# Second obstruction to pseudoisotopy I

Kiyoshi Igusa  Department of Mathematics, Brandeis University, Waltham, MA
02454<EMAIL_ADDRESS>
###### Abstract.

A pseudoisotopy of $M$ is a diffeomorphism of $M\times I$ which is the
identity on $M\times 0$. We give an explicit construction of pseudoisotopies
of 4-manifolds which realize certain elements of the “second obstruction to
pseudoisotopy”. The next paper will give precise statements of our results for
5-manifolds which had been announced long ago.

###### Key words and phrases:

Morse theory, 4-manifolds, diffeomorphisms

###### 2020 Mathematics Subject Classification:

19J10: 58K60

Supported by the Simons Foundation

###### Contents

  1. 1 Definition of second obstruction to pseudoisotopy
    1. 1.1 Basic setup
    2. 1.2 The graphic
    3. 1.3 Second obstruction
    4. 1.4 Computation of second obstruction
    5. 1.5 Dependence on markings
    6. 1.6 Involution
  2. 2 Constructing pseudoisotopies using two disks
  3. 3 Realizing the framing obstruction
  4. 4 Construction of the two disks

## Introduction

Recall that, for any smooth compact manifold $M$, a _pseudoisotopy_ of $M$ is
a diffeomorphism of $M\times I$ which is the identity on $M\times
0\cup\partial M\times I$. The restriction of this diffeomorphism to $M\times
1$ gives a diffeomorphism of $M$ which is the identity on $\partial M$. This
diffeomorphism is said to be _pseudoisotopic_ to the identity diffeomorphism
of $M$ rel $\partial M$. Let ${\mathcal{C}}(M)$ be the group of
pseudoisotopies of $M$. Thus

${\mathcal{C}}(M)=Dif\\!f(M\times I\,rel\,M\times 0\cup\partial M\times I).$

We give ${\mathcal{C}}(M)$ a suitable topology so that, up to canonical
homotopy, a path in ${\mathcal{C}}(M)$ is given by a diffeomorphism of
$M\times I\times I$ which preserves the last coordinate. (Section 1.) Since
${\mathcal{C}}(M)$ is a topological group, $\pi_{0}{\mathcal{C}}(M)$ will be a
group.

For $M$ simply connected with $\dim M\geq 5$, Jean Cerf [1] showed that
${\mathcal{C}}(M)$ is connected. For $\dim M\geq 6$, Hatcher and Wagoner [2]
showed that there is an exact sequence

(0.1)
$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)\to\pi_{0}{\mathcal{C}}(M)\to
Wh_{2}(\pi)\to 0.$

When $\dim M\leq 5$, we apply a “suspension functor”
$\sigma:{\mathcal{C}}(M)\to{\mathcal{C}}(M\times I)$ several times to get to
the _stable pseudoisotopy space_
${\mathcal{P}}(M)=\operatorname{colim}{\mathcal{C}}(M\times I^{N})$ [6] which
should be used instead of ${\mathcal{C}}(M)$ in the above exact sequence.

In [5], [4] and [3] the sequence (0.1) was extended to an exact sequence

(0.2)
$K_{3}{\mathbb{Z}}[\pi_{1}M]\xrightarrow{\chi}Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)\xrightarrow{\theta}\pi_{0}{\mathcal{P}}(M)\to
Wh_{2}(\pi)\to 0.$

The question arises: Which elements of $\pi_{0}{\mathcal{P}}(M)$ are realized
when $\dim M<6$, i.e., which elements of $\pi_{0}{\mathcal{P}}(M)$ lie in the
image of the stabilization map
$\pi_{0}{\mathcal{C}}(M)\to\pi_{0}{\mathcal{P}}(M)$? This question was
answered in dimension 5 by the author in a preliminary version of [7] and
partially answered in the topological category in dimension 4 by Kwasik in [8]
using the techniques of [2] and one idea from [7]. In both [7] and [8] we need
an assumption which trivializes the image of $\chi$ in (0.2).

This series of papers discusses this question for $\dim M=4,5$. In the current
paper we show that, for certain $4$-dimensional manifolds $M$, some elements
of the second obstruction group

$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)=Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})\oplus
Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$

can be realized by pseudoisotopies of $M$. Given a group $G$ and $G$-module
$A$, $Wh_{1}^{+}(G;A)$ is a quotient of
$A[G]/A[e]=A\otimes{\mathbb{Z}}[G]/{\mathbb{Z}}[e]$ (Definition 1.7). In
particular, $Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ is generated by elements
$\alpha[\sigma]$ where $\alpha\in\pi_{2}M$ and $\sigma$ is a nontrivial
element of $\pi_{1}M$.

######  Theorem A (Theorem 2.3).

Let $M$ be the connected sum of ${\mathbb{C}}P^{2}$ with a nonsimply connected
$4$-manifold $X$. Then the elements of $Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ of the
form $\alpha[\sigma]$ where $\alpha$ is represented by the $2$-sphere
${\mathbb{C}}P^{1}\subset{\mathbb{C}}P^{2}$ lift to $\pi_{0}{\mathcal{C}}(M)$
and map to nontrivial elements of the kernel of
$\pi_{0}{\mathcal{P}}(M)\twoheadrightarrow Wh_{2}(\pi)$.

The proof relies on the fact that ${\mathbb{C}}P^{1}\subset{\mathbb{C}}P^{2}$
is an embedded $2$-sphere which gives an element of $\pi_{2}M$ which is not
hit by the Postnikov invariant $k_{1}\in H^{3}(\pi_{1}M;\pi_{2}M)$. We know by
the formula for $\chi$ given in [5] that this element of $\pi_{2}M$, together
with any nontrivial element of $\pi_{1}M$, gives an element of the second
obstruction group $Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ whose image in
$\pi_{0}{\mathcal{P}}(M)$ is nontrivial. The assertion in this paper is that
this second obstruction element of $\pi_{0}{\mathcal{P}}(M)$ lifts to
$\pi_{0}{\mathcal{C}}(M)$ where $\dim M=4$.

The proof relies on the space ${\mathcal{D}}(M)$ of “lens-space models” for
pseudoisotopies of $M$ (Definition 1.4). This is the space considered by
Hatcher and Wagoner to determine the second obstruction to pseudoisotopy. We
also consider the space ${\mathcal{D}}_{0}(M)$ of “marked lens-space models”
for $M$ since $\pi_{0}{\mathcal{D}}_{0}(M)$ forms a group (Lemma 1.6).

We recall that $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$, is
${\mathbb{Z}}_{2}[\pi_{1}M]/{\mathbb{Z}}_{2}[e]$ modulo the conjugation action
of $\pi_{1}M$. For example, when $\pi_{1}M$ is finite,

$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})=\bigoplus_{c-1}{\mathbb{Z}}_{2}$

is the direct sum of $c-1$ copies of ${\mathbb{Z}}_{2}$ where $c$ is the
number of conjugacy classes of elements of $\pi_{1}M$. In dimension 4, we do
not know whether all elements of $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$ can
be realized by pseudoisotopies of $M$ and any realization might have infinite
order. We show in this paper that there are homomorphisms
$\widetilde{\theta}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to\pi_{0}{\mathcal{C}}(M)$
and $\overline{\lambda}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to
Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)$ making the following
diagram commute. Furthermore, when $M=X\\#{\mathbb{C}}P^{2}$,
$\overline{\lambda}_{0}$ maps onto $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$.

$\textstyle{\pi_{0}{\mathcal{D}}_{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\lambda}_{0}}$$\scriptstyle{\widetilde{\theta}_{0}}$$\textstyle{\pi_{0}{\mathcal{C}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{\pi_{0}{\mathcal{P}}(M)}$

This proves the second main theorem:

######  Theorem B (Theorem 3.1).

Let $M$ be the connected sum of ${\mathbb{C}}P^{2}$ with a nonsimply connected
$4$-manifold $X$. Then every elements in the image of
$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})\to\pi_{0}{\mathcal{P}}(M)$ lift to
$\pi_{0}{\mathcal{C}}(M)$.

The same proof works in a slightly more general case, when $M$ has an embedded
$2$-sphere with odd self-intersection number. (Theorem 3.2.) A topological
version of this result for smooth $4$-manifolds was obtained by Kwasik [8,
Proposition 4.1].

##  1\. Definition of second obstruction to pseudoisotopy

We will review the basic theory, due to Cerf [1], relating diffeomorphisms of
$M\times I$ and Morse theory. Then we focus on the second obstruction, due to
Hatcher and Wagoner [2]. We assume in this section that $M$ is a compact
connected smooth manifold of dimension $n\geq 4$.

