}\hat{X}(\hat{v}))$, let $\beta_{\ast}$ denote
the differential of $\beta$.

$\text{ Let \;}\mathcal{S}^{(1)}(\beta)=_{\mathsf{def}}\big{\\{}x\in\partial
X\big{|}\;\mathsf{rk}(\beta_{\ast})\leq n-1\big{\\}}.$

If $S^{(1)}(\beta)$ is a smooth manifold, then the locus

$\mathcal{S}^{(1,1)}(\beta)=_{\mathsf{def}}\big{\\{}x\in\mathcal{S}^{(1)}(\beta)\big{|}\;\mathsf{rk}(\beta_{\ast}|_{\mathcal{S}^{(1)}(\beta)})\leq
n-2\big{\\}}$

is well-defined. Continuing this way, the filtration of $\partial X$ by the
loci
$\big{\\{}\mathcal{S}_{[k]}(\beta)=_{\mathsf{def}}\mathcal{S}^{\omega}(\beta)\big{\\}}_{k}$,
where $k\leq n+1$ and $\omega=(\underbrace{1,1,\ldots,1}_{k})$ is introduced.
Applying Boardman’s Theorems 15.1-15.3, [Bo], the subset $\mathcal{B}$ of maps
$\beta\in C^{\infty}(\partial X,\partial_{1}^{+}\hat{X}(\hat{v}))$, for which
all the strata
$\\{\mathcal{S}_{[k]}=_{\mathsf{def}}\mathcal{S}_{[k]}(\beta)\\}$ are smooth
manifolds and all the maps

$\big{\\{}\beta|:\mathcal{S}_{[k]}^{\circ}=_{\mathsf{def}}\mathcal{S}_{[k]}\setminus\mathcal{S}_{[k+1]}\to\mathsf{int}(\partial_{1}^{+}\hat{X}(\hat{v}))\big{\\}}_{k}$

are immersions, is open ($\partial X$ is compact) and dense. Note that each
$x\in\partial X$ belongs to a unique pure stratum
$\mathcal{S}_{[k(x)]}^{\circ}$. Moreover, using the Thom Multijet
Transversality Theorem (see [GG], Theorem 4.13), the subspace of
$\mathcal{B}$, formed maps $\beta$ for which all the spaces
$\\{\beta_{\ast}(T_{x}\mathcal{S}_{[k(x)]})\\}_{x\in(\pi\circ\beta)^{-1}(y)}$
are in general position in $T_{y}\partial_{1}^{+}\hat{X}(\hat{v})$ for all
$k(x)\geq 2$ and all $y\in\mathsf{int}(\partial_{1}^{+}\hat{X}(\hat{v}))$,
form an open and dense subset $\mathcal{B}_{\mathsf{NC}}\subset\mathcal{B}$
(see [GG], Theorem 5.2). Here “$\mathsf{NC}$” abbreviates the condition known
as “normal crossings”. Note that if $k(x)=1$, then
$\beta_{\ast}(T_{x}S_{[k(x)]})$ coincides with
$T_{\beta(x)}\partial_{1}^{+}\hat{X}(\hat{v})$.

We stress that, applying the previous arguments to the map
$\beta=\pi\circ\alpha^{\partial}$, the strata
$\bigcap_{\,b\,\in\alpha^{-1}(a)}\alpha\big{(}\partial_{j(b)}X(v)\big{)}$ that
involve $b$’s with $j(b)=1$ become “$\beta$-invisible”, thanks to the
“erasing” action of $\pi_{\ast}$ on all $n$-dimensional spaces
$\alpha\big{(}T_{a}(\partial_{1}X(v))\big{)}$. Because of this short comming,
we cannot claim that the traversally generic $\alpha$ to $\hat{v}$ (see
Definition 4.7) are dense in $\mathsf{Sub}(X,\hat{X})$.

(iii) Note that the map
$\pi:\hat{X}\to\mathsf{int}(\partial_{1}^{+}\hat{X}(\hat{v}))$, defined by the
$(-\hat{v})$-flow, is smooth due to $\hat{v}$ being convex; moreover, its
restriction to $\mathsf{int}(\hat{X})$ is a submersion.

For a given $\alpha\in\mathsf{Sub}(X,\hat{X})$, we form the composition
$\alpha^{\flat}=_{\mathsf{def}}\pi\circ\alpha^{\partial}$.

By (i), we can approximate $\alpha$ by a new submersion
$\alpha_{1}=\alpha^{\sharp}(\beta,\epsilon)$ such that
$\alpha_{1}^{\partial}\in\mathcal{N}$.

By (ii), we can approximate
$\alpha_{1}^{\flat}=_{\mathsf{def}}\pi\circ\alpha_{1}^{\partial}$ by a smooth
map $\beta_{1}^{\flat}:\partial
X\to\mathsf{int}(\partial_{1}^{+}\hat{X}(\hat{v}))$ such that
$\beta_{1}^{\flat}\in\mathcal{B}_{\mathsf{NC}}$.

Let us fix a Lyapunov function $\hat{f}:\hat{X}\to\mathbb{R}$ for $\hat{v}$.
In the spirit of Lemma 4.3, the Lyapunov function $\hat{f}$ for $\hat{v}$ and
the projection $\pi:\hat{X}\to\partial_{1}^{+}\hat{X}(\hat{v})$ define global
smooth “coordinates” in the interior of $\hat{X}$. That is, each pair $(t,y)$,
where $y\in\partial_{1}^{+}\hat{X}(\hat{v})$ and $t\in\hat{f}(\pi^{-1}(y))$,
determines a unique point $x\in\hat{X}$ such that $\hat{f}(x)=t$ and
$\pi(x)=y$. Let $f_{1}=\alpha^{\ast}_{1}(\hat{f})$.

For the map $\beta_{1}^{\flat}\in\mathcal{B}_{\mathsf{NC}}$, we define a
smooth map
$\beta_{1}=_{\mathsf{def}}\beta_{1}(\beta_{1}^{\flat},\hat{f}):\partial
X\to\hat{X}$ by the formula $\beta_{1}(x)=y$, where $x\in\partial X$ and $y$
is the unique point on the $\hat{v}$-trajectory $\hat{\gamma}$ through
$\beta_{1}^{\flat}(x)\in\partial_{1}^{+}\hat{X}(\hat{v})$ such that
$\hat{f}(y)=f_{1}(x)$. Note that by choosing $\beta_{1}^{\flat}$ sufficiently
close to $\alpha_{1}^{\flat}$, we insure that $\beta_{1}$ is sufficiently
close to $\alpha_{1}^{\partial}$. Thus we may assume that
$\beta_{1}\in\mathcal{N}$.

Recycling the argument that revolves around the construction of
$\alpha^{\sharp}(\beta,\epsilon):X\to\hat{X}$ in part (i) of the proof, we
form the submersion
$\alpha_{2}=_{\mathsf{def}}\alpha_{1}^{\sharp}(\beta_{1},\epsilon):X\to\hat{X}$.
By the construction of $\alpha_{2}$, we have
$\alpha_{2}^{\flat}=\beta_{1}^{\flat}\in\mathcal{B}_{\mathsf{NC}}$ and
$\beta_{1}\in\mathcal{N}$.

