# Higher homotopy normalities in topological groups

Mitsunobu Tsutaya  Faculty of Mathematics, Kyushu University, Fukuoka
819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to introduce $N_{k}(\ell)$-maps ($1\leq
k,\ell\leq\infty$), which describe higher homotopy normalities, and to study
their basic properties and examples. $N_{k}(\ell)$-map is defined with higher
homotopical conditions. It is shown that a homomorphism is an
$N_{k}(\ell)$-map if and only if there exists fiberwise maps between fiberwise
projective spaces with some properties. Also, the homotopy quotient of an
$N_{k}(k)$-map is shown to be an $H$-space if its LS category is not greater
than $k$. As an application, we investigate when the inclusions
$\operatorname{SU}(m)\to\operatorname{SU}(n)$ and
$\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$ are $p$-locally
$N_{k}(\ell)$-maps.

###### 2010 Mathematics Subject Classification:

55P45 (primary), 55R70 (secondary)

The author was supported by JSPS KAKENHI 19K14535.

##  1\. Introduction

A normal subgroup $H\subset G$ of a topological group $G$ is defined to be a
subgroup closed under the conjugation by $G$. Crossed module is a natural
generalization of normal subgroup, which plays crucial roles in homotopy
theory.

######  Definition 1.1.

A (topological) crossed module is topological groups $H$ and $G$ equipped with
continuous group homomorphisms

$f\colon H\to G\quad\text{and}\quad\rho\colon G\to\operatorname{Aut}(H)$

satisfying the following conditions:

  1. (1)

for any $h,x\in H$, we have $\rho(f(h))(x)=hxh^{-1}$,

  2. (2)

for any $x\in H$ and $g\in G$, we have $f(\rho(g)(x))=gf(x)g^{-1}$.

In this paper, we propose and investigate higher homotopy variants of crossed
module, which sit between homomorphism (without any special property) and
crossed module.

Ordinary and higher homotopy normalities have been extensively studied.
McCarty [McC64] defined a homotopy normality as follows: a subgroup $H\subset
G$ is homotopy normal if the conjugation $G\times H\to G$, $(g,h)\mapsto
ghg^{-1}$ is homotopic to a map into $H$ through a homotopy of maps between
topological pairs $(G\times H,H\times H)\to(G,H)$. James [Jam67] defined a
weaker homotopy normality as follows: a subgroup $H\subset G$ is homotopy
normal if the conjugation $G\times H\to G$ is homotopic to a map into $H$
through an ordinary homotopy of continuous maps. These conditions are
recognized to be “the first order” homotopy normality in our viewpoint.

There have been several works on investigating these kinds of homotopy
normalities of Lie groups. The methods adopted so far have been applications
of the (relative) Samelson products [McC64, Jam67, Jam71, Jam76, Kac82, Fur85,
Fur87, KT18] and the Hopf algebra structures on (generalized) cohomology
groups [Fur95, KY98, KY01, KN03, Nis06].

For higher homotopy normality, it has been focused on where a given
homomorphism is placed in a homotopy fiber sequence [FS10, Pre12]. In general,
a homomorphism $f\colon H\to G$ induces the homotopy fiber sequence

(1) $\displaystyle\cdots\to H\xrightarrow{f}G\to EH\times_{f}G\to
BH\xrightarrow{Bf}BG,$

where $EH\times_{f}G$ is the Borel construction by the action of $H$ on $G$
through $f$. If we suppose $H\subset G$ is a normal subgroup, then the above
sequence can be extended as follows:

$\cdots\to H\to G\to G/H\to BH\to BG\to B(G/H).$

Actually, this is obtained by applying the construction of the homotopy fiber
sequence (1) to the homomorphism $G\to G/H$. This construction is generalized
to crossed modules by Farjoun and Segev [FS10]. Although their construction is
for simplicial groups, it also works for topological groups. They conversely
called $f\colon H\to G$ a homotopy normal map if the homotopy fiber sequence
(1) can be extended as follows by some map $BG\to W$:

$\displaystyle\cdots\to H\xrightarrow{f}G\to EH\times_{f}G\to
BH\xrightarrow{Bf}BG\to W.$

In particular, the homomorphism $H\to\ast$ to the trivial group is homotopy
normal in the sense of Farjoun and Segev if and only if $BH$ is a loop space.

The main objective of this paper is to introduce another class of higher
homotopy normality called $N_{k}(\ell)$-map for $1\leq k,\ell\leq\infty$
(Definition 6.1). In particular, we see that $N_{1}(1)$-map is an equivalent
notion of McCarty’s homotopy normal map (Remark 6.4). Also, we show that a
homomorphism $H\to\ast$ to the trivial group is an $N_{k}(\ell)$-map if and
only if $G$ is a $C(k,\ell)$-space (Theorem 7.3), which is introduced in
[KK10]. Note that $H$ is a $C(\infty,\infty)$-space if and only if $BH$ is an
$H$-space. This implies that our homotopy normality is much weaker than that
of Farjoun and Segev.

The advantage of our homotopy normality is a good connection with fiberwise
homotopy theory. Actually, our main theorem is roughly as follows: given
homomorphism $H\to G$ is an $N_{k}(\ell)$-map if and only if the induced
fiberwise homomorphism of group bundles
$E_{k}H\times_{\operatorname{conj}}H\to E_{k}G\times_{\operatorname{conj}}G$
associated to conjugation action on itself admits a factorization as a
fiberwise $A_{\ell}$-map

$E_{k}H\times_{\operatorname{conj}}H\to E\to
E_{k}G\times_{\operatorname{conj}}G$

through some fiberwise $A_{\ell}$-space $E$ over $B_{k}G$ with fiber
equivalent to $H$ (Theorem 6.2). The latter condition can be checked by the
obstruction theory of fiberwise projective spaces, which provides us a method
to study $N_{k}(\ell)$-maps (Sections 9 and 10).

We also discuss when the homotopy quotient $X=EH\times_{f}G$ of a homomorphism
$f\colon H\to G$ is an $H$-space. We show that $X$ is an $H$-space if $f$ is
an $N_{k}(k)$-space and the LS-category $\operatorname{cat}X$ of $X$ is
estimated as $\operatorname{cat}X\leq k$ (Theorem 8.2).

Our argument is based on the category of topological monoids and $A_{n}$-maps
between them constructed in [Tsu16]. This is just a choice of a model among
possible higher categorical settings for $A_{n}$-maps between
$A_{\infty}$-spaces. Our definitions and results in the present work should be
justified in any other settings as well.

This paper is constructed as follows. In Section 2, we recall the category of
topological monoids and $A_{n}$-maps between them introduced in [Tsu16]. In
Sections 3 and 4, we reformulate $A_{n}$-space and $A_{n}$-map from an
$A_{n}$-space to a topological monoid. Our composition of $A_{n}$-maps between
topological monoids with an $A_{n}$-map from an $A_{n}$-space to a topological
monoid is defined to be associative. In Section 5, we recall the
classification theorem of fiberwise $A_{n}$-spaces established in [Tsu12,
Tsu15], which plays a central role in the proof of the main theorem. In
Section 6, we introduce $N_{k}(\ell)$-maps and prove our main result Theorem
6.2. In Section 7, we discuss the relation between $C(k,\ell)$-space and
$N_{k}(\ell)$-map. In Section 8, we discuss the $H$-structure on homotopy
quotient. In Section 9, we recall the basics of fiberwise projective spaces
and give some criterion for non-$N_{k}(\ell)$-maps (Theorem 9.4). In Section
10, we investigate when the inclusions
$\operatorname{SU}(m)\to\operatorname{SU}(n)$ and
$\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$ are $p$-locally
$N_{k}(\ell)$-maps.

In this paper, we always work in the category of compactly generated spaces
$\mathbf{CG}$. So the adjunction between products and mapping spaces is always
available.

##  2\. Category of topological monoids and $A_{n}$-maps

We begin with recalling the construction of the topological category of
topological monoids and $A_{n}$-maps introduced in [Tsu16, Section 4]. Proofs
will be omitted here. Let $[0,\infty]$ be the one point compactification of
$[0,\infty)$, which is homeomorphic to the unit interval $[0,1]$.