###  1.1. Basic setup

Let ${\mathcal{F}}(M)$ be the space of all smooth functions $f:M\times I\to I$
so that $f$ is equal to projection to $I$ in some neighborhood of

$\partial(M\times I)=\partial M\times I\cup M\times 0\cup M\times 1.$

Then ${\mathcal{F}}(M)$ is clearly contractible: ${\mathcal{F}}(M)\simeq\ast$.
Let ${\mathcal{E}}(M)$ be the space of all $f\in{\mathcal{F}}(M)$ with no
singularities. Then, Cerf observed that

${\mathcal{E}}(M)\simeq{\mathcal{C}}(M)$

where the homotopy equivalence ${\mathcal{C}}(M)\to{\mathcal{E}}(M)$ is given
by sending $g:M\times I\to M\times I$ to $\pi_{I}\circ g:M\times I\to I$ where
$\pi_{I}:M\times I\to I$ is the projection map. Given $f\in{\mathcal{E}}(M)$,
the corresponding element of ${\mathcal{C}}(M)$ is constructed uniquely up to
contractible choice by integrating the gradient of $f$. As a consequence

$\pi_{0}{\mathcal{C}}(M)\cong\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M)).$

For clarity we repeat: This bijection sends the isotopy class of
$g\in{\mathcal{C}}(M)$ to the homotopy class of the 1-parameter family $f_{t}$
if and only if $f_{1}=\pi_{I}\circ g$ (and $f_{0}=\pi_{I}$).

######  Proposition 1.1.

Let $(f_{t}^{a}),(f_{t}^{b})$ be elements of
$\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ corresponding to
$g^{a},g^{b}\in\pi_{0}{\mathcal{C}}(M)$. Thus $f_{1}^{a}=\pi_{I}\circ g^{a}$,
$f_{1}^{b}=\pi_{I}\circ g^{b}$. Then the product in $\pi_{0}{\mathcal{C}}(M)$,
given by composition: $g^{a}\circ g^{b}$ is $f_{t}^{ab}$ given by putting
$f_{t}^{b}$ to the left of $f_{t}^{a}$:

$f_{t}^{ab}=\begin{cases}f_{2t}^{b}&\text{if }t\leq\frac{1}{2}\\\
f_{2t-1}^{a}\circ g^{b}&\text{if }t>\frac{1}{2}.\end{cases}$

###### Proof.

First note that the family of function $f_{t}^{ab}$ lies in
$\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ since, at $t=0$,
$f_{0}^{b}=\pi_{I}$ and, at $t=\frac{1}{2}$, $f_{1}^{b}=\pi_{I}\circ
g^{b}=f_{0}^{a}\circ g^{b}$. It is also easy to see that $f^{ab}$ corresponds
to $g^{a}g^{b}$ since $f_{1}^{ab}=f_{1}^{a}\circ g^{b}=\pi_{I}\circ g^{a}\circ
g^{b}$. ∎

######  Corollary 1.2.

The inverse of $(f_{t})\in\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ is the
1-parameter family

$f_{t}^{\prime}=f_{1-t}\circ g^{-1}:M\times I\to I$

where $g\in{\mathcal{C}}(M)$ is a diffeomorphism of $M\times I$ whose
composition with projection $\pi_{I}:M\times I\to I$ is
$f_{1}\in{\mathcal{E}}(M)$ (so that $f_{0}^{\prime}\circ g=f_{1}=\pi_{I}\circ
g$, making $f_{0}^{\prime}=\pi_{I}$.)

###### Proof.

Let $f^{a}_{t}=f_{t}^{\prime}$ and $f^{b}_{t}=f_{t}$. By Proposition 1.1, the
product $f^{ab}=f^{a}f^{b}$ at $t=1$ is

$f_{1}^{ab}=f_{1}^{a}\circ g^{b}=f^{b}_{0}\circ(g^{b})^{-1}\circ
g^{b}=f_{0}^{b}=\pi_{I}.$

This corresponds to the identity in $\pi_{0}{\mathcal{C}}(M)$. So,
$f^{a}=(f_{t}^{\prime})$ is the inverse of $f^{b}=(f_{t})$. ∎

Here is an alternate version of composition in $\pi_{0}{\mathcal{C}}(M)$ in
terms of functions.

######  Proposition 1.3.

The group structure (composition) on $\pi_{0}{\mathcal{C}}(M)$ corresponds to
stacking of functions $f_{a},f_{b}:M\times I\to I$ in ${\mathcal{E}}(M)$ given
by “putting $f_{b}$ on top of $f_{a}$”:

$(f_{a}\bullet f_{b})(x,t)=\begin{cases}\frac{1}{2}f_{a}(x,2t)&\text{if
}t\leq\frac{1}{2}\\\ \frac{1}{2}+\frac{1}{2}f_{b}(x,2t-1)&\text{if
}t>\frac{1}{2}\end{cases}$

###### Proof.

Let $g_{a},g_{b}\in{\mathcal{C}}(M)$ be given by
$g_{a}(x,t)=(h_{a}(x,t),f_{a}(x,t))$ and $g_{b}=(h_{b},f_{b})$. Then
$g_{a},g_{b}$ are isotopic to
$g_{a}^{\prime},g_{b}^{\prime}\in{\mathcal{C}}(M)$ given by “pushing $g_{a}$
down and $g_{b}$ up”:

$g_{a}^{\prime}(x,t)=\begin{cases}(h_{a}(x,2t),\frac{1}{2}f_{a}(x,2t))&\text{if
}t\leq\frac{1}{2}\\\ (r(x),t)&\text{if }t>\frac{1}{2}\end{cases}$

where $r\in Dif\\!f(M)$ is the restriction of $g_{a}$ to $M\times 1$, i.e.,
$g_{a}(x,1)=(r(x),1)$ and

$g_{b}^{\prime}(x,t)=\begin{cases}(x,t)&\text{if }t\leq\frac{1}{2}\\\
(h_{b}(x,2t-1),\frac{1}{2}+\frac{1}{2}f_{b}(x,2t-1))&\text{if
}t>\frac{1}{2}\end{cases}$

Then $g_{a}\circ g_{b}$ is isotopic to $g_{a}^{\prime}\circ g_{b}^{\prime}$
given by

$(g_{a}^{\prime}\circ
g_{b}^{\prime})(x,t)=\begin{cases}(h_{a}(x,2t),\frac{1}{2}f_{a}(x,2t))&\text{if
}t\leq\frac{1}{2}\\\
(rh_{b}(x,2t-1),\frac{1}{2}+\frac{1}{2}f_{b}(x,2t-1))&\text{if
}t>\frac{1}{2}\end{cases}$

Taking the second coordinate gives $f_{a}\bullet f_{b}$. ∎

The group structure on $\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ is also
given by stacking functions.

###  1.2. The graphic

The _fiberwise singular set_ $\Sigma(f)$ of $f$ is the set

$\Sigma(f):=\\{(x,s,t)\in M\times I^{2}\,:\,(x,s)\in M\times I\text{ is a
singularlity of }f_{t}\\}.$

Since each function $f_{t}$ will generically have a finite singular set, and
$f_{0},f_{1}$ are nonsingular, $\Sigma(f)$ will, generically, be a closed
1-dimensional manifold. If $\Sigma(f)$ is empty or if $f$ is homotopic
relative to $\partial(M\times I^{2})$ to a family of function with empty
singular set, then $f$ represents the trivial element of
$\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$. Thus,
$\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ is the obstruction to the
elimination of the singular set $\Sigma(f)$ by deformation of $f$.

We visualize $\Sigma(f)$ via its _graphic_ which is defined to be the set of
all $(s,t)\in I^{2}$ so that $s$ is a critical value of $f_{t}$. Two examples
are given in Figure 1. The graphic on the right side of Figure 1 is called an
“eye” or “lens”.

######  Definition 1.4.

A _lens-shaped graphic_ or _eye_ is, by definition, the graphic of a family of
functions $f_{t}:M\times I\to I$ so that $f_{t}$ is Morse with two critical
points $p_{t},q_{t}$ with $f_{t}(p_{t})<f_{t}(q_{t})$ for $t_{0}<t<t_{1}$,
$f_{t_{0}},f_{t_{1}}$ have only one critical point called a _birth-death
point_ and $f_{t}$ is nonsingular for $t\notin[t_{0},t_{1}]$. A family of
functions $(f_{t})$ having a lens-shaped graphic will be called a _lens-shaped
model_ for $M$.

$t_{0}$$t_{1}$ Figure 1. Two possible graphics of 1-parameter families of
functions on $M\times I$. The one on the right is called an “eye” or “lens”.