Therefore, we have shown that any given submersion $\alpha$ admits an
approximation by some $\alpha_{2}\in\mathsf{Sub}(X,\hat{X})$ such that
$\alpha_{2}^{\partial}\in\mathcal{N}$ and
$\alpha_{2}^{\flat}\in\mathcal{B}_{\mathsf{NC}}$. These are exactly the two
properties that describe the space $\mathcal{O}$ in the theorem. ∎

######  Corollary 4.2.

Let $(\hat{X},\hat{v})$ be a convex pair, and $X$ a compact smooth manifold
with boundary, $\dim(X)=\dim(\hat{X})=n+1$.

The regular _embeddings_ $\alpha:X\subset\hat{X}$ that are traversally generic
with respect to $\hat{v}$ (see Definition 4.7) form an open and dense set in
the space of all regular smooth embeddings.

###### Proof.

Since, for a regular embedding $\alpha:X\hookrightarrow\hat{X}$,
$\alpha^{\partial}$ is an embedding, the first claim of Theorem 4.1 is
vacuous, and the second claim insures that $\alpha$ is traversally generic. ∎

######  Remark 4.3.

_Consider the $k$-multiple self-intersection manifolds
$\\{\Sigma_{k}^{\alpha^{\partial}}\\}_{k}$ of $k$-normal (see [LS]) immersions
$\alpha^{\partial}:\partial X\to\hat{X}$. By definition,
$\Sigma_{k}^{\alpha^{\partial}}$ is a submanifold of the $k$-fold product
$(\partial X)^{k}$, the preimage of the diagonal $\Delta\subset(\hat{X})^{k}$
under the transversal to it map $(\alpha^{\partial})^{k}$. The projection
$p_{1}$ of $\Sigma_{k}^{\alpha^{\partial}}$ on the first factor $\partial X$
of the product $(\partial X)^{k}$ is an immersion [LS]. By composing $p_{1}$
with $\alpha^{\partial}$, we get an immersion of $\alpha^{\partial}\circ
p_{1}:\Sigma_{k}^{\alpha^{\partial}}\to\hat{X}$. By using the convex
$(-\hat{v})$-flow, we get a map
$\pi:\hat{X}\to\partial_{1}^{+}\hat{X}(\hat{v})$. Finally, we obtain a smooth
composite map $\pi\circ\alpha^{\partial}\circ
p_{1}:\Sigma_{k}^{\alpha^{\partial}}\to\partial_{1}^{+}\hat{X}(\hat{v})$._

_We notice that, under the hypotheses and notations of Theorem 4.1, if
$\hat{v}$ is tangent at $a$ to the intersection
$\bigcap_{b\in(\alpha^{\partial})^{-1}(a)}\;\alpha_{\ast}\big{(}\partial_{j(b)}X(v)\big{)}$,
then, evidently, each local branch
$\alpha_{\ast}(\partial_{j(b_{\star})}X(v))$,
$b_{\star}\in(\alpha^{\partial})^{-1}(a),$ is tangent to $\hat{v}$ at $a$.
Therefore, $j(b_{\star})\geq 2$. In other words, for $k\geq 2$, the singular
locus of the map $\pi\circ\alpha^{\partial}\circ
p_{1}:\Sigma_{k}^{\alpha^{\partial}}\to\partial_{1}^{+}\hat{X}(\hat{v})$ is
always _contained_ in the singular locus of the map
$\pi\circ\alpha^{\partial}\circ p_{1}:(\partial
X)^{k}\to\partial_{1}^{+}\hat{X}(\hat{v})$._ $\diamondsuit$

###  4.4. Quasitopies of convex envelops and pseudo-envelops

Now, let us modify Definition 3.7 from [K9] and Definition 3.1 from this
paper, so that they apply to convex pseudo-envelops of traversing flows (see
Fig. 3). This modification is central to our efforts.

######  Definition 4.9.

Fix natural even numbers $d\leq d^{\prime}$ and consider a closed subposets
$\Theta^{\prime}\subset\Theta$ of the universal poset $\mathbf{\Omega}$ from
Section 2, such that $(\emptyset)\notin\Theta$ .

Let $\hat{X}$ be a $(n+1)$-dimensional compact manifold and $\hat{v}$ a convex
traversing vector field on it. Let
$\hat{Z}=_{\mathsf{def}}\hat{X}\times[0,1]$. We denote by $\hat{v}^{\bullet}$
the vector field on $\hat{Z}$ that is tangent to each slice
$\hat{X}\times\\{t\\}$, $t\in[0,1]$, and is equal to $\hat{v}$ there.

We say that a two convex pseudo-envelops, $\alpha_{0}:X_{0}\to\hat{X}$ and
$\alpha_{1}:X_{1}\to\hat{X}$, are
$(d,d^{\prime};\mathbf{c}\Theta,\mathbf{c}\Theta^{\prime})$-quasitopic in
$\hat{X}$, if there exists a compact smooth orientable $(n+2)$-manifold
$W$,444with corners $\partial X_{0}\coprod\partial X_{1}$ whose boundary
$\partial W=(X_{0}\coprod X_{1})\bigcup_{\\{\partial X_{0}\coprod\partial
X_{1}\\}}\delta W$, and a smooth submersion $A:W\to\hat{Z}$ so that:

  * •

$A|_{X_{0}}=\alpha_{0}$ and $A|_{X_{1}}=\alpha_{1}$;

  * •

for each $z\in\hat{X}\times\,\partial[0,1]$, the total multiplicity
$m_{A}(\hat{\gamma}_{z})$ of the $\hat{v}^{\bullet}$-trajectory
$\hat{\gamma}_{z}$ through $z$, relatively to $A(\partial X_{0}\coprod\partial
X_{1})$, satisfies the constraints $m_{A}(\hat{\gamma}_{z})\leq d$,
$m_{A}(\hat{\gamma}_{z})\equiv 0\mod 2$, and the combinatorial tangency
pattern $\omega^{A}(\hat{\gamma}_{z})$ of $\hat{\gamma}_{z}$ with respect to
$A(\partial X_{0}\coprod\partial X_{1})$ does not belong to $\Theta$;

  * •

for each $z\in\hat{Z}$, the total multiplicity $m_{A}(\hat{\gamma}_{z})$ of
$\hat{\gamma}_{z}$ with respect to $A(\delta W)$ satisfies the constraints
$m_{A}(\hat{\gamma}_{z})\leq d^{\prime}$, $m_{A}(\hat{\gamma}_{z})\equiv 0\mod
2$, and the combinatorial tangency pattern $\omega^{A}(\hat{\gamma}_{z})$ of
$\hat{\gamma}_{z}$ with respect to $A(\delta W)$ does not belong to
$\Theta^{\prime}$;

We denote by
$\mathcal{QT}_{d,d^{\prime}}^{\mathsf{sub}}(\hat{X},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
the set of quasitopy classes of such convex pseudo-envelops
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$.

If we insist that $\alpha_{0},\alpha_{1}$, and $A$ are embeddings, then we get
$\mathcal{QT}_{d,d^{\prime}}^{\mathsf{emb}}(\hat{X},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$,
the set of quasitopy classes of convex envelops. $\diamondsuit$

It is easy to check that the quasitopy of convex pseudo-envelops (convex
envelops) $\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$ is an
equivalence relation.