######  Definition 2.1.

Let $f\colon G\to G^{\prime}$ be a pointed map between topological monoids. A
pair $(\\{f_{i}\\}_{i},L)$ of a family of maps
$\\{f_{i}\colon[0,\infty]^{i-1}\times G^{i}\to G^{\prime}\\}_{i=1}^{n}$ and
$L\in[0,\infty]$ is called an $A_{n}$-form on $f$ if the following conditions
hold:

  1. (1)

for any $x\in G$, we have $f_{1}(x)=f(x)$,

  2. (2)

for any $2\leq i\leq n$, $1\leq k\leq i-1$,
$t_{1},\ldots,t_{i-1}\in[0,\infty]$ and $x_{1},\ldots,x_{i}\in G$, we have

$\displaystyle f_{i}(t_{1},\ldots,t_{i-1};x_{1},\ldots,x_{i})$
$\displaystyle=\begin{cases}f_{i-1}(t_{1},\ldots,t_{k-1},t_{k+1},\ldots,t_{i-1};x_{1},\ldots,x_{k-1},x_{k}x_{k+1},x_{k+2},\ldots,x_{i})&\text{if
$t_{k}=0$},\\\
f_{k}(t_{1},\ldots,t_{k-1};x_{1},\ldots,x_{k})f_{i-k}(t_{k+1},\ldots,t_{i-1};x_{k+1},\ldots,x_{i})&\text{if
$t_{k}\geq L$},\end{cases}$

  3. (3)

for any $2\leq i\leq n$, $1\leq k\leq i$, $t_{1},\ldots,t_{i-1}\in[0,\infty]$
and $x_{1},\ldots,x_{i-1}\in G$, we have

$\displaystyle
f_{i}(t_{1},\ldots,t_{i-1};x_{1},\ldots,x_{k-1},\ast,x_{k},\ldots,x_{i-1})$
$\displaystyle=\begin{cases}f_{i-1}(t_{2},\ldots,t_{i-1};x_{1},\ldots,x_{i-1})&\text{if
$k=1$},\\\
f_{i-1}(t_{1},\ldots,t_{k-2},\max\\{t_{k-1},t_{k}\\},t_{k+1},\ldots,t_{i-1};x_{1},\ldots,x_{k-1},x_{k},\ldots,x_{i-1})&\text{if
$1<k<i$},\\\ f_{i-1}(t_{1},\ldots,t_{i-2};x_{1},\ldots,x_{i-1})&\text{if
$k=i$}.\end{cases}$

A pair $(f,(\\{f_{i}\\}_{i},L))$ is called an $A_{n}$-map. The space of
$A_{n}$-maps between topological monoids $G$ and $G^{\prime}$ is written as
$\mathcal{A}_{n}(G,G^{\prime})$.

The composition is given as follows.

######  Definition 2.2.

The composition $(h,\\{h_{i}\\}_{i},L+L^{\prime})=g\circ f$ of
$f=(f,\\{f_{i}\\}_{i},L)\in\mathcal{A}_{n}(G,G^{\prime})$ and
$g=(g,\\{g_{i}\\}_{i},L^{\prime})\in\mathcal{A}_{n}(G^{\prime},G^{\prime\prime})$
defined as follows: for any $r,i_{1},\ldots,i_{r}\geq 1$ with
$i_{1}+\cdots+i_{r}\leq n$, $t_{1},\ldots,t_{r-1}\in[0,\infty]$,
$\mathbf{s}_{k}\in[0,\infty]^{i_{k}-1}$, $\mathbf{x}_{k}\in G^{i_{k}}$, we
have

$\displaystyle
h_{i_{1}+\cdots+i_{r}}(\mathbf{s}_{1},t_{1}+L,\mathbf{s}_{2},t_{2}+L,\ldots,t_{r-1}+L,\mathbf{s}_{r};\mathbf{x}_{1},\ldots,\mathbf{x}_{k})=g_{r}(t_{1},\ldots,t_{r-1};f_{i_{1}}(\mathbf{s}_{1};\mathbf{x}_{1}),\ldots,f_{i_{r}}(\mathbf{s}_{r};\mathbf{x}_{r})).$

A homomorphism $f\colon G\to H$ can be seen as the $A_{n}$-map equipped with
the standard $A_{n}$-form $(\\{f_{i}\\}_{i},0)$ given as

$f_{i}(t_{1},\ldots,t_{i-1};g_{1},\ldots,g_{i})=f(g_{1}\cdots
g_{i})=f(g_{1})\cdots f(g_{i}).$

In particular, the identity map $\operatorname{id}_{G}\in\mathcal{A}_{n}(G,G)$
works as the identity with respect to the above composition.

A topological category in our sense is a category equipped with topologies on
the sets of morphisms such that the compositions are continuous maps. The
following is proved in [Tsu16, Section 4].

######  Theorem 2.3.

Topological monoids and the spaces of $A_{n}$-maps between them form a
topological category $\mathcal{A}_{n}$. Moreover, the category of topological
monoids and homomorphisms can be embedded in $\mathcal{A}_{n}$ with the
standard $A_{n}$-forms.

Our bar construction functor is defined on the category of $A_{n}$-equivariant
maps.

######  Definition 2.4.

Let $G$ and $G^{\prime}$ be topological monoids acting from the left on $X$
and $X^{\prime}$, respectively, and $f=(f,(\\{f_{i}\\}_{i},L))\colon G\to
G^{\prime}$ be an $A_{n}$-map between topological monoids. A family of maps
$\\{\phi_{i}\colon[0,\infty]^{i}\times G^{i}\times X\to
X^{\prime}\\}_{i=0}^{n}$ is an $A_{n}$-form on a map $\phi\colon X\to
X^{\prime}$ if the following conditions hold:

  1. (1)

for any $x\in X$, we have $\phi_{0}(x)=\phi(x)$,

  2. (2)

for any $1\leq i\leq n$, $1\leq k\leq i$, $t_{1},\ldots,t_{i}\in[0,\infty]$,
$g_{1},\ldots,g_{i}\in G$ and $x\in X$, we have

$\displaystyle\phi_{i}(t_{1},\ldots,t_{i};g_{1},\ldots,g_{i};x)$
$\displaystyle=\begin{cases}\phi_{i-1}(t_{1},\ldots,t_{k-1},t_{k+1},\ldots,t_{i};g_{1},\ldots,g_{k-1},g_{k}g_{k+1},g_{k+2},\ldots,g_{i};x)&\text{if
$t_{k}=0$ and $k<i$},\\\
\phi_{i-1}(t_{1},\ldots,t_{i-1};g_{1},\ldots,g_{i-1},g_{i}x)&\text{if
$t_{i}=0$},\\\
f_{k}(t_{1},\ldots,t_{k-1};g_{1},\ldots,g_{k})\phi_{i-k}(t_{k+1},\ldots,t_{i};g_{k+1},\ldots,g_{i};x)&\text{if
$t_{k}\geq L$},\end{cases}$

  3. (3)

for any $1\leq i\leq n$, $1\leq k\leq i$, $t_{1},\ldots,t_{i}\in[0,\infty]$,
$g_{1},\ldots,g_{i-1}\in G$ and $x\in X$, we have

$\displaystyle\phi_{i}(t_{1},\ldots,t_{i};g_{1},\ldots,g_{k-1},\ast,g_{k},\ldots,g_{i-1};x)$
$\displaystyle=\begin{cases}\phi_{i-1}(t_{2},\ldots,t_{i};g_{1},\ldots,g_{i-1};x)&\text{if
$k=1$},\\\
\phi_{i-1}(t_{1},\ldots,t_{k-2},\max\\{t_{k-1},t_{k}\\},t_{k+1},\ldots,t_{i};g_{1},\ldots,g_{k-1},g_{k},\ldots,g_{i-1};x)&\text{if
$1<k\leq i$}.\end{cases}$

A triple $(f,(\\{f_{i}\\}_{i},L),\\{\phi_{i}\\}_{i})$ is called an
$A_{n}$-equivariant map. The space of $A_{n}$-equivariant maps between a left
$G$-space $X$ and a left $G^{\prime}$-space $X^{\prime}$ is written as
$\mathcal{A}^{\mathrm{L}}_{n}((G,X),(G^{\prime},X^{\prime}))$. Moreover, the
topological category $\mathcal{A}^{\mathrm{L}}_{n}$ of spaces with left
actions of topological monoids and $A_{n}$-equivariant maps is similarly
defined. Equivariant $A_{n}$-maps between spaces with right actions of
topological monoids are similarly defined. The category and a mapping space in
it are written as $\mathcal{A}^{\mathrm{R}}_{n}$ and
$\mathcal{A}^{\mathrm{R}}_{n}((X,G),(X^{\prime},G^{\prime}))$, respectively.