###  1.3. Second obstruction

There are two steps in the process of simplification of the singular set
$\Sigma(f)$ of $f$. Hatcher and Wagoner [2] showed that, for $\dim M\geq 6$,
there is a surjective homomorphism

$p:\pi_{0}{\mathcal{C}}(M)\to Wh_{2}(\pi_{1}M)$

where, for any group $\pi$, $Wh_{2}(\pi)$ is a quotient of
$K_{2}({\mathbb{Z}}[\pi])$. They called this invariant the “first obstruction
to pseudo-isotopy”.

Hatcher and Wagoner also showed that the kernel of the map $p$ is generated by
“lens-shaped models” (Definition 1.4) in the middle indices $k,k+1$ where
$k=\lfloor\frac{n}{2}\rfloor$ is the greatest integer $\leq\frac{n}{2}$ where
$n=\dim M$. The “second obstruction to pseudo-isotopy” is an invariant defined
on this set of 1-parameter families of function.

To define these invariants, we will add extra structure, called “markings”, to
these families of functions, define invariants on the marked lens-shaped
models, then determine how these invariants change when we change the
markings.

######  Definition 1.5.

Let ${\mathcal{D}}(M)$ denote the space of lens-shaped models $(f_{t})$ for
$M$ with critical points $p_{t},q_{t}$ in indices $k,k+1$ where
$k=\lfloor\frac{n}{2}\rfloor$. Let ${\mathcal{D}}_{0}(M)$ denote the space of
_“marked” lens-shaped models_ for $M$. These are defined to be lens-shaped
models $(f_{t})$ together with:

  1. (1)

A homotopy class of paths $\gamma$ from a fixed base point
$\ast=(x_{0},0,0)\in M\times I^{2}$ to the birth point $b_{0}$ of $f_{t}$ (at
$t=t_{0}$ in Fig. 1).

  2. (2)

An orientation of the negative eigenspace of $D^{2}f_{t}$ at the birth point
$b_{0}$.

The group $\pi_{1}(M)\times{\mathbb{Z}}_{2}$ acts on the set of marking,
$\pi_{1}(M)$ acting on the paths by composition and ${\mathbb{Z}}_{2}$ acting
by reversing the orientation. We denote by
$\widetilde{\theta}:\pi_{0}{\mathcal{D}}(M)\to\pi_{0}{\mathcal{C}}(M)$ the set
map which sends a lens-shaped model $(f_{t})$ to the corresponding element of
$\pi_{0}{\mathcal{C}}(M)\cong\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$.
Composing with the surjective mapping
$\pi_{0}{\mathcal{D}}_{0}(M)\twoheadrightarrow\pi_{0}{\mathcal{D}}(M)$ we get
a mapping

$\widetilde{\theta}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to\pi_{0}{\mathcal{C}}(M)$

which we will show is a homomorphism of groups (Lemma 1.6).

The group structure on $\pi_{0}{\mathcal{D}}_{0}(M)$ is given as follows
where, for convenience of notation, we double the size of the parameter
interval to $[0,2]$. Let $f^{a},f^{b}$ be two marked lens-shaped models for
$M$. These are 1-parameter families of functions $f_{t}^{a},f_{t}^{b}:M\times
I\to I$. The new function $f^{a}\ast f^{b}$ will be given by rescaling a
family of function $f_{t}^{ab}:M\times I\to I$, $t\in[0,2]$ by $(f^{a}\ast
f^{b})_{t}=f^{ab}_{2t}$. The functions $f_{t}^{ab}$ are constructed in two
stages.

  1. (1)

Starting with $f^{c}=f^{a}\cdot f^{b}$ given by putting $f^{b}$ to the left of
$f^{a}$ as in Proposition 1.1 but with the parameter domain stretched out.
Thus $f^{c}_{t}=f^{b}_{t}$ for $t\leq 1$ and $f^{c}_{t}=f_{t-1}^{a}\circ
g^{b}$ for $t>1$ where $g^{b}\in{\mathcal{C}}(M)$ so that
$f^{b}_{1}=\pi_{I}\circ g^{b}$. See Fig. 2.

  2. (2)

“Merging” the birth point $b_{2}$ of $f^{a}\circ g^{b}$ (related to the birth
point $b_{0}^{\prime}$ of $f^{a}$ by $b_{2}=(g^{b})^{-1}(b_{0}^{\prime})$)
with the death point $b_{1}$ of $f^{b}$ along the dashed path $w_{t}$ in Fig.
2 using the markings. The result is $f_{t}^{ab}$ for $t\in[0,2]$.

$b_{1}$$0$$f^{b}$$t$$s$$t_{1}$$b_{0}$$p_{t}$$w_{t}$$f^{a}\circ
g^{b}$$b_{2}$$1$$t_{2}$$2$ Figure 2. The family of functions $f^{c}_{t}$ on
$M\times I$ for $t\in[0,2]$ is given to be $f^{b}_{t}$ on the left side where
$t\in[0,1]$ and $f_{t-1}^{a}\circ g^{b}$ on the right side where $t\in[1,2]$.
The death point $b_{1}$ at $t_{1}$ is cancelled with the birth point $b_{2}$
at $t_{2}$ along the path $w_{t}$ to produce the marked lens shaped model
$f_{t}^{ab}$.

The family of functions $f_{t}^{c}$, $t\in[0,2]$, is indicated in Figure 2.
The “merging” process is given as follows. We have functions $f^{c}_{t}$ for
$t\in[t_{1},t_{2}]$ which are nonsingular for $t_{1}<t<t_{2}$ and so that each
of $f^{c}_{t_{1}},f^{c}_{t_{2}}$ has exactly one critical point $b_{1},b_{2}$,
resp., which is a birth-death point of index $k=\lfloor\frac{n}{2}\rfloor$.
Furthermore, the markings on $f^{b},f^{a}$ (when we compose with
$(g^{b})^{-1}$ in the case of $f^{a}$) give paths from $b_{1},b_{2}$ to a
common basepoint. In the case of $b_{1}$ this path goes first along the path
$p_{t}$ of index $k$ critical points of $f_{t}^{b}$ to the birth point $b_{0}$
of $f^{b}$ and then along the path to the basepoint given by the marking. We
combine these to give a path $w_{t}\in M\times I$ for $t\in[t_{1},t_{2}]$ so
that $w_{t}=b_{i}$ for $t=t_{i}$ and $f^{c}_{t}$ is nonsingular at $w_{t}$ for
all $t\in(t_{1},t_{2})$.

At each point $w_{t}$ for $t\in[t_{1},t_{2}]$ we let $v_{t}$ be the tangent
vector which points in the direction of the gradient of $f^{c}_{t}$ for
$t\in(t_{1},t_{2})$ and which points in the positive cubic direction of
$f_{t}^{c}$ in the kernel of $D^{2}f_{t}^{c}$ for $t=t_{1},t_{2}$. Using the
second part of the markings for $f^{a},f^{b}$ (composed with $D(g^{b})^{-1}$
in the case of $f^{a}$), we have an oriented tangent $k$ plane at
$b_{1},b_{2}$ which is orthogonal to $v_{t}$. For $n\geq 3$ the space of
oriented $k$-planes in ${\mathbb{R}}^{n}$ is simply connected. So, this can be
extended uniquely up to homtopy to a family of oriented tangent $k$ planes
along $w_{t}$ orthogonal to $v_{t}$ at each $w_{t}$.

We can choose local coordinates
$(x,y,z)\in{\mathbb{R}}^{k}\times{\mathbb{R}}^{n-k}\times{\mathbb{R}}$ for
$M\times I$ centered at $w_{t}$ so that

$f^{c}_{t}(x,y,z)=h_{t}(z)+||y||^{2}-||x||^{2}$

for all $t\in[t_{1},t_{2}]$ where $h_{t}(z)$ is a smooth family of functions
in one variable $z$ so that $h_{t}(0)=f^{c}_{t}(w_{t})$ for each
$t\in[t_{1},t_{2}]$. Working only with $h_{t}$ we can modify the family of
function $f_{t}^{c}$ in a small neighborhood of the path $w_{t}$ so that the
new function has a birth-death point at $w_{t}$ for all $t\in[t_{1},t_{2}]$.
We can then resolve these degenerate critical points into two Morse critical
points in “cancelling position”. This gives the family of functions
$f_{t}^{ab}$ which is well-defined up to homotopy. The product is $(f^{a}\ast
f^{b})_{t}=f_{2t}^{ab}$. The marking for $f^{a}\ast f^{b}$ is given by the
marking for $f_{t}^{b}$.