Figure 3. The ingredients of Definition 4.9. For simplicity of the depiction,
the maps $\alpha_{0},\alpha_{1},A$ are shown as embeddings.

We are in position to state one of the main results of this paper.

######  Theorem 4.2.

Let $\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}$ be closed subposets
that do not contain the element $(\emptyset)$. Assume that $d^{\prime}\geq d$
and $d^{\prime}\equiv d\equiv 0\mod 2$. Let $\hat{X}$ be a smooth compact
connected $(n+1)$-dimensional manifold, equipped with a convex traversing
vector field $\hat{v}$.

Then, under the notations of Definition 3.2, there is a canonical bijection

$\displaystyle\Phi^{\mathsf{emb}}:\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}$
(4.5)
$\displaystyle\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\big{[}\big{[}(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v})),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})\big{]}\big{]}$

and a canonical surjection

$\displaystyle\Phi^{\mathsf{sub}}:\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle
epi}}{{\longrightarrow}}$ (4.6) $\displaystyle\stackrel{{\scriptstyle
epi}}{{\longrightarrow}}\big{[}\big{[}(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v})),\;\epsilon_{d,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})\big{]}\big{]}.$

The map $\Phi^{\mathsf{sub}}$ admits a right inverse.

###### Proof.

By Lemma 4.3, any convex pseudo-envelop $\alpha:X\to\hat{X}$ may be
incapsulated into a convex pseudo-envelop
$\tilde{\alpha}:X\to\hat{X}\subset[0,1]\times Y$ of the product type (we
called a capsule $[0,1]\times Y$ “a box”), where $Y$ is homeomorphic to
$\partial_{1}^{+}\hat{X}(\hat{v})$, and its boundary $\partial Y$ to
$\partial_{2}^{-}\hat{X}(\hat{v})$.

Conversely, any submersion $\tilde{\alpha}:X\to[0,1]\times Y$, where $Y$ is a
smooth compact $n$-manifold, by rounding the corners
$\partial[0,1]\times\partial Y$ of $[0,1]\times Y$ produces a convex pseudo-
envelop $(\hat{X},\hat{v})\subset([0,1]\times Y,\tilde{v})$, where
$\tilde{v}\neq 0$ is tangent to the fibers of $[0,1]\times Y\to Y$. Similarly,
the cobordisms $A$ between pairs of quasitopies $\alpha_{0},\alpha_{1}$ can be
incapsulated in boxes of the form $([0,1]\times Y)\times[0,1]$. Therefore, all
the results from Section 3 in [K9] apply to $\tilde{\alpha}^{\partial}$; in
particular, the pivotal Theorem 3.2 from [K9] applies. With its help, the maps
$\Phi^{\mathsf{sub}}$ from (4.2) and $\Phi^{\mathsf{emb}}$ from (4.2) are
generated as compositions of maps
$p(\tilde{v}):(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))\to(Y,\partial
Y)$ with the maps $\\{\Phi^{\tilde{\alpha}^{\partial}}:(Y,\partial
Y)\to\mathcal{(}\mathcal{P}_{d}^{\mathbf{c}\Theta},\mathcal{P}_{d}^{(\emptyset)})\\}$
from Theorem 3.2 in [K9]; the latter map being generated by the locus
$\tilde{\alpha}(\partial X)\subset\mathbb{R}\times Y$. Note that the cell
$\mathcal{P}_{d}^{(\emptyset)}\subset\mathcal{P}_{d}$ contracts to a singleton
$pt\in\mathcal{P}_{d}^{(\emptyset)}$.

Similarly, for any quasitopy $A:W\to\hat{X}\times[0,1]$ as in Definition 4.9,
the product $Y\times[0,1]$ that encapsulates $\hat{X}\times[0,1]$ is mapped to
$\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}}$ with the help of
$A(\delta W)$, while $\partial Y\times[0,1]$ are mapped to the cell
$\mathcal{P}_{d^{\prime}}^{(\emptyset)}$, which contracts to the singleton
$\epsilon_{d,d^{\prime}}(pt)\in\mathcal{P}_{d^{\prime}}^{(\emptyset)}$.

As in Theorem 3.2 from [K9], and by similar transversality arguments,
$\Phi^{\mathsf{emb}}$ is a 1-to-1 map. Here we need to use Lemma 3.4 from [K9]
to conclude that, under a $\partial\mathcal{E}_{d}$-regular map (see
Definition 3.4 from [K9]), the preimage of the hypersurface
$\partial\mathcal{E}_{d}\subset\mathbb{R}\times\mathcal{P}_{d}$ (see (2))
bounds a compact manifold $X$ in $\hat{X}$, provided $d\equiv 0\mod 2$. Again,
as in Proposition 3.5 from [K9], the bijective map in (4.2) helps to prove
that the map in (4.2) is a split surjective one. ∎

######  Corollary 4.3.

The sets
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
are trivial when
$\dim(\hat{X})<\max_{\omega\in\Theta}\\{|\omega|^{\prime}\\}$.

###### Proof.

Since
$\textup{codim}(\mathcal{P}_{d}^{\Theta},\mathcal{P}_{d})=\max_{\omega\in\Theta}\\{|\omega|^{\prime}\\}$
and $\mathcal{P}_{d}$ is contractible, by Theorem 4.2, the claim follows by a
general position argument, applied to homotopies of maps from the pair
$(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))$ to the
pair $(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)$. ∎

######  Corollary 4.4.

The sets
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$,
$\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
are invariants of the path-connected component of the vector field $\hat{v}$
in the space $\mathsf{conv}(\hat{X})$ of convex traversing vector fields.

###### Proof.

By their definitions, the sets
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
and
$\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
depend only of the smooth topological types of the oriented $1$-dimensional
foliations $\mathcal{L}(\hat{v})$, and
$\mathcal{L}^{\bullet}(\hat{v}^{\bullet})$ on $\hat{X}$ and
$\hat{X}\times[0,1]$, respectively. By Lemma 4.1, the smooth topological types
of these foliations do not change along any path in the space
$\mathsf{conv}(\hat{X})$, that contains the point $\hat{v}$. ∎

######  Corollary 4.5.

The set
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
depends only on the homotopy type of the pair
$(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))$.

###### Proof.

The claim follows instantly from the bijection in formula (4.2). ∎

The next theorem is our main result about the stability of
$\mathcal{QT}^{\mathsf{emb}}_{d,\,d+2}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta)$
as a function of $d$ in terms of the function $\eta_{\Theta}(d+2)$ from (2.8).