We take a model of the $i$-dimensional simplex $\Delta^{i}$ as

$\Delta^{i}=\\{(t_{0},\ldots,t_{i})\in[0,\infty]^{i}\mid\text{$t_{k}=\infty$
for some $k$}\\}.$

The face $\partial_{k}\colon\Delta^{i-1}\to\Delta^{i}$ and degeneracy
$\epsilon_{k}\colon\Delta^{i+1}\to\Delta^{i}$ ($k=0,\ldots,i$) are given as

$\displaystyle\partial_{k}(t_{0},\ldots,t_{i-1})$
$\displaystyle=(t_{0},\ldots,t_{k-1},0,t_{k},\ldots,t_{i-1}),$
$\displaystyle\epsilon_{k}(t_{0},\ldots,t_{i+1})$
$\displaystyle=(t_{0},\ldots,t_{k-1},\max\\{t_{k},t_{k+1}\\},t_{k+2},\ldots,t_{i+1}).$

Consider the fiber product category
$\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}$
in the obvious sense, where an object is a triple $(X,G,Y)$ of a topological
monoid $G$, a right $G$-space $X$ and a left $G$-space $Y$ and a morphism is
an $A_{n}$-equivariant map between them.

######  Definition 2.5.

For a triple
$(X,G,Y)\in\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}$,
the space $B_{n}(X,G,Y)$ is defined to be the quotient space

$B_{n}(X,G,Y)=\left(\coprod_{0\leq i\leq n}\Delta^{i}\times X\times
G^{i}\times Y\middle)\right/{\sim}$

by the usual simplicial relation. For a morphism

$(\phi,f,\psi)=(\phi,\\{\phi_{i}\\}_{i},f,\\{f_{i}\\}_{i},\psi,\\{\psi_{i}\\}_{i},L)\colon(X,G,Y)\to(X^{\prime},G^{\prime},Y^{\prime})$

in
$\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}$,
the induced map

$B_{n}(\phi,f,\psi)\colon B_{n}(X,G,Y)\to
B_{n}(X^{\prime},G^{\prime},Y^{\prime})$

is defined by

$\displaystyle
B_{n}(\phi,f,\psi)[\mathbf{s}_{1},t_{1}+L,\mathbf{s}_{2},t_{2}+L,\ldots,t_{r-1}+L,\mathbf{s}_{r};x,\mathbf{g}_{1},\ldots,\mathbf{g}_{r},y]$
$\displaystyle=[t_{1},\ldots,t_{r-1};\phi_{i_{1}}(\mathbf{s}_{1};x,\mathbf{g}_{1}),f_{i_{2}}(\mathbf{s}_{2};\mathbf{g}_{2}),\ldots,f_{i_{r-1}}(\mathbf{s}_{r-1};\mathbf{g}_{r-1}),\psi_{i_{r}}(\mathbf{s}_{r};\mathbf{g}_{r},y)].$

for $i=i_{1}+\cdots+i_{r}\leq n$, $t_{1},\ldots,t_{r-1}\in[0,\infty]$,
$\mathbf{s}_{k}\in[0,L]^{i_{k}-1}$, $x\in X$, $\mathbf{g}_{k}\in G^{i_{k}}$,
$y\in Y$. This construction defines a continuous functor

$B_{n}\colon\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}\to\mathbf{CG}.$

In particular, the correspondence $G\mapsto B_{n}G=B_{n}(\ast,G,\ast)$ defines
the $n$-th projective space functor

$B_{n}\colon\mathcal{A}_{n}\to\mathbf{CG}_{\ast}.$

For
$(X,G,Y)\in\mathcal{A}_{\infty}^{\mathrm{R}}\times_{\mathcal{A}_{\infty}}\mathcal{A}_{\infty}^{\mathrm{L}}$,
let

$\iota_{n}\colon B_{n}(X,G,Y)\to B(X,G,Y)=B_{\infty}(X,G,Y)$

denote the natural inclusion.

Note that our bar construction functor coincides with the bar construction for
usual equivariant maps through the obvious embedding into the category
$\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}$.

The following is a technical lemma found in [May75, Theorem 7.6].

######  Lemma 2.6.

Let $G$ be a grouplike topological monoid and
$(X,G,Y)\in\mathcal{A}_{n}^{\mathrm{R}}\times_{\mathcal{A}_{n}}\mathcal{A}_{n}^{\mathrm{L}}$.
Then the maps

$B_{n}(X,G,Y)\to B_{n}(X,G,\ast)\quad\text{and}\quad B_{n}(X,G,Y)\to
B_{n}(\ast,G,Y)$

are quasifibrations.

The following is the main theorem in [Tsu16].

######  Theorem 2.7.

Let $G$ be a topological monoid, of which the underlying space is a CW
complex, and $G^{\prime}$ be a grouplike topological monoid, of which the
basepoint has the homotopy extension property. Then the following composite is
a weak homotopy equivalence:

$\mathcal{A}_{n}(G,G^{\prime})\xrightarrow{B_{n}}\operatorname{Map}_{\ast}(B_{n}G,B_{n}G^{\prime})\xrightarrow{(\iota_{n})_{\sharp}}\operatorname{Map}_{\ast}(B_{n}G,BG^{\prime}).$

This homotopy equivalence establishes the one-to-one correspondence between
the homotopy classes of $A_{n}$-maps $G\to G^{\prime}$ and the basepoint-
preserving maps $B_{n}G\to BG^{\prime}$. In this correspondence, the map
corresponding to given $A_{n}$-map or basepoint-preserving map is called its
adjoint. This is because the homotopy equivalence can be understood as the
adjunction

$\pi_{0}(\mathcal{A}_{n}(G,\Omega^{\mathrm{M}}X))\cong\pi_{0}(\operatorname{Map}_{\ast}(B_{n}G,X))$

between the projective space functor $B_{n}$ and the Moore based loop space
functor $\Omega^{\mathrm{M}}$ in the homotopy categories.

##  3\. Planar rooted tree and associahedron

Let us recall the construction of associahedra in [BV73]. But our construction
slightly different from it since we need to make the composition of
$A_{n}$-maps associative.

We set some notions related to planar rooted trees.

  * •

A rooted tree in our sense is a contractible finite graph with distinguished
vertex called the root such that the root is of degree $1$ and any vertex is
not of degree $2$.

  * •

A planar rooted tree is an equivalence class of embeddings of a rooted tree
into the upper half plane $\\{(x,y)\in\mathbb{R}^{2}\mid y\geq 0\\}$ such that
the root is mapped to the origin. Such two embeddings of rooted tree said to
be equivalent if they are isotopic through an isomorphism between rooted
trees.

  * •

In a planar rooted tree, a vertex of degree $1$ different from the root is
called a leaf. We will write $\mathcal{T}_{n}$ for the set of planar rooted
trees with $n$ leaves. For $\tau\in\mathcal{T}_{n}$, we assign numbers
$1,\ldots,n$ to the leaves of $\tau$ from left to right and call a leaf like
$k$-th leaf.

  * •

We will call the edge connected to the root the root edge, an edge connected
to a leaf a leaf edge and other edges internal edges. We will write $I(\tau)$
for the set of internal edges of a planar rooted tree $\tau$.

  * •

Along the shortest paths from leaves to the root, an orientation is assigned
to each edge in a planar rooted tree.

Consider the space

$\mathcal{LT}_{n}=\coprod_{\tau\in\mathcal{T}_{n}}\\{\tau\\}\times[0,\infty]^{I(\tau)},$

where $[0,\infty]^{I(\tau)}$ is the set of maps $I(\tau)\to[0,\infty]$. An
elementary collapse of a planar rooted tree $\tau$ at $e\in I(\tau)$ is a
planar rooted tree obtained by just collapsing $e$ to a point. Define the
equivalence relation on $\mathcal{LT}_{n}$ generated by
$(\tau,\ell)\sim(\tau^{\prime},\ell^{\prime})$ such that $\tau^{\prime}$ is an
elementary collapse of $\tau$ at $e\in I(\tau)$, $\ell(e)=0$, and, considering
$I(\tau^{\prime})\subset I(\tau)$, the function $\ell$ restricts to
$\ell^{\prime}$ on $I(\tau^{\prime})$. Denote by $\mathcal{K}_{n}$ the
quotient space of $\mathcal{LT}_{n}$, called the $n$-th associahedron. In the
rest of the paper, an element of $\mathcal{LT}_{n}$ or $\mathcal{K}_{n}$ will
be written simply as $\tau$. The function $I(\tau)\to[0,\infty]$ will be
expressed by $\ell$ for any $\tau$. Such a convention does not cause a
confusion.

Figure 1. Equivalence by elementary collapse.