######  Lemma 1.6.

For $n\geq 3$, $\pi_{0}{\mathcal{D}}_{0}(M)$ forms a groups as outlined above
and the mapping
$\widetilde{\theta}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to\pi_{0}{\mathcal{C}}(M)$,
given in Definition 1.5 is a group homomorphism.

###### Proof.

The multiplication rule is $(f^{a})(f^{b})=f^{a}\ast f^{b}$ as described
above. It is straightforward to show that this is associative. It is well-
defined since the operation is given by the path in the space of oriented
$k$-planes in the kernel of $df_{t}$ which is given by the marking. Since this
multiplication rule is a deformation of the product given in Proposition 1.1,
the mapping
$\widetilde{\theta}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to\pi_{0}{\mathcal{C}}(M)$
is a homomorphism of semigroups.

The identity in $\pi_{0}{\mathcal{D}}_{0}(M)$ is given by any lens-shaped
model $(f_{t})$ which is symmetrical in the sense that $f_{t}=f_{1-t}$ for all
$t$. The left inverse of $(f_{t}^{b})$ is $f_{t}^{a}$ given by

$f_{t}^{a}=f_{1-t}^{b}\circ g^{-1}:M\times I\to I$

where $g\in{\mathcal{C}}(M)$ is given up to isotopy by $f_{1}^{b}=\pi_{I}\circ
g:M\times I\to I$. The marking on $f^{a}$ is given by $g^{b}(\gamma)\beta$
where $\gamma$ is the path from the basepoint to the birth point of $f^{b}$
which maps by $g^{b}$ to the death point of $f^{a}$ and $\beta$ is the path
along the Morse point set $p_{t}$ back to the birth point of $f^{a}$. This
represents the inverse of $f^{b}$ in
$\pi_{1}({\mathcal{F}}(M),{\mathcal{E}}(M))$ by Proposition 1.2. However,
$f_{t}^{ab}$ is symmetrical since $f^{c}_{t}=f_{b}$ and
$f^{c}_{2-t}=f^{a}_{1-t}\circ g^{b}=f^{b}_{t}$ and the markings match. So,
$\pi_{0}{\mathcal{D}}_{0}(M)$ is a group and $\widetilde{\theta}_{0}$ is a
homomorphism. ∎

###  1.4. Computation of second obstruction

To compute $\pi_{0}{\mathcal{D}}_{0}(M)$, we need the following. For any group
$\pi$ and left $\pi$-module $A$, we consider
$A[\pi]:=A\otimes{\mathbb{Z}}[\pi]$, the direct sum of copies of $A$, one for
each element of $\pi$. This is a $\pi$-module with the diagonal action of
$\pi$ where $\pi$ acts on ${\mathbb{Z}}[\pi]$ by conjugation:

(1.1) $\sigma\left(\sum a_{i}\otimes g_{i}\right)=\sum\sigma
a_{i}\otimes\sigma g_{i}\sigma^{-1}.$

Each conjugacy class in $\pi$ gives a submodule of $A[\pi]$, namely the sum of
all $A\otimes\tau$ indexed by the conjugates $\tau$ of some fixed
$\sigma\in\pi$. Let $A[e]=A\otimes e$ be the submodule indexed by the identity
$e\in\pi$. We consider the quotient module $A[\pi]/A[e]$.

######  Definition 1.7.

We define $Wh_{1}^{+}(\pi;A)$ to be the additive group of coinvariants of the
action of $\pi$ on $A[\pi]/A[e]$:

$Wh_{1}^{+}(\pi;A):=(A[\pi]/A[e])_{\pi}.$

When the action of $\pi$ on $A$ is trivial, this is the direct sum of copies
of $A$, one for every nonidentity conjugacy class in $\pi$. More generally

$Wh_{1}^{+}(\pi;A)\cong\bigoplus A_{C(\sigma_{\alpha})}$

This is the direct sum over all nonidentity conjugacy classes in $\pi$ of the
coinvariants of the action of $C(\sigma_{\alpha})$ on $A$ where
$C(\sigma_{\alpha})$ is the centralizer of a representative $\sigma_{\alpha}$
of the conjugacy class.

The main example is $\pi=\pi_{1}M$ and $A={\mathbb{Z}}_{2}\oplus\pi_{2}M$. For
example, if $\pi$ is finite with trivial action on $\pi_{2}M$ then

$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)=\bigoplus_{c-1}\
({\mathbb{Z}}_{2}\oplus\pi_{2}M)$

where $c$ is the number of conjugacy classes of elements of $\pi_{1}M$.

Then the theorem of Hatcher and Wagoner can be restated as follows.

######  Theorem 1.8.

[2] For $\dim M=n\geq 6$ we have an isomorphism of groups:

(1.2)
$\lambda_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\cong({\mathbb{Z}}_{2}\oplus\pi_{2}M)\otimes{\mathbb{Z}}[\pi_{1}M]/{\mathbb{Z}}[e]=({\mathbb{Z}}_{2}\oplus\pi_{2}M)[\pi_{1}M]/({\mathbb{Z}}_{2}\oplus\pi_{2}M)[e].$

Furthermore, this isomorphism is equivariant with respect to the action of
$\pi_{1}M\times{\mathbb{Z}}_{2}$ given by changing the markings on
${\mathcal{D}}_{0}(M)$ and with the diagonal action of $\pi_{1}M$ given in
(1.1) and the trivial action of ${\mathbb{Z}}_{2}$. Thus
$\pi_{0}{\mathcal{D}}_{0}(M)_{\pi_{1}M}=Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)$.
Furthermore, we have an exact sequence

$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)\xrightarrow{\overline{\theta}_{0}}\pi_{0}{\mathcal{C}}(M)\to
Wh_{2}(\pi_{1}M)\to 0$

where $\overline{\theta}_{0}$ is the homomorphism induced by
$\widetilde{\theta}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to\pi_{0}{\mathcal{C}}(M)$.

The idea is that, by Morse theory, a 1-parameter family of functions gives a
family of chain complexes. In this case the chain complexes are acyclic.
Hatcher and Wagoner show that this family can be deformed until all critical
points are in two indices (the “Two-index Theorem”). This gives a family of
invertible matrices. There is a K-theoretic obstruction to simplifying this
family into the constant family of an $m\times m$ identity matrix. The
resulting graphic is a disjoint union of $m$ “eyes” which can be deformed into
one “eye” by choosing markings and using the product structure of
$\pi_{0}{\mathcal{D}}_{0}(M)$.

We now recall the definition of the invariant $\lambda_{0}$ in (1.2). Consider
a marked lens-shaped model $(f_{t})$ for $M^{n}$ with birth-death points at
parameter values $t_{0},t_{1}$. For each $t_{0}<t<t_{1}$, $f_{t}$ has critical
points $p_{t},q_{t}$ in indices $k,k+1$ where $k=\lfloor\frac{n}{2}\rfloor$
with critical values $f_{t}(p_{t})<\frac{1}{2}<f_{t}(q_{t})$. The level
surface $V_{t}^{n}=f_{t}^{-1}(\frac{1}{2})$ has an embedded $k$-sphere
$S_{q}^{k}$ which is the stable manifold of $q_{t}$ and an embedded $n-k$
sphere $S^{n-k}_{p}$ which is the unstable manifold of $p_{t}$. For $t$
slightly greater than $t_{0}$ and slightly less than $t_{1}$, these spheres
meet transversely at one point. However, for values of $t$ between $t_{0}$ and
$t_{1}$, these spheres may meet in more than one point.