######  Theorem 4.3.

(short stabilization: $\mathbf{\\{d\Rightarrow d+2\\}}$) Let $\Theta$ be a
closed subposet of $\mathbf{\Omega}$. If

$\dim(\hat{X})<d+2-\psi_{\Theta}(d+2)$

and $|\omega|^{\prime}>2$ for all $\omega\in\Theta_{\langle d+2]}$, then there
exists a bijection

$\displaystyle\Phi^{\mathsf{emb}}:\mathcal{QT}^{\mathsf{emb}}_{d,\,d+2}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta)\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\big{[}(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v})),\,(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\big{]}$

###### Proof.

Let $k=_{\mathsf{def}}d+2-\psi_{\Theta}(d+2)$. If $|\omega|^{\prime}>2$ for
all $\omega\in\Theta_{\langle d+2]}$, then both
$\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and
$\mathcal{P}_{d+2}^{\mathbf{c}\Theta}$ are simply-connected, since
$\textup{codim}(\bar{\mathcal{P}}_{d+2}^{\Theta},\bar{\mathcal{P}}_{d+2})>2$
and $\textup{codim}(\bar{\mathcal{P}}_{d}^{\Theta},\bar{\mathcal{P}}_{d})>2$.
By combining Theorem 2.2 with the Alexander duality as in [KSW2], we get an
isomorphism
$\epsilon_{\ast}:H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})\approx
H_{j}(\mathcal{P}_{d+2}^{\mathbf{c}\Theta};\mathbb{Z})$ for all $j<k$. By the
Whitehead Theorem (the inverse Hurewicz Theorem) (see Theorem (7.13) in [Wh]
or Theorem 10.1 in [Hu]), the map
$\epsilon:\mathcal{P}_{d}^{\mathbf{c}\Theta}\to\mathcal{P}_{d+2}^{\mathbf{c}\Theta}$
is $(k-1)$-connected. Thus, if $n=\dim(\partial_{1}^{+}\hat{X}(\hat{v}))\leq
k-1$, then, by the standard application of the obstruction theory, no map
$\Phi:(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))\to(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)$,
which is not null-homotopic, becomes null-homotopic in
$(\mathcal{P}_{d+2}^{\mathbf{c}\Theta},pt^{\prime})$. Now the claim follows
from Theorem 4.2. ∎

Theorem 4.3 leads to the following straightforward, but important implication.

######  Corollary 4.6.

(long stabilization $\mathbf{d\Rightarrow\infty}$) Let $\Theta$ be a closed
profinite (see Definition 2.1) poset such that $|\omega|^{\prime}>2$ for all
$\omega\in\Theta$.

Then, given a convex pair $(\hat{X},\hat{v})$, the quasiptopy set
$\mathcal{QT}^{\mathsf{emb}}_{d,\,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta)$
stabilizes towards the set of homotopy classes
$\big{[}(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v})),\,(\mathcal{P}_{\infty}^{\mathbf{c}\Theta},pt)\big{]}$
for all sufficiently big $d$ relative to $\dim(\hat{X})$. $\diamondsuit$

###  4.5. Group structure on the ball-shaped convex pseudo-envelops

The connected sum $\alpha_{1}\uplus\alpha_{2}$ of convex pseudo-envelops
$\alpha_{1}:X_{1}\to\hat{X}_{1},\alpha_{2}:X_{2}\to\hat{X}_{2}$ can be
introduced in a fashion, similar to the operation $\uplus$ in formulae (3.14)
and (3.15) from [K9] and formula (3) above.

Let $D^{n-1}_{+}$ denote the Southern hemisphere in $\partial D^{n}$. As with
the connected sums of submersions from Section 3, there is an ambiguity about
how to attach a $1$-handle $D^{n-1}_{+}\times[0,1]$ to
$\partial_{1}^{+}\hat{X}_{1}(\hat{v}_{1})\coprod\partial_{1}^{+}\hat{X}_{2}(\hat{v}_{2})$
and a $1$-handle $H=D^{n}\times[0,1]$ to $\hat{X}_{1}\coprod\hat{X}_{2}$ to
form the “coordinated” connected sums
$\partial_{1}^{+}\hat{X}_{1}(\hat{v}_{1})\,\\#_{\partial}\,\partial_{1}^{+}\hat{X}_{2}(\hat{v}_{2})$
and $\hat{X}_{1}\,\\#_{\partial}\,\hat{X}_{2}$. The ambiguity arises if
$\partial_{2}^{-}\hat{X}_{1}(\hat{v}_{1})$ or
$\partial_{2}^{-}\hat{X}_{2}(\hat{v}_{2})$ has more than a single connected
component. To avoid it, as in [K9], formula (3.14), we need to pick a
preferred connected component of $\partial_{2}^{-}\hat{X}_{1}(\hat{v}_{1})$
and $\partial_{2}^{-}\hat{X}_{2}(\hat{v}_{2})$. Using the local models of
convex vector fields (as in the proof of Lemma 4.3), the vector fields
$\hat{v}_{1}$ and $\hat{v}_{2}$ extend across $H$ so that the convexity of the
extended traversing vector field $\hat{v}_{\\#_{\partial}}$ is enforced. For
example, the lower diagram in Fig.2 is the connected sum of the upper diagram
with itself.

######  Theorem 4.4.

Let $\Theta^{\prime}\subset\Theta\subset\mathbf{\Omega}$ be closed subposets
which do not contain the element $(\emptyset)$. Assume that $d^{\prime}\geq d$
and $d^{\prime}\equiv d\equiv 0\mod 2$. Let $\hat{v}$ be a convex traversing
vector field on the standard ball $D^{n+1}$ such that
$\partial_{1}^{+}D^{n+1}(\hat{v})$ is diffeomorphic to the standard ball
$D^{n}$.555The constant vector field will do.

The group operation in the sources of the maps (4.4) and (4.4) below is the
connected sum $\uplus$ of convex envelops/convex pseudo-envelops.

$\bullet$ There is a group isomorphism

$\displaystyle\Phi^{\mathsf{emb}}:\mathcal{QT}^{\mathsf{emb}}_{d,\,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}$
(4.7)
$\displaystyle\stackrel{{\scriptstyle\approx}}{{\longrightarrow}}\big{[}\big{[}(D^{n},\,S^{n-1}),\;\epsilon_{d,\,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})\big{]}\big{]}.$

The group homomorphism

$\displaystyle\Phi^{\mathsf{sub}}:\mathcal{QT}^{\mathsf{sub}}_{d,\,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle}}{{\longrightarrow}}$
(4.8) $\displaystyle\stackrel{{\scriptstyle
epi}}{{\longrightarrow}}\big{[}\big{[}(D^{n},\,S^{n-1}),\;\epsilon_{d,\,d^{\prime}}:(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})\big{]}\big{]}.$

is an split epimorphism. Moreover, we have a split group extension

(4.9) $\displaystyle\qquad
1\to\ker(\Phi^{\mathsf{sub}})\to\mathcal{QT}^{\mathsf{sub}}_{d,\,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\stackrel{{\scriptstyle\mathcal{R}}}{{\rightarrow}}\mathcal{QT}^{\mathsf{emb}}_{d,\,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\to
1.$

$\bullet$ For $n>1$ all these groups are abelian.

$\bullet$ For $n<\max_{\omega\in\Theta_{\langle d]}}\\{|\omega|^{\prime}\\}$,
the groups
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
are trivial.