Let $\rho\in\mathcal{T}_{r}$, $\sigma\in\mathcal{T}_{s}$ and $1\leq k\leq r$
($k\in\mathbb{Z}$). Then the grafting
$\partial_{k}(\rho,\sigma)\in\mathcal{T}_{r+s-1}$ of $\sigma$ to $\rho$ at the
$k$-th leaf is obtained by identifying the root edge of $\sigma$ and the
$k$-th leaf edge respecting the orientation. Here we note:

$I(\partial_{k}(\rho,\sigma))=I(\rho)\sqcup I(\sigma)\sqcup\\{\text{the new
internal edge}\\},$

where the new internal is the identified edge in the construction. We can
extend the grafting construction to

$\partial_{k}^{L}\colon\mathcal{LT}_{r}\times\mathcal{LT}_{s}\to\mathcal{LT}_{r+s-1}\quad\text{and}\quad\partial_{k}^{L}\colon\mathcal{K}_{r}\times\mathcal{K}_{s}\to\mathcal{K}_{r+s-1}$

for $L\in[0,\infty]$ by defining $\ell\colon
I(\partial_{k}^{L}(\rho,\sigma))\to[0,\infty]$ as

$\ell(e)=\begin{cases}\ell(e)&\text{if $e\in I(\rho)\sqcup I(\sigma)$},\\\
L&\text{if $e$ is the new internal edge}.\end{cases}$

When $L=\infty$, we will simply write $\partial_{k}=\partial_{k}^{\infty}$.

Let $\tau\in\mathcal{T}_{n}$ ($n\geq 3$) and $1\leq k\leq n$. Let us construct
the degeneracy $s_{k}(\tau)\in\mathcal{T}_{n-1}$ as follows. Let $s_{k}(\tau)$
be a planar rooted tree obtained from just removing the $k$-th leaf edge of
$\tau$ if the resulting tree does not have a vertex of degree $2$ (i.e. the
$k$-th leaf edge is not connected to a vertex of degree $3$). When it has a
vertex $v$ of degree $2$, define a planar rooted tree $s_{k}(\tau)$
identifying its edges connected to $v$ respecting the orientation. To extend
the degeneracy to the maps

$s_{k}\colon\mathcal{LT}_{n}\to\mathcal{LT}_{n-1}\quad\text{and}\quad
s_{k}\colon\mathcal{K}_{n}\to\mathcal{K}_{n-1},$

we define
$\ell(e_{0})=\max\\{\ell(e_{0}^{\prime}),\ell(e_{0}^{\prime\prime})\\}$ for
the edge $e_{0}\in I(s_{k}(\tau))$ obtained from identifying
$e_{0}^{\prime},e_{0}^{\prime\prime}\in I(\tau)$ if it exists and is internal.
The same value $\ell(e)$ as in $\tau$ is assigned for any other internal edge
$e\in I(s_{k}(\tau))$.

Figure 2. Degeneracy $s_{2}$ on $\mathcal{LT}_{4}$.

##  4\. $A_{n}$-space and $A_{n}$-map

In this section, we recall the definitions of $A_{n}$-spaces and of
$A_{n}$-maps from $A_{n}$-spaces to topological monoids.

######  Definition 4.1.

Let $H$ be a pointed space. A family of maps
$\\{m_{i}\colon\mathcal{K}_{i}\times H^{i}\to H\\}_{i=2}^{n}$ is called an
$A_{n}$-form on $X$ if the following conditions hold:

  1. (1)

for any $r,s\geq 2$ with $r+s-1\leq n$, $1\leq k\leq r$,
$\rho\in\mathcal{K}_{r}$, $\sigma\in\mathcal{K}_{s}$ and
$x_{1},\ldots,x_{r+s-1}\in H$, we have

$\displaystyle
m_{r+s-1}(\partial_{k}(\rho,\sigma);x_{1},\ldots,x_{r+s-1})=m_{r}(\rho;x_{1},\ldots,x_{k-1},m_{s}(\sigma;x_{k},\ldots,x_{k+s-1}),x_{k+s},\ldots,x_{r+s-1}),$

  2. (2)

for any $x\in H$, we have

$m_{2}(\ast;x,\ast)=m_{2}(\ast;\ast,x)=x,$

  3. (3)

for any $3\leq i\leq n$, $1\leq k\leq i$, $\tau\in\mathcal{K}_{i}$ and
$x_{1},\ldots,x_{i-1}\in H$, we have

$\displaystyle
m_{i}(\tau;x_{1},\ldots,x_{k-1},\ast,x_{k},\ldots,x_{i-1})=m_{i-1}(s_{k}(\tau);x_{1},\ldots,x_{k-1},x_{k},\ldots,x_{i-1}).$

A pair $(H,\\{m_{i}\\}_{i})$ of a pointed space and an $A_{n}$-form on it is
called an $A_{n}$-space.

Similarly, we can define an $A_{n}$-form
$\\{\mu_{i}\colon\mathcal{K}_{i+1}\times H^{i}\times X\to X\\}_{i=1}^{n}$ of
$A_{n}$-actions on $X$ by $H$. Taking the adjoint, this can be considered as
an “$A_{n}$-map” from $H$ to $\operatorname{Map}(X,X)$. This is an idea to
define $A_{n}$-map from an $A_{n}$-map to a topological monoid as follows
while our definition includes the parameter $L$ not appeared in the one by
Stasheff [Sta70].

######  Definition 4.2.

Let $(H,\\{m_{i}\\}_{i})$ be an $A_{n}$-space, $G$ a topological monoid and
$f\colon H\to G$ a pointed map. A pair $(\\{f_{i}\\}_{i},L)$ of a family of
maps $\\{f_{i}\colon\mathcal{K}_{i+1}\times H^{i}\to G\\}_{i=1}^{n}$ and
$L\in[0,\infty]$ is called an $A_{n}$-form on $f$ if the following conditions
hold:

  1. (1)

for any $r\geq 1$, $s\geq 2$ with $r+s-1\leq n$, $1\leq k\leq r$,
$\rho\in\mathcal{K}_{r+1}$, $\sigma\in\mathcal{K}_{s}$ and
$x_{1},\ldots,x_{r+s-1}\in H$, we have

$\displaystyle
f_{r+s-1}(\partial_{k}(\rho,\sigma);x_{1},\ldots,x_{r+s-1})=f_{r}(\rho;x_{1},\ldots,x_{k-1},m_{s}(\sigma;x_{k},\ldots,x_{k+s-1}),x_{k+s},\ldots,x_{r+s-1}),$

  2. (2)

for any $r,s\geq 1$ with $r+s\leq n$, $\rho\in\mathcal{K}_{r+1}$,
$\sigma\in\mathcal{K}_{s+1}$, $x_{1},\ldots,x_{r+s}\in H$ and $\ell\geq L$, we
have

$f_{r+s}(\partial_{r+1}^{\ell}(\rho,\sigma);x_{1},\ldots,x_{r+s})=f_{r}(\rho;x_{1},\ldots,x_{r})f_{s}(\sigma;x_{r+1},\ldots,x_{r+s}),$

  3. (3)

for any $x\in H$, we have

$f_{1}(\ast;x)=f(x),$

  4. (4)

for any $2\leq i\leq n$, $1\leq k\leq i$, $\tau\in\mathcal{K}_{i+1}$ and
$x_{1},\ldots,x_{i-1}\in H$, we have

$\displaystyle
f_{i}(\tau;x_{1},\ldots,x_{k-1},\ast,x_{k},\ldots,x_{i-1})=f_{i-1}(s_{k}(\tau);x_{1},\ldots,x_{k-1},x_{k},\ldots,x_{i-1}).$

A pair $(f,(\\{f_{i}\\}_{i},L))$ of a pointed map and an $A_{n}$-form on it is
called an $A_{n}$-map. The space of $A_{n}$-maps from an $A_{n}$-space $H$ to
a topological monoid $G$ is written as

$\mathcal{A}^{\prime}_{n}(H,G)\subset\left(\prod_{i=1}^{n}\operatorname{Map}(\mathcal{K}_{i+1}\times
H^{i},G)\right)\times[0,\infty].$

Note that we have two types of $A_{n}$-maps between topological monoids since
any topological monoid $G$ is equipped with the standard $A_{n}$-form
$\\{m_{i}\\}_{i}$ given as

$m_{i}(\tau;x_{1},\ldots,x_{i})=x_{1}\cdots x_{i}.$

We will see later that $\mathcal{A}^{\prime}_{n}(G,G^{\prime})$ and
$\mathcal{A}_{n}(G,G^{\prime})$ are naturally homotopy equivalent. So the
difference between two types of $A_{n}$-maps is not essential in homotopical
sense.

Now we define the composition between $\mathcal{A}^{\prime}_{n}(H,G)$ and
$\mathcal{A}_{n}(G,G^{\prime})$. Let $\tau\in\mathcal{LT}_{n}$ and
$L\in[0,\infty]$. Consider the subsets $I^{\prime}(\tau)\subset I(\tau)$ of
the internal edges lie in the shortest path between the $n$-th leaf and the
root and

$I^{\prime\prime}_{L}(\tau)=\\{e\in I^{\prime}(\tau)\mid\ell(e)\geq L\\}.$

Cutting $\tau$ at internal edges in $I^{\prime\prime}_{L}(\tau)$, we obtain
the planar rooted trees $\tau_{1},\ldots,\tau_{r}$ when $\sharp
I^{\prime\prime}_{L}(\tau)=r-1$ such that $\tau_{i}$ is closer to the root
than $\tau_{j}$ for $i<j$.