######  Definition 1.9.

[2] The _framed bordism invariant_ of the marked lens-shaped model $f_{t}$ is
defined as follows. The orientation of the negative eigenspace of $D^{2}f_{t}$
at the birth point $b_{0}$ of $f_{t_{0}}$ gives an orientation for $S^{k}_{q}$
and a normal framing of $S_{p}^{n-k}$. Thus, their intersection is a framed
1-dimensional submanifold $J^{1}\subset S^{k}_{q}\times(t_{0},t_{1})$ which
has one open component going from $t_{0}$ to $t_{1}$. Let $J_{0}$ be the union
of the closed components of $J$. Each point in $J$ represents a trajectory of
the negative gradient $-\nabla f_{t}$ from $q_{t}$ down to $p_{t}$. Together
with the paths $p_{t},q_{t}$ to the birth point $b_{0}$ then the marking which
is a path from the basepoint of $M\times I^{2}$ to $b_{0}$, we obtain a
mapping

$J^{1}\to\Omega M.$

Restricting this to $J_{0}$ gives an element in the framed bordism group

$[J_{0}]\in\Omega_{1}^{fr}(\Omega
M)=({\mathbb{Z}}_{2}\oplus\pi_{2}M)[\pi_{1}M].$

The open component of $J$ maps to the identity component of $\Omega M$. This
makes any framed bordism class in $({\mathbb{Z}}_{2}\oplus\pi_{2}M)[e]$ not
well-defined since it could be absorbed into the open component of $J$. Modulo
this we obtain the _framed bordism invariant_

$\lambda_{0}(f_{t})\in({\mathbb{Z}}_{2}\oplus\pi_{2}M)[\pi_{1}M]/({\mathbb{Z}}_{2}\oplus\pi_{2}M)[e].$

If we take the product of two families $f_{t},f_{t}^{\prime}$ in
${\mathcal{D}}_{0}(M)$, we obtain the disjoint union of the closed components
$J_{0},J_{0}^{\prime}$ of the framed manifolds $J,J^{\prime}$ for
$f_{t},f_{t}^{\prime}$. Therefore, $\lambda_{0}$ gives a homomorphism of
groups:

$\lambda_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to({\mathbb{Z}}_{2}\oplus\pi_{2}M)[\pi_{1}M]/({\mathbb{Z}}_{2}\oplus\pi_{2}M)[e].$

######  Lemma 1.10.

[2] For $\dim M\geq 6$, $\lambda_{0}$ is an isomorphism.

###### Proof.

In this range of dimensions, $n-k\geq k\geq 3$. So,
$S_{q}^{k}\times(t_{0},t_{1})$ has dimension $\geq 4$ and there is no
obstruction to deforming framed $1$-manifolds in
$S_{q}^{k}\times(t_{0},t_{1})$. If the framed bordism invariant of $f_{t}$ is
trivial, i.e., if it lies in $({\mathbb{Z}}_{2}\oplus\pi_{2}M)[e]$, each
component of $J$ can be absorbed into the open component by connected sum.
Since $\dim V\geq 6$, the open component can be straightened making $f_{t}$
into the identity element of $\pi_{0}{\mathcal{D}}_{0}(M)$. Thus $\lambda_{0}$
is a monomorphism.

To show that $\lambda_{0}$ is surjective, take any generator
$(a,b)[\sigma]\in({\mathbb{Z}}_{2}\oplus\pi_{2}M)[\pi_{1}M]$. This is
represented by a framed circle $J_{0}$ in $S_{q}^{k}\times(t_{0},t_{1})$
together with a map $J_{0}\to\Omega M$. Since $\dim M\geq 6$ this can be
realized as a one parameter family of Morse functions. We create two critical
points $p_{t},q_{t}$ of index $k,k+1$ in cancelling position. Then create two
extra geometric incidences which add and subtract $\sigma$ from the incidence
number. Then cancel them again after moving them around using
$(a,b)\in{\mathbb{Z}}_{2}\oplus\pi_{2}M$. The result has only one geometric
incidence between the two critical points and thus they can be cancelled. This
creates the desired lens-space module showing that $\lambda_{0}$ is
surjective. ∎

We want to extend the framed bordism invariant for
$\pi_{0}{\mathcal{D}}_{0}(M)$ to the case $n=4$ (and $n=5$). In both $n=4,5$,
we have $k=2$. So, $S_{q}^{k}\times(t_{0},t_{1})$ has dimension $3$ and the
framed 1-dimensional manifold $J_{0}$ in Definition 1.9 has integer framing
invariant (for the same reason that $\pi_{3}S^{2}={\mathbb{Z}}$). In dimension
$n=4$ we cannot prove the analogue of Lemma 1.10. (The case $n=5$ will be
discussed further in [7].) However, the framed bordism invariant of Definition
1.9 is still defined and can be refined as follows.

######  Proposition 1.11.

For $n=4$ or $5$, the framed bordism invariant factors through a well-defined
integer framed bordism invariant which is a homomorphism

$\widetilde{\lambda}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to({\mathbb{Z}}\oplus\pi_{2}M)[\pi_{1}M]/({\mathbb{Z}}\oplus\pi_{2}M)[e].$

###  1.5. Dependence on markings

We show that the integer framed bordism invariant $\widetilde{\lambda}_{0}$ is
$\pi_{1}M\times{\mathbb{Z}}_{2}$ equivariant.

######  Lemma 1.12.

For $n=4,5$, the integer framed bordism invariant $\widetilde{\lambda}_{0}$
from Proposition 1.11 is equivariant with respect to the action of
$\pi_{1}M\times{\mathbb{Z}}_{2}$ given on ${\mathcal{D}}_{0}(M)$ by changing
the markings and on $({\mathbb{Z}}\oplus\pi_{2}M)[\pi_{1}M]$ by the diagonal
action of $\pi_{1}M$ as in (1.1) and the sign action of ${\mathbb{Z}}_{2}$ on
${\mathbb{Z}}$.

To clarify the statement, the group of coinvariants of the action of
$\pi_{1}M\times{\mathbb{Z}}_{2}$ on
$({\mathbb{Z}}\oplus\pi_{2}M)[\pi_{1}M]/({\mathbb{Z}}\oplus\pi_{2}M)[e]$ is
equal to $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)$. In other
words, the sign of the integer framing number of $J_{0}$ is not intrinsic. It
just comes from the marking. This implies the following.

######  Proposition 1.13.

For $n=4,5$ the integer framed bordism invariant induces a well-defined
mapping

$\overline{\lambda}:\pi_{0}{\mathcal{D}}(M)\to
Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M).$

To understand the proof of Lemma 1.12, it helps to consider a concrete
equivalent problem. What happens to the generator of $\pi_{3}S^{2}$ when we
reverse the orientation for both source and target? When we reverse the
orientation of $S^{2}$, nothing happens! This is because the Hopf map
$S^{3}\to{\mathbb{C}}P^{1}=S^{2}$ commutes with complex conjugation which is
homotopic to the identity on $S^{3}$ but reverses the orientation of
${\mathbb{C}}P^{1}=S^{2}$. Reversing the orientation of $S^{3}$ is negation on
$\pi_{3}S^{2}$ by definition of its group structure. Therefore, reversing the
orientation of both $S^{2}$ and $S^{3}$ acts on $\pi_{3}S^{2}={\mathbb{Z}}$ by
changing sign.

###### Proof of Lemma 1.12.

Each closed component of the framed 1-manifold $J_{0}$ gives an oriented cycle
in the loop space of $M\times I^{2}$ at the birth point of $f_{t}$. This
contributes $(n,a)[\sigma]$ to the integer framing invariant where
$n\in{\mathbb{Z}}$, $a\in\pi_{2}M$ and $\sigma\in\pi_{1}M$. When we change the
orientation of the descending plane of the birth point, we change the
orientation of the descending sphere $S^{2}_{q}\times(t_{0},t_{1})$ and the
normal orientation of the ascending sphere $S^{n-2}_{p}\times(t_{0},t_{1})$.
The result is that tangential orientation of the intersection $J\subset
S^{2}_{q}\times(t_{0},t_{1})$ is unchanged. However, the self-intersection
number of $J$ in $S^{2}_{q}\times(t_{0},t_{1})$ changes sign. This will change
the invariant from $(n,a)[\sigma]$ to $(-n,a)[\sigma]$. This is similar to
what happens with $\pi_{3}S^{2}$.

The other part of the marking is the path from the basepoint of $M\times
I^{2}$ to the birth point of $f_{t}$. is changed by, say $\gamma\in\pi_{1}M$,
this invariant will change to $(n,\gamma a)[\gamma\sigma\gamma^{-1}]$. The
framing number $n$ does not change since it does not use the path to the
basepoint. ∎

###  1.6. Involution

We also have a very useful involution $\varepsilon$ which flips the function
$f$ “upside-down”. It is given by

$\varepsilon f(x,t)=1-f(x,1-t).$

This is projection to the second coordinate of the function $\varepsilon g$
given for each $g(x,t)=(h(x,t),f(x,t))$ in ${\mathcal{C}}(M)$ by

$\varepsilon g(x,t)=(r^{-1}h(x,1-t),1-f(x,1-t))$

where $r$ is the automorphism of $M$ given by $g(x,1)=(r(x),1)$. Since
$\varepsilon$ switches the stacking direction for functions (Proposition 1.3)
we have

$\varepsilon(f\bullet
f^{\prime})=\varepsilon(f^{\prime})\bullet\varepsilon(f).$

The involution $\varepsilon$ turns the graphic of a function upside-down.
Also, it changes the index $k$ of a Morse critical point $(x,t)$ to index
$n+1-k$ (and the critical point moves to $(x,1-t)$). Thus, in the case when
$\dim M=n$ is even, this involution sends ${\mathcal{D}}(M)$ to
${\mathcal{D}}(M)$.