###### Proof.

The main observation is that the construction of the boxes
$(\hat{X},\hat{v})\subset([0,1]\times Y,\tilde{v})$ in Lemma 4.3 is amenable
to the connected sum operation for convex pseudo-envelops of traversing flows.
That is, given two boxes $([0,1]\times
Y_{1},\tilde{v}_{1})\supset(\hat{X}_{1},\hat{v}_{1})$ and $([0,1]\times
Y_{2},\tilde{v}_{2})\supset(\hat{X}_{2},\hat{v}_{2})$ as in Lemma 4.3, we get
that

$(\hat{X}_{1}\,\\#_{\partial}\,\hat{X}_{2},\hat{v}_{1}\,\\#_{\partial}\,\hat{v}_{2})\subset([0,1]\times(Y_{1}\,\\#_{\partial}\,Y_{2}),\tilde{v}_{1}\,\\#_{\partial}\,\tilde{v}_{2})$

is also a box as in that lemma. Therefore, all the constructions and arguments
from Section 3 in [K9] (like Proposition 3.4, Corollary 3.3, and Theorem 3.2)
apply to the convex pseudo-envelops/envelops
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(D^{n+1},\hat{v})$, where the
traversing vector field $\hat{v}$ is convex with respect to $\partial
D^{n+1}$. Thus, as in formula (3) above (see formula (3.15) and Proposition
3.2 from [K9]), for a convex vector field $\hat{v}$ with
$\partial_{1}^{+}D^{n+1}(\hat{v})$ being a smooth $n$-ball, the quasitopies
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
and
$\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
are groups. For $n>1$ they are commutative by arguments as in Proposition 3.2
from [K9].

The last claim follows by the general position argument. ∎

Recall that, by Corollary 4.5, the set
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
depends only on the homotopy type of the pair
$(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))$;
however, the corollary does not make any claims about
$\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$.
The next proposition, in line with Proposition 4.1 (which deals with the
homology $n$-spheres, $n\geq 6$), is a hint that
$\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
may also depend only on the homotopy type of the pair
$(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))$.

######  Proposition 4.2.

For $n\geq 5$, assuming that the locus $\partial_{2}^{-}D^{n+1}(\hat{v})$ is
simply-connected, the groups
$\mathcal{QT}^{\mathsf{sub/emb}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
do not depend on the choice of the convex traversing vector field $\hat{v}$ on
$D^{n+1}$.

For $n\leq 3$, the groups
$\mathcal{QT}^{\mathsf{sub/emb}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
also do not depend on the convex $\hat{v}$.

###### Proof.

For a convex $\hat{v}$, the locus $\partial_{1}^{+}\hat{X}(\hat{v})$ is a
deformation retract of $D^{n+1}$ and thus has a homotopy type of a point. By
the Poincaré duality,
$\partial_{2}^{-}D^{n+1}(\hat{v})=\partial(\partial_{1}^{+}\hat{X}(\hat{v}))$
is a homology $(n-1)$-sphere $\Sigma^{n-1}\subset\partial D^{n+1}$. Let us
delete a small smooth ball $B^{n}$ from the interior of
$\partial_{1}^{+}\hat{X}(\hat{v})$. We denote its complement by $W^{n}$. For
$n\geq 5$, assuming that $\pi_{1}(\partial_{2}^{-}D^{n+1}(\hat{v}))=1$, we may
apply the smooth $h$-cobordism theorem (see [Mi]) to $W$ to conclude that it
is diffeomorphic to the product $S^{n-1}\times[0,1]$. Therefore,
$\partial_{1}^{+}D^{n+1}(\hat{v})$ is a smooth ball $D^{n}\subset S^{n}$. Any
two regular embeddings $D^{n}\hookrightarrow S^{n}$ are diffeotopic. By the
proof of Lemma 4.1, any diffeotopy of $S^{n}$ which maps
$\partial_{1}^{+}D^{n+1}(\hat{v}_{1})$ to
$\partial_{1}^{+}D^{n+1}(\hat{v}_{2})$ extends to a diffeotopy of $D^{n+1}$
that maps the $\hat{v}_{1}$-trajectories to $\hat{v}_{2}$-trajectories, while
preserving their orientation. As a result, for $n\geq 5$, the group
$\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(D^{n+1},\hat{v};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
does not depend on the choice of $\hat{v}$, as long as
$\pi_{1}(\partial_{2}^{-}D^{n+1}(\hat{v}))=1$.

In small dimensions $n\leq 3$, the contractibility of
$\partial_{1}^{+}D^{n+1}(\hat{v})$ implies that it is the standard smooth
$n$-ball. The case $n=1$ is obvious. For $n=2$, if the domain
$\partial_{1}^{+}\hat{X}(\hat{v})\subset S^{2}$ has a boundary that is a
homology $1$-sphere, then the domain is the $2$-ball, and any $2$-balls in
$S^{2}$ are isotopic. For $n=3$, if the domain
$\partial_{1}^{+}D^{4}(\hat{v})\subset S^{3}$ has a smooth boundary that is a
homology $2$-sphere. By the classification of $2$-surfaces, it follows that
$\partial_{2}^{-}D^{4}(\hat{v})$ is the standard $2$-sphere. By the solution
of the $3$-dimensional Poincaré Conjecture [P1]-[P3], the contractible domain
$\partial_{1}^{+}D^{4}(\hat{v})$ is with the spherical boundary is the
$3$-ball, and any two $3$-balls in $S^{3}$ are isotopic. Thus, by Lemma 4.1,
the smooth isotopy type of the locus $\partial_{1}^{+}D^{n+1}(\hat{v})\subset
S^{n}$ determines the smooth topological type of the foliation
$\mathcal{L}(\hat{v})$. Since, any two standard $n$-balls
$\partial_{1}^{+}\hat{X}(\hat{v}_{1})$ and
$\partial_{1}^{+}\hat{X}(\hat{v}_{2})$ are isotopic in $S^{n}$, the two
statements of the proposition are validated. The difficult case $n=4$ is wide
open. ∎

Combining Theorem 4.2 and Theorem 4.4, we get the following claim.

######  Corollary 4.7.

Let $\hat{v}^{\parallel}$ be a constant vector field on the ball
$D^{n+1}\subset\mathbb{R}^{n+1}$, and let $\hat{v}$ be a convex vector field
on $\hat{X}$. Put $r=\\#(\pi_{0}(\partial_{2}^{-}\hat{X}(\hat{v})))$. With the
help of the maps from (4.4) and (4.4), the groups

$\mathcal{G}^{\mathsf{emb}}:=\big{(}\mathcal{QT}^{\mathsf{emb}}_{d,d^{\prime}}(D^{n+1},\hat{v}^{\parallel};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\big{)}^{r}\;\text{
and
}\;\mathcal{G}^{\mathsf{sub}}:=\big{(}\mathcal{QT}^{\mathsf{sub}}_{d,d^{\prime}}(D^{n+1},\hat{v}^{\parallel};\,\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})\big{)}^{r}$

are represented in the group

$\big{(}\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\big{/}\ker\big{\\{}(\epsilon_{d,d^{\prime}})_{\ast}:\pi_{n}(\mathcal{P}_{d}^{\mathbf{c}\Theta},pt)\to\pi_{n}(\mathcal{P}_{d^{\prime}}^{\mathbf{c}\Theta^{\prime}},pt^{\prime})\big{\\}}\big{)}^{r}.$

We denote by $\Psi^{\mathsf{emb}}$ and $\Psi^{\mathsf{sub}}$ these two
representations (the first one is an isomorphism).