Figure 3. Cutting $\tau$ at edges in $I^{\prime\prime}_{L}(\tau)$ when
$l_{3},l_{4}\geq L$.

######  Definition 4.3.

Let $(H,\\{m_{i}\\}_{i})$ be an $A_{n}$-space and $G,G^{\prime}$ be
topological monoids. The composition
$(h,(\\{h_{i}\\}_{i},L+L^{\prime}))=g\circ f$ of
$f=(f,(\\{f_{i}\\}_{i},L))\in\mathcal{A}^{\prime}_{n}(H,G)$ and
$g=(g,(\\{g_{i}\\}_{i},L^{\prime}))\in\mathcal{A}_{n}(G,G^{\prime})$ is
defined as follows: for any $1\leq i\leq n$, $\tau\in\mathcal{LT}_{i}$ with
$\tau_{j}\in\mathcal{LT}_{i_{k}}$ ($k=1,\ldots,r$) obtained by cutting $\tau$
as above, $\mathbf{x}_{k}\in H^{i_{k}}$, we have

$\displaystyle
h_{i}(\tau;\mathbf{x}_{1},\ldots,\mathbf{x}_{r})=g_{r}(\ell(e_{1})-L,\ldots,\ell(e_{r-1})-L;f_{i_{1}}(\tau_{1};\mathbf{x}_{1}),\ldots,f_{i_{r}}(\tau_{r};\mathbf{x}_{r})),$

where $I^{\prime\prime}_{L}(\tau)=\\{e_{1},\ldots,e_{r}\\}$ and $e_{j}$ is
closer to the root than $e_{j^{\prime}}$ for $j<j^{\prime}$.

By a similar argument to that in [Tsu16], the following theorem holds.

######  Theorem 4.4.

Let $(H,\\{m_{i}\\}_{i})$ be an $A_{n}$-space and
$G,G^{\prime},G^{\prime\prime}$ be topological monoids. The composition

$\circ\colon\mathcal{A}_{n}(G,G^{\prime})\times\mathcal{A}^{\prime}_{n}(H,G)\to\mathcal{A}_{n}(H,G^{\prime})$

is continuous and the following associativity and unitality hold:

  1. (1)

for any $f\in\mathcal{A}^{\prime}_{n}(H,G)$,
$g\in\mathcal{A}_{n}(G,G^{\prime})$ and
$g^{\prime}\in\mathcal{A}_{n}(G^{\prime},G^{\prime\prime})$, we have
$g^{\prime}\circ(g\circ f)=(g^{\prime}\circ g)\circ f$,

  2. (2)

for any $f\in\mathcal{A}^{\prime}_{n}(H,G)$, we have
$\operatorname{id}_{G}\circ f=f$.

Figure 4. Associativity of composition $g^{\prime}\circ g\circ f$ in
$\mathcal{K}_{4}$.

An $A_{n}$-map is said to be weak $A_{n}$-equivalence if the underlying map is
a weak homotopy equivalence. Let
$\mathcal{A}^{\prime}_{n}(H,G)_{\operatorname{eq}}\subset\mathcal{A}^{\prime}_{n}(H,G)$
and
$\mathcal{A}_{n}(G,G^{\prime})_{\operatorname{eq}}\subset\mathcal{A}_{n}(G,G^{\prime})$
denote the subspaces of weak $A_{n}$-equivalences. The following proposition
can be shown by a proof similar to [Tsu16, Proposition 4.9].

######  Proposition 4.5.

Let $H$ be an $A_{n}$-space and $G,G^{\prime}$ be topological monoids. Assume
that all of them have the homotopy extension property of the basepoint. The
composition with weak $A_{n}$-equivalences
$f\in\mathcal{A}_{n}^{\prime}(H,G)_{\operatorname{eq}}$ and
$g\in\mathcal{A}_{n}(G,G^{\prime})_{\operatorname{eq}}$ induce the weak
homotopy equivalences

$\displaystyle
f^{\sharp}\colon\mathcal{A}_{n}(G,G^{\prime})\xrightarrow{\simeq}\mathcal{A}_{n}^{\prime}(H,G^{\prime})\quad\text{and}\quad
g_{\sharp}\colon\mathcal{A}^{\prime}_{n}(H,G)\xrightarrow{\simeq}\mathcal{A}_{n}^{\prime}(H,G^{\prime}).$

In particular, the canonical inclusion
$(\operatorname{id}_{G})^{\sharp}\colon\mathcal{A}_{n}(G,G^{\prime})\to\mathcal{A}_{n}^{\prime}(G,G^{\prime})$
defined to be the composite with the identity is a weak homotopy equivalence
for topological monoids $G$ and $G^{\prime}$.

##  5\. Fiberwise $A_{n}$-space and classification theorem

Let $B$ be a space. We follow the terminology of Crabb–James [CJ98] as
follows. A fiberwise space is just a map $\pi\colon E\to B$ called the
projection and a fiberwise pointed space is a fiberwise space $\pi\colon E\to
B$ equipped with a section $\sigma\colon B\to E$ assigning the basepoint on
each fiber. Fiberwise maps and fiberwise pointed maps are maps $E\to
E^{\prime}$ compatible with projections and sections as usual. Let
$\operatorname{Map}_{B}(E,E^{\prime})$ and
$\operatorname{Map}_{B}^{B}(E,E^{\prime})$ denote the space of fiberwise and
fiberwise pointed maps, respectively. The fiber product
$E\times_{B}E^{\prime}$ of $E$ and $E^{\prime}$ is exactly the categorical
product with respect to fiberwise (pointed) maps.

A fiberwise $A_{n}$-space $(E,\\{m_{i}\\}_{i})$ for a fiberwise pointed space
$E$ over $B$ and a family of fiberwise maps
$\\{m_{i}\colon\mathcal{K}_{i}\times E^{i}\to E\\}_{i}$, where $E^{i}$ means
the $i$-fold fiber product $E\times_{B}\cdots\times_{B}E$, is defined in the
same way as in Section 4. Fiberwise $A_{n}$-maps between a fiberwise
$A_{n}$-space and a fiberwise topological monoid and between topological
monoids are also similarly defined.

Let
$\mathcal{A}_{n}(G,G^{\prime})_{\operatorname{eq}}\subset\mathcal{A}_{n}(G,G^{\prime})$
and
$\mathcal{A}^{\prime}_{n}(H,G^{\prime})_{\operatorname{eq}}\subset\mathcal{A}_{n}(H,G^{\prime})$
denote submonoids of weak $A_{n}$-equivalences, namely, $A_{n}$-maps of which
the underlying maps are weak equivalences. We take a product-preserving
functor $\mathcal{W}\colon\mathbf{CG}\to\mathbf{CG}$ as assigning a CW
replacement. For example, it is sufficient to take $\mathcal{W}X$ to be the
geometric realization of the singular simplices of a space $X$.

######  Theorem 5.1.

Let $E$ be a fiberwise $A_{n}$-space over a connected pointed CW complex $B$
such that the projection $E\to B$ is a Hurewicz fibration, the section $B\to
E$ has the homotopy extension property and the fiber over the basepoint is
$A_{n}$-equivalent to a topological monoid $G$ where $G$ is a CW complex. Then
there exists a map $B\to
B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$,
which we will call a classifying map, unique up to homotopy such that the
pullback $f^{\ast}\tilde{E}$ of the universal fiberwise $A_{n}$-space
$\tilde{E}$ is fiberwise $A_{n}$-equivalent to given $E$.

######  Remark 5.2.

We take the classifying space as
$B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$
rather than the usual one
$B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$. This is because we
defined only the composition of $A_{n}$-maps between topological monoids from
the left.