######  Proposition 1.14.

Suppose that $n\geq 4$ is even and $(f_{t})\in{\mathcal{D}}(M)$ with
$\overline{\lambda}(f_{t})=(n,\alpha)[\sigma]\in
Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}\oplus\pi_{2}M)$ where
$n\in{\mathbb{Z}}_{2}$, $\alpha\in\pi_{2}M$ and $\sigma\in\pi_{1}M$. Then

$\overline{\lambda}(\varepsilon(f_{t}))=(n+w_{2}(\alpha),-\alpha)[\sigma^{-1}].$

where $w_{2}\in H^{2}(M;{\mathbb{Z}}_{2})$ is the second Stiefel-Whitney class
of $M$.

###### Proof.

First, choose a marking for $(f_{t})$. By linearity of the invariant, we can
restrict attention to the contribution of one closed component, say $C$, of
$J$, the intersection of the descending sphere $S^{k}_{q}\times(t_{0},t_{1})$
with the ascending sphere $S^{k}_{p}\times(t_{0},t_{1})$ in
$V^{2k}\times(t_{0},t_{1})$.

To compute the invariant for $\varepsilon(f_{t})$, we take the same family of
functions $f_{t}$ with the same critical points $p_{t},q_{t}$ but with the
negative gradient $-\nabla f_{t}$ replaced with the positive gradient. The
loop representing $\sigma$ in $\overline{\lambda}(f_{t})$ goes from the birth
point $b_{0}$ up to $q_{t}$, then down through $C$ to $p_{t}$ and back to
$b_{0}$. The loop for $\overline{\lambda}(\varepsilon(f_{t}))$ goes the other
way (from $p_{t}$ up to $q_{t}$ through $C$). So, $\sigma$ is replaced with
$\sigma^{-1}$.

The tangential orientation of $J$ is unchanged since, by definition, it points
from the birth point to the death point on the open component. So, the
component $C$ is tangentially oriented in the same direction for both
$(f_{t})$ and $\varepsilon(f_{t})$. But, the $\sigma$ direction is changed.
So, the $\pi_{2}M$ element changes sign from $\alpha$ to $-\alpha$. This
element $\alpha\in\pi_{2}M$ is represented by an embedded $2$-sphere with
equator $C$, upper hemisphere $D^{2}_{+}$ contained in the descending manifold
$D^{k+1}_{q}\times(t_{0},t_{1})$ of $q_{t}$ and lower hemisphere $D^{2}_{-}$
contained in the ascending manifold $D^{k+1}_{p}\times(t_{0},t_{1})$ of
$p_{t}$. For $\varepsilon(f_{t})$ the roles of $D^{2}_{+}$, $D^{2}_{-}$ are
switched.

The framing number $n$ of $C$ in $\overline{\lambda}(f_{t})$ is given by
comparing the normal framing of $S_{p}^{k}\times(t_{0},t_{1})$ along $C$,
which extends to a normal framing of $D^{2}_{-}$ in
$D_{p}^{k+1}\times(t_{0},t_{1})$, with the tangential framing of $D^{2}_{+}$
in $D_{q}^{k+1}\times(t_{0},t_{1})$. The framing number $n^{\prime}$ of $C$ in
$\overline{\lambda}(\varepsilon(f_{t}))$ is given by comparing the normal
framing of $D^{2}_{+}$ in $D_{q}^{k+1}\times(t_{0},t_{1})$ with the tangential
framing of $D^{2}_{-}$ in $D_{p}^{k+1}\times(t_{0},t_{1})$. So, the sum
$n+n^{\prime}$ compares the normal bundle of $D^{2}_{+}$ in $M\times I^{2}$
with the normal bundle of $D^{2}_{-}$. In other words,
$n+n^{\prime}=w_{2}(D^{2}_{+}\cup D^{2}_{-})=w_{2}(\alpha)$. ∎

Proposition 1.14 will guide us in the remainder of this paper. We will assume
that $M$ is a $4$-manifold with an embedded $2$-sphere with odd $w_{2}$. For
example, we could take connected sum with ${\mathbb{C}}P^{2}$. Using an
embedded $2$-sphere we will construct a lens-shaped model for $M$ with
invariant $\alpha[\sigma]$ where $\alpha\in\pi_{2}M$ is the class of the
embedded $2$-sphere and $\sigma$ is any element of $\pi_{1}M$. The explicit
construction will allow us to compute the framing number. It will be the
degree of the normal bundle of the embedded $2$-sphere in $M$, i.e., its self-
intersection number. We do the same using $\sigma^{-1}$ and apply the
involution $\varepsilon$. By Proposition 1.14, the sum will give all
nontrivial elements of $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$.

##  2\. Constructing pseudoisotopies using two disks

To prove Theorems A and B we need embedded disks: $D_{q}^{3},D_{p}^{2}\subset
M^{4}$ as shown in Figure 3. We use the fact that the _self-intersection
number_ of the $2$-sphere ${\mathbb{C}}P^{1}$ in ${\mathbb{C}}P^{2}$, i.e.,
the degree of its normal bundle, is 1. Conversely, if $M$ contains an embedded
$2$-sphere with self-intersection number $1$, it must have the form
$M=X\\#{\mathbb{C}}P^{2}$.

######  Lemma 2.1.

Let $S_{0}^{2}\subset M$ be an embedded $2$-sphere with self-intersection
number $w$. Given any $\sigma\in\pi_{1}M$, there are embedded disks
$D_{q}^{3}\subset M^{4}$ and $D_{p}^{2}\subset M^{4}$ with boundaries
$S_{q}^{2}=\partial D_{q}^{3}$, $S_{p}^{1}=\partial D_{p}^{2}$ with the
following properties (Figure 3).

  1. (1)

$S_{q}^{2}\cap D_{p}^{2}=\\{a\\}\subset int\,D_{p}^{2}$

  2. (2)

$D_{q}^{3}\cap S_{p}^{1}=\\{b\\}\subset int\,D_{q}^{3}$

  3. (3)

$D_{q}^{3}\cap D_{p}^{2}$ is the disjoint union of

    1. (a)

a line segment from $a$ to $b$ (denoted $J$)

    2. (b)

a circle $S_{\ast}^{1}$.

  4. (4)

Any path from $b$ to $S_{\ast}^{1}$ in $D_{q}^{3}$ followed by any path from
$S_{\ast}^{1}$ to $b$ in $D_{p}^{2}$ is homotopic to (a conjugate of) the
chosen element $\sigma\in\pi_{1}M$.

  5. (5)

$S_{\ast}^{1}$ bounds two $2$-disks $B_{q}^{2}\subset D_{q}^{3}$ and
$B_{p}^{2}\subset D_{p}^{2}$ whose union is homotopic to $S_{0}^{2}$.

  6. (6)

The normal framing of $S_{\ast}^{1}$ in $D_{q}^{3}$ given by any
trivialization of the normal bundle of $D_{p}^{2}$ in $M$ has winding number
$w$.

$\sigma$$J$$a$$b$$B_{p}^{2}$$B_{q}^{2}$$\partial D_{p}^{2}=S_{p}^{1}$$\partial
D_{q}^{3}=S_{q}^{2}$$D_{q}^{3}$$D_{p}^{2}$$S^{1}_{\ast}$$\tilde{b}$ Figure 3.
Two embedded disks
$D_{p}^{2},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{q}^{3}}\subset
M$ with
$D_{p}^{2}\cap{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{q}^{3}}=\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S^{1}_{\ast}\coprod
J$. In this figure, $D_{p}^{2}$ is flat and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{q}^{3}}$
is drawn as a boundary connected sum of two $3$-disks along a path given by
$\sigma$. $B_{p}^{2}\subset D_{p}^{2}$ is a $2$-disk with $\partial
B_{p}^{2}=\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S^{1}_{\ast}$.
$\tilde{b}$ is the normal vector field of $S_{p}^{1}$ pointing inward into
$D_{p}^{2}$.

######  Remark 2.2.

We choose a path from $b$ to the base point of $M$ so that the loop at $b$
represents the element $\sigma\in\pi_{1}M$. We also fix an orientation of the
normal bundle of $D_{p}^{2}$ in $M$ (giving a normal orientation of
$S_{p}^{1}$ in $M$) and an orientation of $D_{q}^{3}$ so that these
orientations agree at $D_{q}^{3}\cap S_{p}^{1}=\\{b\\}$. These choices will
give a marking on the lens-shaped model we are constructing (Definition 1.5.)