Then the maps $\Phi^{\mathsf{sub}}(\hat{X},\hat{v})$ in (4.2) and
$\Phi^{\mathsf{emb}}(\hat{X},\hat{v})$ in (4.2) from Theorem 4.2 are
_equivariant_ with respect to the $\mathcal{G}^{\mathsf{sub}}$\- and
$\mathcal{G}^{\mathsf{emb}}$-actions on their source sets and the
$\Psi^{\mathsf{sub}}(\mathcal{G}^{\mathsf{sub}})$\- and
$\Psi^{\mathsf{emb}}(\mathcal{G}^{\mathsf{emb}})$-actions on their target
sets. $\diamondsuit$

###  4.6. Quasitopies of envelops with generic combinatorics $\Theta$ and
$d^{\prime}=d$

From now and until Subsection 4.9, each result about quasitopies
$\mathcal{QT}^{\mathsf{sub/emb}}_{d,d^{\prime}}(\hat{X},\hat{v};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$
of convex envelops is a recognizable immage of a similar result from [K9]
about the quasitopies
$\mathcal{QT}^{\mathsf{imm/emb}}_{d,d^{\prime}}(Y,\partial
Y;\mathbf{c}\Theta,(\emptyset),\mathbf{c}\Theta^{\prime})$ of
immersions/embedding into the products $\mathbb{R}\times Y$.

######  Theorem 4.5.

Let $(\hat{X},\hat{v})$ be a convex pair. Let $d,k$ be natural numbers such
that $2<k<d$ and $d\equiv 0\mod 2$. Put $\mathbf{\Omega}_{|\sim|^{\prime}\leq
k-1}=_{\mathsf{def}}\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k}$. Then
we get a bijection666see (2.12) for the definition of the number
$A(d,k):=A(d,k,0)$

$\displaystyle\Phi:\;\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
k-1};\mathbf{\Omega}_{|\sim|^{\prime}\leq
k-1})\;\approx\;\big{[}\big{(}\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v})\big{)},\;\big{(}\bigvee_{\ell=1}^{A(d,\,k)}S^{k-1}_{\ell},\;\star\big{)}\big{]}.$

###### Proof.

The theorem is based on Proposition 3.10 from [K9]. For $d>k>2$, Proposition
2.1, being combined with the Alexander duality, describes the homotopy type of
$\mathcal{P}_{d}^{\mathbf{\Omega}_{|\sim|^{\prime}\leq
k-1}}=\mathcal{P}_{d}^{\mathbf{c\Omega}_{|\sim|^{\prime}\geq k}}$ as a bouquet
$\bigvee_{\ell=1}^{A(d,\,k)}S^{k-1}_{\ell}$ of $(k-1)$-spheres. The space
$\mathcal{P}_{d}^{(\emptyset)}$ is contractible to the point $\star$. Thus, by
Theorem 4.2 (based on Theorem 3.1 from [K9]), the claim follows. ∎

We consider now the “combinatorially generic” case of convex pseudo-envelops
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$ for which $\hat{v}$
is traversally generic with respect to $\alpha(\partial X)$. By Definition
4.7, any such $\alpha$ has tangency patterns that belong to the poset
$\mathbf{\Omega}_{|\sim|^{\prime}\leq n}$, where $n=\dim\hat{X}-1$.

######  Corollary 4.8.

Let $(\hat{X},\hat{v})$ be a convex pair, $\dim\hat{X}=n+1$. For $d>n>2$, we
get a bijection

$\Phi:\,\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
n};\mathbf{\Omega}_{|\sim|^{\prime}\leq n})\approx\mathbb{Z}^{A(d,\,n+1)}.$

These $\mathbb{Z}$-valued invariants of convex envelops are delivered by the
degrees of the maps

$\\{\Phi_{\ell}:\partial_{1}^{+}\hat{X}(\hat{v})\big{/}\partial_{2}^{-}\hat{X}(\hat{v})\to
S^{n}_{\ell}\\}_{\ell\in[1,\;A(d,n+1)]},$

to the individual spheres, induced by the map $\Phi$ from Theorem 4.5.

In particular, the $(d,d;\mathbf{\Omega}_{|\sim|^{\prime}\leq
n},\mathbf{\Omega}_{|\sim|^{\prime}\leq n})$-quasitopy class of any
_traversally generic_ convex envelop
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$ is determined by
the collection of such degrees.

###### Proof.

Since $\hat{X}$ is connected and $\hat{v}$ is convex, the locus
$\partial_{1}^{+}\hat{X}$ is connected as well.

We repeat the arguments from Proposition 3.6 in [K9]. For $n>2$, the homotopy
classes of the classifying maps
$\Phi^{\alpha}:(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))\to\big{(}\mathcal{P}_{d}^{\mathbf{\Omega}_{|\sim|^{\prime}\leq
n}},\,\mathcal{P}_{d}^{(\emptyset)}\big{)}$ are in $1$-to-$1$ correspondence
with homotopy classes of the corresponding maps
$\tilde{\Phi}^{\alpha}:(\partial_{1}^{+}\hat{X}(\hat{v}),\,\partial_{2}^{-}\hat{X}(\hat{v}))\to\big{(}\bigvee_{\ell=1}^{A(d,\,n+1)}S^{n}_{\ell},\,\star\big{)}$.
For $n\geq 2$, the latter ones are detected by the degrees of the maps
$\\{\Phi^{\alpha}_{\ell}\\}$ to the individual spheres $\\{S^{n}_{\ell}\\}$. ∎

######  Example 4.5.

_Take $d=6$ and $n=3$. Then
$\bar{\mathcal{P}}_{6}^{\mathbf{\Omega}_{|\sim|^{\prime}\geq 4}}$ consist of a
single $0$-dimensional cell (the $``\infty"\in\bar{\mathcal{P}}_{6}$), one
$1$-dimensional cell, labelled by $\omega=(6)$, and five $2$-dimensional
cells, labelled by $\omega=(51),(15),(42),(24),(33)$. Hence,
$A(6,4)=\chi(\bar{\mathcal{P}}_{6}^{\mathbf{\Omega}_{|\sim|^{\prime}\geq
4}})-1=4$. The space
$\bar{\mathcal{P}}_{6}^{\mathbf{\Omega}_{|\sim|^{\prime}\geq 4}}$ has a
homotopy type of a bouquet $\bigvee_{\ell=1}^{4}S^{2}$ of four $2$-spheres. By
the Alexander duality,
$\mathcal{P}_{6}^{\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
4}}=\mathcal{P}_{6}^{\mathbf{\Omega}_{|\sim|^{\prime}\leq 3}}$ has a homotopy
type of a bouquet $\bigvee_{\ell=1}^{4}S^{3}$ of four $3$-spheres. _

_We pick a convex and traversing vector field $\hat{v}$ on $D^{4}$. Recall
that $\partial_{1}^{+}D^{4}(\hat{v})$ is contractible and thus
$\partial_{2}^{-}D^{4}(\hat{v})$ a homology $2$-sphere, which implies that
$\partial_{2}^{-}D^{4}(\hat{v})$ is diffeomorphic to $S^{2}\subset S^{3}$.
Thus $\partial_{1}^{+}D^{4}(\hat{v})$, by [P1], [P2], is diffeomorphic to the
ball $D^{3}$. Therefore, for any convex $\hat{v}$, we get the group
isomorphism_