###### Proof.

In the classification theorem in [Tsu12], fiberwise $A_{n}$-spaces are not
assumed to be unital. But, as stated in [Tsu15, Section 7], our theorem is
proved by a similar argument except the point that the classifying space there
$M_{n}(G)$ coincides with
$B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$ up
to canonical weak homotopy equivalence. By the same argument as in [Tsu12,
Section 5], the space

$\operatorname{Prin}^{\prime}(\tilde{E})^{\operatorname{op}}=\coprod_{b\in
M_{n}(G)}\mathcal{A}^{\prime}_{n}(E_{b},G)_{\operatorname{eq}}$

equipped with an appropriate topology is weakly contractible. Moreover, it
admits a right action by the grouplike topological monoid
$\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$ as in Theorem
4.4, which is the topological monoid
$\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$ equipped with the opposite
multiplication. Together with Proposition 4.5, this implies that
$\operatorname{Prin}^{\operatorname{op}}(\tilde{E})$ is a universal principal
fibration for the grouplike topological monoid
$\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$. Thus the space
$M_{n}(G)$ is weakly homotopy equivalent to
$B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}^{\operatorname{op}}$. ∎

As in [Tsu12, Section 2], locally sliceable fiberwise spaces $E,E^{\prime}$
admit the fiberwise mapping space $\operatorname{map}_{B}(E,E^{\prime})$,
where $\operatorname{map}_{B}(E,-)$ is the right adjoint functor to
$E\times_{B}-$. Since it is compatible with pullback by maps of base spaces,
one can see that the projection $\operatorname{map}_{B}(E,E^{\prime})$ is a
Hurewicz fibration when $E\to B$ and $E^{\prime}\to B$ are Hurewicz
fibrations. Together with it, the fiberwise mapping spaces

$\mathscr{A}^{\prime}_{n}(E,E^{\prime})=\coprod_{b\in
B}\mathcal{A}^{\prime}_{n}(E_{b},E^{\prime}_{b})$

between a fiberwise $A_{n}$-space $E$ and a fiberwise topological monoid
$E^{\prime}$ over $B$ is naturally topologized and becomes a fiberwise space
over $B$. Also, if the projections $E\to B$ and $E^{\prime}\to B$ are Hurewicz
fibrations and the sections $B\to E$ and $B\to E^{\prime}$ have the homotopy
extension properties, then the projection
$\mathscr{A}^{\prime}_{n}(E,E^{\prime})\to B$ is a Hurewicz fibration.

The following proposition describes the classifying map of the fiberwise
mapping space $\mathscr{A}^{\prime}_{n}(E,E^{\prime})$. Note that when
$E^{\prime}$ is a fiberwise topological monoid with fibers
$A_{\infty}$-equivalent to a topological monoid $G$, the classifying map is
given as the one $B\to B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$
of the principal fibration

$\operatorname{Prin}(E^{\prime})=\coprod_{b\in
B}\mathcal{A}_{n}(G,E^{\prime}_{b})_{\operatorname{eq}}$

equipped with the usual right action
$\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$.

######  Proposition 5.3.

Let $E$ be a fiberwise $A_{n}$-space over $B$ with fibers $A_{n}$-equivalent
to a topological monoid $H$ and $E^{\prime}$ be a fiberwise topological monoid
over $B$ with fibers $A_{n}$-equivalent to a topological monoid $G$. Suppose
$B,G,H$ are CW complexes and the projections of $E$ and $E^{\prime}$ are
Hurewicz fibrations and the sections of $E$ and $E^{\prime}$ have the homotopy
extension property. If $E$ is classified by a map $\alpha\colon B\to
B\mathcal{W}\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}$ and
$E^{\prime}$ is classified by a map $\beta\colon B\to
B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$, then the fiberwise
mapping space $\mathscr{A}^{\prime}_{n}(E,E^{\prime})$ is classified by the
composite

$B\xrightarrow{(\alpha,\beta)}B\mathcal{W}\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\times
B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}\xrightarrow{B\mathcal{W}\Theta}B\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{n}(H,G),\mathcal{W}\mathcal{A}_{n}(H,G))_{\operatorname{eq}}$

where
$\operatorname{Map}(X,Y)_{\operatorname{eq}}\subset\operatorname{Map}(X,Y)$
denotes the subset of weak equivalences and the homomorphism

$\Theta\colon\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\times\mathcal{A}_{n}(G,G)_{\operatorname{eq}}\to\operatorname{Map}(\mathcal{A}_{n}(H,G),\mathcal{A}_{n}(H,G))_{\operatorname{eq}}$

is given by $\Theta(f,g)(h)=g\circ h\circ f$.

###### Proof.

Let $\tilde{E}\to
B\mathcal{W}\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}$ and
$\tilde{E}^{\prime}\to B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$
be the universal fiberwise $A_{n}$-spaces. We write as
$\mathcal{G}=\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\times\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$
and
$\mathcal{B}=B\mathcal{WG}=B\mathcal{W}\mathcal{A}_{n}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\times
B\mathcal{W}\mathcal{A}_{n}(G,G)_{\operatorname{eq}}$ for simplicity. Consider
the commutative diagram

$\textstyle{\mathscr{A}^{\prime}_{n}(\tilde{E},\tilde{E}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\operatorname{Prin}^{\prime}(\tilde{E})^{\operatorname{op}}\times\operatorname{Prin}(\tilde{E}^{\prime}),\mathcal{WG},\mathcal{W}\mathcal{A}_{n}(H,G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\ast,\mathcal{WG},\mathcal{W}\mathcal{A}_{n}(H,G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}}$$\textstyle{B(\operatorname{Prin}^{\prime}(\tilde{E})^{\operatorname{op}}\times\operatorname{Prin}(\tilde{E}^{\prime}),\mathcal{WG},\ast)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\ast,\mathcal{WG},\ast)}$

where the left are arrows induced from the compositions of $A_{n}$-maps and
the squares are homotopy pullback by Lemma 2.6. Since the vertical maps are
quasifibrations and the bottom arrows are weak homotopy equivalences, the top
horizontal arrows are also weak homotopy equivalences. Inverting the bottom
left arrow, we obtain the composite

$\mathcal{B}=B\mathcal{WG}\to
B(\operatorname{Prin}^{\prime}(\tilde{E})^{\operatorname{op}}\times\operatorname{Prin}(\tilde{E}^{\prime}),\mathcal{WG},\ast)\to
B\mathcal{WG},$

which is indeed homotopic to the identity map since it is the classifying map
of the universal principal fibration. Consider the homotopy pullback square

$\textstyle{B(\ast,\mathcal{WG},\mathcal{W}\mathcal{A}_{n}(H,G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B(\ast,\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{n}(H,G),\mathcal{W}\mathcal{A}_{n}(H,G))_{\operatorname{eq}},\mathcal{W}\mathcal{A}_{n}(H,G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\mathcal{WG}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B\mathcal{W}\Theta}$$\textstyle{B\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{n}(H,G),\mathcal{W}\mathcal{A}_{n}(H,G))_{\operatorname{eq}}.}$

This implies that the Hurewicz fibration
$\mathscr{A}^{\prime}_{n}(\tilde{E},\tilde{E}^{\prime})\to\mathcal{B}$ is
classified by $B\mathcal{W}\Theta$. Since
$\mathscr{A}^{\prime}_{n}(E,E^{\prime})$ is the pullback of
$\mathscr{A}^{\prime}_{n,\mathcal{B}}(\tilde{E},\tilde{E}^{\prime})\to\mathcal{B}$
by the map $(\alpha,\beta)\colon B\to\mathcal{B}$, the proposition follows. ∎

##  6\. $N_{k}(\ell)$-map

We introduce $N_{k}(\ell)$-map mimicking Definition 1.1 of crossed module as
follows.

######  Definition 6.1.

Let $f\colon H\to G$ be a homomorphism between topological groups and $k,\ell$
be positive integers or infinity. We say the homomorphism $f$ is an
$N_{k}(\ell)$-map if an $A_{k}$-map $\rho\colon G\to\mathcal{A}_{\ell}(H,H)$
is given and the following conditions hold:

  1. (1)

the composite of $A_{k}$-maps

$H\xrightarrow{f}G\xrightarrow{\rho}\mathcal{A}_{\ell}(H,H)$

is homotopic to $\operatorname{conj}\colon H\to\mathcal{A}_{\ell}(H,H)$
($\operatorname{conj}(h)(x)=hxh^{-1}$) as an $A_{k}$-map,

  2. (2)

the pair of a map

$\text{point}\to\mathcal{A}_{\ell}(H,G),\quad\ast\mapsto f$

and the $A_{k}$-map

$\theta\colon
G\to\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G)),\quad\theta(g)(\alpha)=\operatorname{conj}(g)\circ\alpha\circ\rho(g^{-1}).$

extends to an $A_{k}$-equivariant map
$(G,\ast)\to(\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}},\mathcal{A}_{\ell}(H,G))$,

  3. (3)

the composite of the $A_{k}$-form of the previous $A_{k}$-equivariant map with
$f\colon(H,\ast)\to(G,\ast)$ is homotopic to the trivial one coming from the
equality $\operatorname{conj}(f(h))\circ f\circ\operatorname{conj}(h^{-1})=f$
for any $h\in H$.