We shall now construct a 1-parameter family of functions $(f_{t})$ on
$M^{4}\times I$ assuming we are given embedded disks
$D_{q}^{3},D_{p}^{2}\subset M$ as described above and shown in Figure 3. Start
with $f_{0}=\pi_{I}$, the projection to $I=[0,1]$. To show that the result is
a nontrivial element of $\pi_{0}{\mathcal{C}}(M)$ we need to assume that
$w=1$.

We introduce a pair of cancelling Morse critical points $p_{t},q_{t}$ of
indices $2,3$ for $t>t_{0}$ where $f_{t_{0}}$ has a birth-death point of index
$2$. Choose a path from this birth-death point to the basepoint of $M\times
I^{2}$.

$p_{t}$2$q_{t}$3$t$$t_{0}$

$M\times 1$$S_{p}^{1}$$q_{t}$$V$$S_{L}^{2}$$S_{p}^{2}$$p_{t}$

For $t$ slightly greater than $t_{0}$, the regular levels of the Morse
function $f_{t}$ start at the top: $M^{4}=M\times 1$ as shown in Figure 3,
with an embedded $2$-disk $D_{p}^{2}\subset M^{4}$ (shaded on the left),
performs surgery on $\partial D_{p}^{2}=S_{p}^{1}$ at $q_{t}$ to produce the
intermediate level surface $V^{4}$ shown in Figure 4 with an embedded sphere
$S_{L}^{2}$ which is the stable sphere of $q_{t}$. $D_{p}^{2}\subset M$
descends to $S_{p}^{2}\subset V$, the unstable sphere of $p_{t}$.

$\widetilde{J}$$a$$\tilde{b}$$\widetilde{D}_{q}^{3}$$S_{q}^{2}$$S_{L}^{2}$$S_{p}^{2}$$B_{p}^{2}$$S_{\ast}^{1}$$B_{q}^{2}$
Figure 4. The level surface $V^{4}$ obtained by doing surgery on
$S_{p}^{1}\subset M$. The disk $D_{p}^{2}\subset M$ becomes the sphere
$S_{p}^{2}\subset V$ which meets $S_{L}^{2}\subset V$, the sphere dual to
$S_{p}^{1}$, transversely at $\tilde{b}$. In $V$, the disk
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{q}^{3}}\subset
M$ becomes
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\widetilde{D}_{q}^{3}=S^{2}\times\widetilde{J}}$
with boundary
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\partial\widetilde{D}_{q}^{3}=S_{L}^{2}\coprod
S_{q}^{2}$. In this figure,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\widetilde{D}_{q}^{3}$
is flat.

Thus $S_{L}^{2}$ is the stable sphere of $q_{t}$, the set of all points in
$V^{4}$ which are carried by trajectories of $\nabla f_{t}$ to $q_{t}$. Points
on the surgery circle $S_{p}^{1}=\partial D_{p}^{2}$ give normal vector fields
for $S_{L}^{2}$ in $V^{4}$ and, in the direction corresponding to the point
$b=\partial D_{p}^{2}\cap D_{q}^{3}$, a small $2$-sphere in $D_{q}^{3}$
centered $b$ in $M$, under the flow generated by $-\nabla f_{t}$ goes to a
sphere $\tilde{S}_{L}^{2}\subset V^{4}$ parallel to $S_{L}^{2}$.

Let $S_{p}^{2}\subset V^{4}$ be the unstable sphere of $p_{t}$. Since
$p_{t},q_{t}$ are a cancelling pair of critical points, the two spheres
$S_{L}^{2},S_{p}^{2}$ intersect transversely at one point. We can take
$S_{p}^{2}$ to be the image of $D_{p}^{2}$ in $V^{4}$.

For $t_{1}\leq t\leq t_{2}$ we do an isotopy of the descending sphere of the
critical point $q_{t}$ from $S_{L}^{2}$ to $S_{q}^{2}$ along the cylinder
$\widetilde{D}_{q}^{3}=S^{2}\times\widetilde{J}$. This intersects $S_{p}^{2}$
along the circle $S_{\ast}^{1}$.

$t_{2}$$t_{1}$

By construction, any normal framing of $\widetilde{D}_{q}^{3}$ in $V\times I$
gives a trivial framing for $S_{\ast}^{1}$ in $S_{p}^{2}\times I$ and, by
Condition (6), for $n$ odd, any normal framing of $S_{p}^{2}\times I$ gives a
nontrivial framing of $S_{\ast}^{1}$ in $\widetilde{D}_{q}^{3}$, the trace of
the deformation of $S_{L}^{2}$ to $S_{q}^{2}$. By (5), the trajectories of
$-\nabla f_{t}$ through $S_{\ast}^{1}$ gives a $2$-sphere homotopic to
$S_{0}^{2}$. Also, each of these trajectories, together with the paths
$p_{t}$, $q_{t}$ to the birth-death point of $f_{t_{0}}$ followed by the
chosen path to the basepoint of $M\times I^{2}$ gives the chosen element
$\sigma\in\pi_{1}M$.

When $n=1$ and $M=X\\#{\mathbb{C}}P^{2}$, $M$ contains a subspace $Y=X-D^{4}$
with $\pi_{1}Y=\pi_{1}M$ and ${\mathbb{C}}P^{1}$ is disjoint from $Y$. So, the
projection of the first Postnikov invariant $k_{1}\in
H^{3}(\pi_{1}M;\pi_{2}M)$ to
$H^{3}(\pi_{1}M;{\mathbb{Z}})=H^{3}(\pi_{1}M;\pi_{2}{\mathbb{C}}P^{2})$ is
trivial. Thus, in the exact sequence:

$K_{3}({\mathbb{Z}}[\pi_{1}M])\xrightarrow{\chi}Wh_{1}^{+}(\pi_{1};{\mathbb{Z}}_{2}\oplus\pi_{2}M)\to\pi_{1}{\mathcal{P}}(M)\to
Wh_{2}(\pi_{1}M)\to 0$

the boundary map $\chi$ does not hit the element $\tau[\sigma]\in
Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ where $\tau\in\pi_{2}M$ is the class of
${\mathbb{C}}P^{1}$ since, by [5], this factor of $\chi$ is given by:

$K_{3}({\mathbb{Z}}[\pi_{1}M])\to H_{3}(\pi_{1}M;\pi_{2}M)\xrightarrow{\cap
k_{1}}H_{0}(\pi_{1}M;\pi_{2}M)\to Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$

which does not hit the summand $Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ generated by
$\tau[\pi_{1}M]$.

At $t_{2}$ we can cancel the critical points at a birth-death point to give a
marked lens-shaped model with invariant $\tau[\sigma]\in
Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ where $\tau\in\pi_{2}M$ is the homotopy class
of ${\mathbb{C}}P^{1}$.

$t_{2}$

More generally, this same construction proves the following.

######  Theorem 2.3 (Theorem A).

Given a compact $4$-manifold $M$ and $\tau\in\pi_{2}M$ represented by an
embedded $2$-sphere and $\sigma\in\pi_{1}M$, the obstruction element
$\tau[\sigma]\in Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)$ lifts to ${\mathcal{C}}(M)$.
Furthermore, when the embedded $2$-sphere has self-intersection number $\pm
1$, this element of $\pi_{0}{\mathcal{C}}(M)$ is stably nontrivial (survives
to a nontrivial element of $\pi_{0}{\mathcal{P}}(M)$).

######  Remark 2.4.

The fact that the obstruction element $\tau[\sigma]$ lifts to an element of
$\pi_{0}{\mathcal{C}}(M)$ does not imply that its image in
$\pi_{0}{\mathcal{C}}(M)$ is nonzero except in the case when the self-
intersection number of $\tau$ is $n=\pm 1$ since the Postnikov invariant
argument given above only works in that case.

##  3\. Realizing the framing obstruction

We use the construction from the previous section to construct elements
realizing every element of the framing obstruction group

$Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2}).$

However, the comment in Remark 2.4 still applies. Lifting these obstruction
elements to $\pi_{0}{\mathcal{C}}(M)$ does not imply that their images are
nonzero, except in the following case.

Suppose that $\pi_{1}M$ is free abelian of finite nonzero rank. Then, as
observed by Hatcher, $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$ will be an
infinitely generated group. But $K_{3}({\mathbb{Z}}[\pi_{1}M])$ is finitely
generated by the fundamental theorem of K-theory. From the exact sequence

$K_{3}({\mathbb{Z}}[\pi_{1}M])\xrightarrow{\chi}Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})\oplus
Wh_{1}^{+}(\pi_{1}M;\pi_{2}M)\to\pi_{0}{\mathcal{P}}(M)$

we can then conclude that there are infinitely many nontrivial elements of
$\pi_{0}{\mathcal{C}}(M)$ coming from $Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})$.
However, these elements might not be $2$-torsion when $M$ is $4$-dimensional.