$\mathcal{QT}^{\mathsf{emb}}_{6,6}(D^{4},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
3};\mathbf{\Omega}_{|\sim|^{\prime}\leq
3})\approx\pi_{3}(\bigvee_{\ell=1}^{4}S^{3}_{\ell})\approx\mathbb{Z}^{4}.\quad$

_As a result, any element
$[\alpha]\in\mathcal{QT}^{\mathsf{sub}}_{6,6}(D^{4},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
3};\mathbf{\Omega}_{|\sim|^{\prime}\leq 3})$ generates four integer-valued
characteristic invariants. For embeddings $\alpha$, they determine
$[\alpha]$._

_At the same time, by Corollary 4.9 below,
$\mathcal{QT}^{\mathsf{emb}}_{6,8}(D^{4},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
3};\mathbf{\Omega}_{|\sim|^{\prime}\leq 4})=0$._ $\diamondsuit$

The following claim contrasts Corollary 4.8.

######  Corollary 4.9.

Under the hypotheses of Corollary 4.8, including $d>n>2$, the set
$\mathcal{QT}^{\mathsf{emb}}_{d,\,d+2}(\hat{X},\hat{v};\,\mathbf{\Omega}_{|\sim|^{\prime}\leq
n};\mathbf{\Omega}_{|\sim|^{\prime}\leq n+1})$ consists of a single element,
represented by $X=\emptyset$.

In particular, the $(d,d+2;\mathbf{\Omega}_{|\sim|^{\prime}\leq
n},\mathbf{\Omega}_{|\sim|^{\prime}\leq n+1})$-quasitopy class of any
traversally generic convex envelop
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$ is trivial.

###### Proof.

Consider a convex pseudo-envelop
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$ as in Theorem 4.5.

The claim is based on the observation that any map from the $n$-dimensional
$CW$-complex
$\partial_{1}^{+}\hat{X}(\hat{v})/\partial_{2}^{-}\hat{X}(\hat{v})$ to
$\mathcal{P}_{d}^{\mathbf{c\Omega}_{|\sim|^{\prime}\geq n}}$—homotopically a
bouquet of $n$-spheres— is null-homptopic in
$\mathcal{P}_{d+2}^{\mathbf{c\Omega}_{|\sim|^{\prime}\geq n+1}}$, since the
latter space has the homotopy type of a bouquet of $(n+1)$-spheres. Again, by
Theorem 4.2, the pseudo-envelop $\alpha$ is null-quasitopic. ∎

###  4.7. From inner framed cobordisms of $\partial_{1}^{+}\hat{X}(\hat{v})$
to quasitopies of $k$-flat convex envelops

Theorem 4.5 suggests a somewhat unexpected connection between quasitopies of
certain convex envelops $\alpha:X\to\hat{X}$ and inner framed cobordisms of
the manifold $\partial\hat{X}$.

Let $Y$ be a compact smooth $n$-manifold. For $k-1\leq n$, we associate with
$Y$ the set of inner framed cobordisms $\mathcal{FB}^{k-1}_{\Xi}(Y)$. These
cobordisms are based on codimension $k-1$ smooth closed submanifolds $Z$ of
$Y$ of the form $Z=\coprod_{\sigma=1}^{A(d,k,q)}Z_{\sigma}$, where the number
$A(d,k,q)$ is introduced in (2.12). The normal $(k-1)$-bundle $\nu(Z,Y)$ is
required to be framed. The disjoint “components” $\\{Z_{\sigma}\\}$ of $Z$ are
marked with different colors $\sigma$ from a pallet $\Xi$ of cardinality
$A(d,k,q)$.

We have seen in [K9], Proposition 3.11, that the inner framed $\Xi$-colored
codimension $k-1$ bordisms of the $n$-dimensional manifold $Y$ produce, via
the Thom construction, quite special $k$-flat embeddings
$\beta:M\to\mathbb{R}\times Y$, where $\dim M=\dim Y$. The analogous
mechanism, with the help of Lemma 4.3 and Theorem 4.5, generates special
$k$-flat envelops (see Definition 4.6).

######  Proposition 4.3.

Let $\hat{X}$ be a smooth compact connected $(n+1)$-dimensional manifold,
equipped with a convex traversing vector field $\hat{v}$. Let $k\in[3,n+1]$,
$k<d$, and $d\equiv 0\mod 2$. Then the Thom construction delivers a bijection

$Th:\mathcal{FB}_{\Xi}^{k-1}(\partial_{1}^{+}\hat{X}(\hat{v}))\approx\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
k};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k}),$

where $\mathcal{FB}_{\Xi}^{k-1}(\partial_{1}^{+}\hat{X}(\hat{v}))$ denotes the
set of inner framed $\Xi$-colored $(n-k+1)$-dimensional bordisms of the space
$\partial_{1}^{+}\hat{X}(\hat{v})$. Here $\\#\Xi=A(d,k)$ (see 2.12).
$\diamondsuit$

######  Example 4.6.

_Let us recycle Example 3.5 from[K9]: take $k=3$, $d=6$, $q=0$, and $n=3$.
Then $\\#\Xi=10$. So we get a homotopy equivalence
$\tau:\mathcal{P}_{6}^{\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3}}\sim\bigvee_{\sigma=1}^{10}S^{2}_{\sigma}$. Let $(\hat{X},\hat{v})$ be a
convex pair, where $\dim\hat{X}=4$, and let
$Z\subset\partial_{1}^{+}\hat{X}(\hat{v})$ be a normally framed
$1$-dimensional closed submanifold, each loop in $Z$ being colored with a
color from the pallet $\Xi$. Using the isomorphism_

$Th:\mathcal{FB}^{2}_{\Xi}(\partial_{1}^{+}\hat{X}(\hat{v}))\approx\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}\hat{X},\hat{v};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)},$

_any _framed link_ $Z\subset\partial_{1}^{+}\hat{X}(\hat{v})$, colored with
$10$ distinct colors at most, produces a quasitopy class of a convex envelop
$\alpha:(X,\alpha^{\dagger}(\hat{v}))\to(\hat{X},\hat{v})$. Its tangency
patterns $\omega$, except for $\omega=(\emptyset)$, have only entries from the
list $\\{1,2,3\\}$, so that no more than one $3$ is present in $\omega$, and
no more than two $2$’s are present, while $|\omega|\leq 6$._

_Now, let us assume that $D^{4}$ carries a constant vector field
$v^{\parallel}$ and that $\partial_{2}^{-}\hat{X}(\hat{v})\neq\emptyset$. Note
that any element of $H_{1}(\partial_{1}^{+}\hat{X}(\hat{v}));\mathbb{Z})$ my
be realized by a disjoint union of framed oriented loops. Then, like in
Example 3.5 from [K9], the orbit-space of the
$\mathsf{G}^{\mathsf{emb}}_{6,6}(D^{4};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3})$-action on
$\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}\hat{X},\hat{v};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)}$ admits a surjection on the
group
$(H_{1}(\partial_{1}^{+}\hat{X}(\hat{v}));\mathbb{Z}))^{10}\approx(H_{1}(\hat{X};\mathbb{Z}))^{10}$,
provided that $\partial\hat{X}$ is orientable._