This definition could be extended to an $A_{n}$-map $f$. To do so, it is
necessary to determine how good coherence can be guaranteed for the action of
$H$ on
$\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$
in the condition (3). We avoid this problem and concentrate on homomorphisms
here.

If a homomorphism $f\colon H\to G$ is an $N_{k}(\ell)$-map and $k^{\prime}\leq
k,\ell^{\prime}\leq\ell$, then $f$ is obviously an
$N_{k^{\prime}}(\ell^{\prime})$-map.

If $f\colon H\to G$ is a crossed module, then $f$ is obviously an
$N_{\infty}(\infty)$-map. This is all the case for which we can verify the
condition of Definition 6.1 directly since it contains very complicated higher
homotopical conditions in the present work. The following theorem provides us
the way to check the conditions from the obstruction theoretic point of view.

######  Theorem 6.2.

Let $f\colon H\to G$ be a homomorphism between topological groups $G,H$, of
which the underlying spaces are CW complexes, and $k,\ell$ are non-negative
integers. Then the homomorphism $f$ is an $N_{k}(\ell)$-map if and only if
there exists a fiberwise $A_{\ell}$-space $E$ over $B_{k}G$ and fiberwise
$A_{\ell}$-maps $\phi\colon(B_{k}f)^{\ast}E\to
E_{k}H\times_{\operatorname{conj}}H$ over $B_{k}H$ and $\psi\colon E\to
E_{k}G\times_{\operatorname{conj}}G$ over $B_{k}G$ satisfying the following
conditions:

  1. (1)

$\phi$ restricts to a weak $A_{\ell}$-equivalence on each fiber,

  2. (2)

for the restrictions to the fiber over the basepoint
$\phi_{\ast},\psi_{\ast}$, the composite $f\circ\phi_{\ast}\colon E_{\ast}\to
G$ is homotopic to $\psi_{\ast}$ as an $A_{\ell}$-map,

  3. (3)

the composite of $\phi$ and the induced map
$E_{k}H\times_{\operatorname{conj}}H\to E_{k}H\times_{\operatorname{conj}\circ
f}G$ of $f$ is homotopic to $(B_{k}f)^{\ast}\psi$ as a fiberwise $A_{k}$-map.

Actually, the condition (2) is immediately implied by (3). But we dare to
write in this way in order to clarify the correspondence of the conditions
between this theorem and Definition 6.1.

###### Proof.

Suppose the condition (1) in the theorem. Let $\rho_{0}^{\prime}\colon
B_{k}G\to
B\mathcal{W}\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}^{\operatorname{op}}$
be the classifying map of
$\operatorname{Prin}^{\prime}(E)^{\operatorname{op}}$ as in Theorem 5.1. By
the assumption (1), the composite $\rho_{0}^{\prime}\circ B_{k}f$ is homotopic
to $\iota_{k}\circ B_{k}(\operatorname{conj}\circ\text{(inversion)})\colon
B_{k}H\to B\mathcal{W}\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}$. Let
$\rho\colon
G\to\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}^{\operatorname{op}}$ be the
composite of the inversion and the $A_{k}$-map adjoint to $\rho_{0}^{\prime}$.
Then the composite $\rho\circ f$ is homotopic to $\operatorname{conj}\colon
H\to\mathcal{A}_{\ell}(H,H)$ as an $A_{k}$-map. This is the condition (1) in
Definition 6.1.

Conversely, suppose the condition (1) in Definition 6.1. Then the composite

$B_{k}G\xrightarrow{B_{k}\text{(inversion)}}B_{k}G^{\operatorname{op}}\xrightarrow{B_{k}\rho}B_{k}\mathcal{W}\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\xrightarrow{\iota_{k}}B\mathcal{W}\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}^{\operatorname{op}}$

classifies a fiberwise $A_{\ell}$-space $E$ over $B_{k}G$. Then the pullback
$(B_{k}f)^{\ast}E$ is weakly fiberwise $A_{\ell}$-equivalent to the fiberwise
topological group $E_{k}H\times_{\operatorname{conj}}H$. This is the condition
(1) in the theorem.

Suppose the condition (1), (2) and (3) in the theorem. The fiberwise
$A_{\ell}$-map $\psi$ defines a section of the fiberwise mapping space
$\mathscr{A}^{\prime}_{\ell}(E,E_{k}G\times_{\operatorname{conj}}G)$ over
$B_{k}G$. By the map $\phi$, we can identify the fiber of
$\mathscr{A}_{\ell}(E,E_{k}G\times_{\operatorname{conj}}G)$ over the base
point with $\mathcal{A}_{\ell}(H,G)$ of which the basepoint is $f$. By
Proposition 5.3, the fiberwise space
$\mathscr{A}^{\prime}_{\ell}(E,E_{k}G\times_{\operatorname{conj}}G)$ is
classified by the composite

$B_{k}G\xrightarrow{(\rho_{0}^{\prime},\iota_{k}\circ
B_{k}\operatorname{conj})}B\mathcal{W}\mathcal{A}_{\ell}(H,H)_{\operatorname{eq}}^{\operatorname{op}}\times
B\mathcal{W}\mathcal{A}_{\ell}(G,G)_{\operatorname{eq}}\xrightarrow{B\mathcal{W}\Theta}B\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}.$

The section induced from $\psi$ determines the lift

$\Psi^{\prime}\colon B_{k}G\to
B\mathcal{W}\operatorname{Map}_{\ast}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$

of the previous composite along the map

$B\mathcal{W}\operatorname{Map}_{\ast}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}\to
B\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}.$

Let $\Psi\colon
G\to\operatorname{Map}_{\ast}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$
be the $A_{k}$-map adjoint to $\Psi^{\prime}$. It defines the obvious
$A_{k}$-equivariant map

$\Psi\colon(G,\ast)\to(\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}},\mathcal{W}\mathcal{A}_{\ell}(H,G)),$

where the underlying map $\ast\to\mathcal{W}\mathcal{A}_{\ell}(H,G)$ is
$\ast\mapsto f$. By the argument so far, the underlying $A_{k}$-map of $\Psi$
is homotopic to the composite of $A_{k}$-maps

$G\xrightarrow{\theta}\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}\xrightarrow{\mathcal{W}}\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}.$

This homotopy determines an $A_{k}$-equivariant map

$(G,\ast)\to(\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}},\mathcal{A}_{\ell}(H,G))$

with the underlying $A_{k}$-map $\theta$. To be precise, it is determined up
to homotopy. Also, the composite

$B_{k}H\xrightarrow{B_{k}f}B_{k}G\xrightarrow{\Psi^{\prime}}B\mathcal{W}\operatorname{Map}_{\ast}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$

is homotopic to the map adjoint to the $A_{k}$-map (actually a homomorphism)

$H\to\mathcal{W}\operatorname{Map}_{\ast}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$

induced from the conjugation on $H$ and the conjugation through $f$ on $G$ by
the assumption (3). Thus the condition (2) and (3) in Definition 6.1 is
verified.

Conversely, suppose the conditions (1), (2) and (3) in Definition 6.1. By
Proposition 5.3, $B_{k}\theta$ classifies the fiberwise mapping space
$\mathscr{A}^{\prime}_{\ell}(E,E_{k}G\times_{\operatorname{conj}}G)$. The bar
construction of the $A_{k}$-equivariant map

$(G,\ast)\to(\operatorname{Map}(\mathcal{A}_{\ell}(H,G),\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}},\mathcal{A}_{\ell}(H,G))$

defines a map

$B_{k}G=B_{k}(\ast,G,\ast)\to
B_{k}(\ast,\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}},\mathcal{W}\mathcal{A}_{\ell}(H,G))$

of which the composition with the projection onto
$B\mathcal{W}\operatorname{Map}(\mathcal{W}\mathcal{A}_{\ell}(H,G),\mathcal{W}\mathcal{A}_{\ell}(H,G))_{\operatorname{eq}}$
is homotopic to $B_{k}\theta$. Then
$\mathscr{A}^{\prime}_{\ell}(E,E_{k}G\times_{\operatorname{conj}}G)$ admits a
section which restricts to $f$ up to the canonical $A_{k}$-equivalence
$E_{\ast}\simeq H$. This verifies the condition (2) in the theorem. Moreover,
since the $A_{k}$-equivariant map restricts to the trivial one on $H$ as
supposed, we can verify the condition (3). ∎

Let us consider the special case when $k=\ell=1$. For pointed spaces $A,B$
with nondegenerate basepoints and a topological group $G$, we have the natural
split exact sequence

$1\to[A\wedge B,G]\xrightarrow{q^{\ast}}[A\times
B,G]\xrightarrow{i^{\ast}}[A\vee B,G]\to 1,$

where $i\colon A\vee B\to A\times B$ is the inclusion and $q\colon A\times
B\to A\wedge B$ is the quotient map. The Samelson product $\langle
f,g\rangle\in[A\wedge B,G]$ of based maps $f\colon A\to G$ and $g\colon B\to
G$ is the homotopy class determined by the commutator map

$A\times B\to G,\quad(a,b)\mapsto f(a)g(b)f(a)^{-1}g(b)^{-1}$

and the previous split exact sequence.