For nontrivial $\sigma\in\pi_{1}M$ and $\tau\in\pi_{2}M$ represented by an
embedded $2$-sphere in $M$ with self-intersection number $\pm 1$ we realize
the obstruction $\tau[\sigma]$ as in Theorem 2.3 and do it again for
$\tau[\sigma^{-1}]$ where we choose markings for our lens-shaped models
(lifting them to elements of ${\mathcal{D}}_{0}(M)$) to make these
obstructions well-defined (Definition 1.5). However, the second time, we apply
the involution $\varepsilon$ which turns the Morse functions upside down to
make the intersection circle $S_{\ast}^{1}$ trivially framed in
$\widetilde{D}_{q}^{2}$ and nontrivially framed in $S_{p}^{2}\times I$. As in
the proof of Proposition 1.14, the product of these two elements of
${\mathcal{D}}_{0}(M)$ will map to zero in $\pi_{2}M[\pi_{1}M]/\pi_{2}M[e]$
since $\tau[\sigma]-\tau[\sigma]=0$. The integer framed bordism invariant
(Proposition 1.11) will be $\pm[\sigma]$, a generator of
${\mathbb{Z}}[\pi_{1}M]/{\mathbb{Z}}[e]$. This gives:

######  Theorem 3.1 (Theorem B).

Let $M=X\\#{\mathbb{C}}P^{2}$ be the connected sum of a nonsimply connected
$4$-manifold $X$ with ${\mathbb{C}}P^{2}$. Then the integer framed bordism
invariant
$\widetilde{\lambda}_{0}:\pi_{0}{\mathcal{D}}_{0}(M)\to{\mathbb{Z}}[\pi_{1}M]/{\mathbb{Z}}[e]$
is surjective. In particular, every element in the image of the second
obstruction map
$\theta:Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})\to\pi_{0}{\mathcal{P}}(M)$ lifts
to $\pi_{0}{\mathcal{C}}(M)$.

More generally, an analogous construction gives the following.

######  Theorem 3.2.

Suppose $M^{4}$ contains an embedded $2$-sphere with odd self-intersection
number. Then every element of the image of
$\theta:Wh_{1}^{+}(\pi_{1}M;{\mathbb{Z}}_{2})\to\pi_{0}{\mathcal{P}}(M)$ lifts
to $\pi_{0}{\mathcal{C}}(M)$.

##  4\. Construction of the two disks

We prove Lemma 2.1 by constructing the two disks $D_{p}^{2},D_{q}^{3}$ first
in the special case when the embedded $2$-sphere is ${\mathbb{C}}P^{1}$, then
more generally for any embedded $2$-sphere by modifying the construction in
the special case.

The disk $D_{q}^{3}$ will be the boundary connected sum of two $3$-disks
$D_{0}^{3}$ and $D_{1}^{3}$ where $D_{p}^{2},D_{0}^{3},D_{1}^{3}$ will all be
contained in a punctured ${\mathbb{C}}P^{2}$ contained in $M$.

Let $D_{p}^{2}$ be the set of all point in ${\mathbb{C}}P^{2}$ with
coordinates $[1,z,0]$ where $|z|\leq 3$ and let $B_{p}^{2}$ be the subset of
those points $[1,z,0]$ with $|z|\leq 1$. These are $2$-disks with $\partial
B_{p}^{2}=S_{\ast}^{1}$ being the set of all $[1,x,0]$ where $|x|=1$.

Let $S_{\ast}^{1}\times D^{2}$ denote the set of all points in
${\mathbb{C}}P^{2}$ of the form $[1,x,z]$ where $|x|=1$ and $|z|\leq 1$. This
is the restriction to $S_{\ast}^{1}$ of the normal disk bundle of $D_{p}^{2}$
in ${\mathbb{C}}P^{2}$. Thus $S_{\ast}^{1}\times D^{2}$ meets $D_{p}^{2}$
transversely along $S_{\ast}^{1}$ as shown in Figure 5.

$J$$D_{p}^{2}$$D_{0}^{3}$$S^{1}_{\ast}$$S_{\ast}^{1}\times D^{2}$ Figure 5.
The $2$-disk $D_{p}^{2}\subset{\mathbb{C}}P^{1}\subset{\mathbb{C}}P^{2}$ with
subdisk $B_{p}^{2}$. The boundary
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\partial
B_{p}^{2}=S_{\ast}^{1}$ is in red. In blue are
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}S_{\ast}^{1}\times
D^{2}$, meeting $D_{p}^{2}$ transversely along
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}S_{\ast}^{1}$
and
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}D_{0}^{3}$
meeting $\partial D_{p}^{2}$ tranversely at one point with
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}J=D_{0}^{3}\cap
D_{p}^{2}$ in red.

Let $D_{0}^{3}$ be the $3$-disk in ${\mathbb{C}}P^{2}$ which meets $\partial
D_{p}^{2}=S^{1}_{p}$ transversely at one point and so that $D_{0}^{3}\cap
D_{p}^{2}$ is the line segment $J$ as shown in Figure 5.

To construct $D_{1}^{3}$, we take the twisted circle
$\widetilde{S}_{\ast}^{1}$ in $\partial(S_{\ast}^{1}\times D^{2})$ given by

$\widetilde{S}_{\ast}^{1}=\\{[1,x,x]\in{\mathbb{C}}P^{2}\,|\,|x|=1\\}.$

This bounds a $2$-disk in ${\mathbb{C}}P^{2}$ in the complement of
${\mathbb{C}}P^{1}$ by taking the set of all $[t,x,x]\in{\mathbb{C}}P^{2}$
with $|x|=1$ and $t\in[0,1]$. This is a $2$-disk since
$[t,x,x]=[t\overline{x},1,1]$. The union of $B_{p}^{2}$ with this $2$-disk is
a $2$-sphere homotopic to ${\mathbb{C}}P^{1}$. Use this $2$-disk to attach a
handle to $S_{\ast}^{1}\times D^{2}$ to obtain a $3$-disk $D_{1}^{3}$ which
meets $D_{p}^{2}$ transversely along $S_{\ast}^{1}$.

Finally, let $D_{q}^{3}$ be given by taking the boundary connect sum of the
two $3$-disks $D_{1}^{3},D_{0}^{3}$ along a path in $M$ given by the chosen
element $\sigma\in\pi_{1}M$. The result is as shown in Figure 3 proving Lemma
2.1 in the special case then $S_{0}^{2}={\mathbb{C}}P^{1}$.

###### Proof of Lemma 2.1 in general.

Let $S_{0}^{2}\subset M$ be an embedded $2$-sphere with self-intersection
number $n$. We follow the same outline as above using Figure 5.

Let $D_{p}^{2}$ in Figure 5 be the southern hemisphere of $S_{0}^{2}$ with
subdisk $B_{p}^{2}\subset D_{p}^{2}\subset S_{0}^{2}$ and
$S_{\ast}^{1}=\partial B_{p}^{2}$. For the circle bundle $\partial
S_{\ast}^{1}\times D^{2}\to S_{\ast}^{1}$ take the section
$\widetilde{S}_{\ast}^{1}$ with winding number $n$. This is the section which
extends to a section $B_{q}^{2}$ of the normal circle bundle over
$S_{0}^{2}\backslash B_{p}^{2}$. We can use this section to attach a
$3$-dimensional $2$-handle to $S_{\ast}^{1}\times D^{2}$. This gives a
$3$-disk $D_{1}^{3}$ which intersects $D_{p}^{2}$ transversely along
$S_{\ast}^{1}$ so that $B_{q}^{2}\cup B_{p}^{2}\subset D_{1}^{3}\cup
D_{p}^{2}$ is homotopic to $S_{0}^{2}$. Let $D_{q}^{3}$ be the boundary
connected sum of $D_{1}^{3}$ with a $3$-disk $D_{0}^{3}$ given as before using
the path $\sigma$ to do the connected sum. This produces the disks $D_{p}^{2}$
and $D_{q}^{3}$ with all the required properties. ∎

## Acknowledgements

The author would like to thank Sander Kupers for encouraging him to write up
these results and for commenting on earlier versions of this manuscript. He
also thanks Bjørn Jahren and Tom Goodwillie for useful correspondences. The
author is indebted to Allen Hatcher for explaining pseudoisotopy theory to him
and for posing the question of what happens in low dimensions. The author is
supported by the Simons Foundation.

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