_Let us give a couple of specific examples of this fact. Consider the box
$T^{3}_{\circ}\times[0,1]$, where $T^{3}_{\circ}$ is the compliment in the
$3$-torus $T^{3}$ to a ball $D^{3}$. By rounding the corners of the box, we
get a convex pair $(\hat{X},\hat{v})$, where $\hat{X}$ is homeomorpfic to
$T^{3}_{\circ}\times[0,1]$. Then the orbit-space of the
$\mathsf{G}^{\mathsf{emb}}_{6,6}(D^{4};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3})$-action on
$\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}\hat{X},\hat{v};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)}$ is mapped onto the lattice
$\mathbb{Z}^{30}$._

_Let $M=\mathbb{H}^{3}/\Gamma$ be a compact hyperbolic $3$-manifold and
$M_{\circ}:=M\setminus D^{3}$. By rounding the corners of the box
$M_{\circ}\times[0,1]$, we get a convex pair $(\hat{X},\hat{v})$. Then the
orbit-space of the
$\mathsf{G}^{\mathsf{emb}}_{6,6}(D^{4};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3})$-action on
$\mathsf{QT}^{\mathsf{emb}}_{6,6}\big{(}\hat{X},\hat{v};\mathbf{c}\Theta_{|\sim|^{\prime}\geq
3};\mathbf{c}\Theta_{|\sim|^{\prime}\geq 3}\big{)}$ is mapped onto the abelian
group $(\Gamma/[\Gamma,\Gamma])^{10}$, where $[\Gamma,\Gamma]$ denotes the
commutator. $\diamondsuit$_

By Proposition 4.3 and repeating the (obviously modified) arguments in the
proof of Corollary 3.13 from [K9], we get the following corollary.

######  Corollary 4.10.

Let $d>k$, and $d\equiv 0\mod 2$. Let $(\hat{X},\hat{v})$ be a convex pair.

  * •

For $\dim\hat{X}=k\geq 3$ and any choice of
$\kappa\in\pi_{0}(\partial_{2}^{-}\hat{X}(\hat{v}))$, the group
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(D^{k},\hat{v}^{\parallel};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
k};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k})$ acts freely and
transitively on the set
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
k};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k})$. Thus both sets are in
a $1$-to-$1$ correspondence.

  * •

For $\dim\hat{X}=k+1>5$, a simply-connected $\hat{X}$, and any choice of
$\kappa\in\pi_{0}(\partial_{2}^{-}\hat{X}(\hat{v}))$, the group
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(D^{k+1},\hat{v}^{\parallel};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
k};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k})$ acts freely and
transitively on the set
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
k};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq k})$. Again, both sets are
in a $1$-to-$1$ correspondence.

$\diamondsuit$

######  Example 4.7.

_Let $\hat{X}=\mathbb{C}\mathbb{P}^{2}\times D^{2}$ with $\hat{v}$ on
$\mathbb{C}\mathbb{P}^{2}\times D^{2}$ being generated by a constant vector
field on $D^{2}$. Then
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(D^{6},\hat{v}^{\parallel};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
5};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq 5})$ acts freely and
transitively on the set
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(\mathbb{C}\mathbb{P}^{2}\times
D^{2},\hat{v};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
5};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq 5})$ for all even $d\geq 6$.
Thus both sets are in $1$-to-$1$ correspondence. As a result,
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(\mathbb{C}\mathbb{P}^{2}\times
D^{2},\hat{v};\,\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq
5};\mathbf{c}\mathbf{\Omega}_{|\sim|^{\prime}\geq 5})$ acquires the structure
of the abelian group
$\pi_{5}\big{(}\bigvee_{i=1}^{A(d,5)}S^{4}_{i},\,\star\big{)}$._
$\diamondsuit$

###  4.8. Convex envelops with special combinatorics $\Theta$

For a given $\omega\in\mathbf{\Omega}$, we denote by $\langle\omega\rangle$
the minimal closed poset, generated by $\omega$. Combining Theorem 4.2 and
Theorem 3.5 from [K9] with Proposition 4.2, we get the following result, in
which the constraint $d\leq 12$ reflects only the scope of the numerical
experiments in [KSW2].

######  Theorem 4.6.

Assume that $d\leq 12$ and $d\equiv 0\mod 2$. Let
$\omega\in\mathbf{\Omega}_{\leq d}$ be such that $|\omega|^{\prime}>2$. Let
$\hat{X}$ be a smooth compact connected $(n+1)$-dimensional manifold, equipped
with a convex traversing vector field $\hat{v}$.

Then the quasitopy set
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(\hat{X},\hat{v};\mathbf{c}\langle\omega\rangle;\mathbf{c}\langle\omega\rangle)$
either consists of single element (is trivial), or is isomorphic to the
cohomotopy set
$\pi^{k}(\partial_{1}^{+}\hat{X}(\hat{v})/\partial_{2}^{-}\hat{X}(\hat{v}))$,
where $k=k(\omega)\in[|\omega|^{\prime}-1,\,d-1]$.

In particular, for a constant vector field $\hat{v}=v^{\parallel}$, $d\leq
12$, and $\omega$ and $k=k(\omega)$ from the table in [K9], Appendix, the
group
$\mathcal{QT}^{\mathsf{emb}}_{d,d}(D^{n+1},\hat{v};\mathbf{c}\langle\omega\rangle,\mathbf{c}\langle\omega\rangle)\approx\pi_{n}(S^{k})$.
A similar claim is valid for any convex $\hat{v}$ on $D^{n+1}$, provided that
either $n\leq 3$ or $n\geq 5$ and $\partial_{2}^{-}\hat{X}(\hat{v})$ is
simply-connected. $\diamondsuit$

For $d\leq 13$, the table from Appendix in [K9] lists all $\omega$’s and the
corresponding $k=k(\omega)$’s for which
$\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ is homologically
nontrivial. In fact, $\mathcal{P}_{d}^{\mathbf{c}\langle\omega\rangle}$ is a
homology $k$-sphere and even a homotopy $k$-sphere, at least when
$|\omega|^{\prime}>2$ [K9], a quite mysterious phenomenon…

Let us recycle Example 3.4 from [K9], while adapting it to convex envelops.

######  Example 4.8.

_Let $d=8$ and $\omega=(4)$. Then, for any convex pair $(\hat{X},\hat{v})$,
using the list in [K9], Appendix, we get a bijection_

$\mathcal{QT}^{\mathsf{emb}}_{8,8}(\hat{X},\hat{v};\mathbf{c}\langle(4)\rangle,\mathbf{c}\langle(4)\rangle)\approx\pi^{4}\big{(}\partial_{1}^{+}\hat{X}(\hat{v})\big{/}\partial_{2}^{-}\hat{X}(\hat{v})\big{)},$

_where $\pi^{4}(\sim)$ stands for the $4$-cohomotopy set. In particular, using
Proposition 4.2, for any convex $\hat{v}$ on $D^{8}$ such that the locus
$\partial_{2}^{-}D^{8}(\hat{v})$ is simply-connected (the constant $\hat{v}$