######  Theorem 6.3.

Let $f\colon H\to G$ be a homomorphism between topological groups $G,H$, of
which the underlying spaces are CW complexes. The map $f$ is an $N_{1}(1)$-map
if and only if there exists a based map $\rho^{\prime}\colon G\wedge H\to H$
satisfying the following conditions:

  1. (1)

the composite $\rho^{\prime}\circ(f\wedge\operatorname{id}_{H})\colon H\wedge
H\to H$ is homotopic to the Samelson product
$\langle\operatorname{id}_{H},\operatorname{id}_{H}\rangle$,

  2. (2)

the composite $f\circ\rho^{\prime}\colon G\wedge H\to G$ is homotopic to the
Samelson product $\langle\operatorname{id}_{G},f\rangle$

  3. (3)

the composite of above two homotopies is homotopic to the stationary homotopy
on $\langle f,f\rangle$.

$\textstyle{H\wedge
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle\operatorname{id}_{H},\operatorname{id}_{H}\rangle}$$\scriptstyle{f\wedge\operatorname{id}_{H}}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{G\wedge
H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime}}$$\scriptstyle{\langle\operatorname{id}_{G},f\rangle}$$\textstyle{G}$

###### Proof.

For given $\rho^{\prime}\colon G\wedge H\to H$, define

$\rho\colon
G\to\operatorname{Map}_{\ast}(H,H),\quad\rho(g)(h)=\rho^{\prime}(g,h)h.$

Then the condition (1) is equivalent to the one that $\rho\circ f$ is
homotopic to $\operatorname{conj}\colon H\to\operatorname{Map}_{\ast}(H,H)$.
The latter condition is nothing but the condition (1) in Definition 6.1. The
condition (2) is equivalent to the one that the map $(g,h)\mapsto
f(\rho(g)(h))$ is homotopic to the map $(g,h)\mapsto gf(h)g^{-1}$. The latter
condition is equivalent to the first half of the condition (2) in Definition
6.1. The equivalence between the condition (3) and the latter half of the
condition (2) in Definition 6.1 follows similarly. ∎

######  Remark 6.4.

From this result, a homomorphism $f\colon H\to G$ is an $N_{1}(1)$-map if and
only if $f$ is homotopy normal in the sense of McCarty [McC64]. The condition
(2) is exactly the homotopy normality of James [Jam67].

##  7\. $C(k,\ell)$-space

A $C(k,\ell)$-space is a topological monoid with certain higher homotopy
commutativity introduced in [KK10]. We do not recall the precise definition
but instead quote the following characterization [Tsu16, Theorem 8.3], where
the equivalence with fourth condition follows from Theorem 5.1.

######  Theorem 7.1.

Let $G$ be a topological group, of which the underlying space is a CW complex.
Then the following conditions are equivalent:

  1. (1)

$G$ is a $C(k,\ell)$-space,

  2. (2)

the wedge sum of the inclusions $(\iota_{k},\iota_{\ell})\colon B_{k}G\vee
B_{\ell}G\to BG$ extends over the product $B_{k}G\times B_{\ell}G$,

  3. (3)

the homomorphism $\operatorname{conj}\colon\mathcal{A}_{\ell}(G,G)$ is
homotopic to the constant map to the identity as $A_{k}$-map,

  4. (4)

the fiberwise topological group $E_{k}G\times_{\operatorname{conj}}G$ over
$B_{k}G$ is fiberwise $A_{\ell}$-equivalent to the trivial fiberwise
topological group $B_{k}G\times G$.

Note that, in particular, $G$ is a $C(\infty,\infty)$-space if and only if
$BG$ is an $H$-space.

The following is an easy application of the previous theorem.

######  Theorem 7.2.

Let $H$ and $G$ be topological groups of which the underlying spaces are CW
complexes and $f\colon H\to G$ be a homomorphism. If $H$ and $G$ are
$C(k,\ell)$-spaces for some $k,\ell\leq n$, then $f$ is an $N_{k}(\ell)$-map.

###### Proof.

By assumption and Theorem 7.1, the fiberwise topological groups
$E_{k}H\times_{\operatorname{conj}}H$ and
$E_{k}G\times_{\operatorname{conj}}G$ are fiberwise $A_{\ell}$-equivalent to
the trivial bundles. Consider the trivial fiberwise topological group
$E=B_{k}G\times H$. Then we can verify the conditions in Theorem 6.2. Thus $f$
is an $N_{k}(\ell)$-map. ∎

For the homomorphism $H\to\ast$ to the trivial group, we obtain the following.

######  Theorem 7.3.

Let $H$ be a topological group of which the underlying space is a CW complex.
The homomorphism $H\to\ast$ to the trivial group is an $N_{k}(\ell)$-map if
and only if $H$ is a $C(k,\ell)$-space.

###### Proof.

Note that $B_{k}\ast=\ast$ for any $k$. Then by Theorem 6.2, the homomorphism
$H\to\ast$ is an $N_{k}(\ell)$-map if and only if the fiberwise topological
monoid $E_{k}H\times_{\operatorname{conj}}H$ is fiberwise
$A_{\ell}$-equivalent to the trivial fiberwise topological group $B_{k}H\times
H$. Thus the theorem follows from Theorem 7.1. ∎

##  8\. $H$-structure on quotient space

Since the homotopy quotient of a crossed module is known to be a topological
monoid, we expect a similar result for $N_{k}(\ell)$-maps. Let us investigate
the existence of $H$-structures on the homotopy quotient.

######  Proposition 8.1.

Let $f\colon H\to G$ be a homomorphism between topological groups of which the
underlying spaces are CW complexes. Suppose $f$ is an $N_{k}(\ell)$-map. Then,
the topological group $F$ with classifying space $BF\simeq
B(\ast,H,G)=EH\times_{f}G$ is a $C(k,\ell)$-space.

Note that the topological group $F$ is $A_{\infty}$-equivalent to the Moore
based loop space of $BF$. This is the key proposition in our study of
$H$-structure on $EH\times_{f}G$.

###### Proof.

We may suppose that $f$ is a Hurewicz fibration and $F=\ker f$. Let $i\colon
F\to H$ be the inclusion. Note that the results in [Tsu15, Sections 3 and 4]
for $A_{n}$-spaces similarly verified for fiberwise $A_{n}$-spaces. By [Tsu15,
Proposition 4.1], we may suppose that there exists a fiberwise
$A_{\ell}$-space $E$ over $B_{k}G$ and a fiberwise $A_{\ell}$-equivalence
$\phi\colon(B_{k}f)^{\ast}E\to E_{k}H\times_{\operatorname{conj}}H$ and a
fiberwise $A_{\ell}$-homomorphism $\psi\colon E\to
E_{k}G\times_{\operatorname{conj}}G$ as in Theorem 6.2. Let $E^{\prime}$ be
the fiberwise homotopy fiber of $\psi$ over $B_{k}G$. Then by [Tsu15, Theorem
3.1] (the pullback of $A_{n}$-homomorphisms), $E^{\prime}$ admits the
canonical structure of fiberwise $A_{\ell}$-space. Applying [Tsu15, Theorem
3.3] (the universal property of the pullback of $A_{n}$-homomorphisms), a
fiberwise $A_{\ell}$-map $\chi\colon(B_{k}f)^{\ast}E^{\prime}\to
E_{k}H\times_{H}F$ is induced as in the following diagram:

$\textstyle{(B_{k}f)^{\ast}E^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{(B_{k}f)^{\ast}E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{B_{k}H}}$$\scriptstyle{\phi}$$\textstyle{E_{k}H\times_{\operatorname{conj}\circ
f}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{k}H\times_{H}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{k}H\times_{\operatorname{conj}}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{k}H\times_{\operatorname{conj}\circ
f}G}$

We can see that $\chi$ is a fiberwise $A_{\ell}$-equivalence. Pulling buck
along the map $B_{k}i\colon B_{k}F\to B_{k}H$, we obtain the fiberwise
$A_{\ell}$-equivalence

$(B_{k}i)^{\ast}\chi\colon(B_{k}(i\circ
f))^{\ast}E^{\prime}\xrightarrow{\simeq}E_{k}F\times_{H}F.$

Since $i\circ f$ is the constant homomorphism, $E_{k}F\times_{H}F$ is
fiberwise $A_{\ell}$-equivalent to the trivial fiberwise topological group
$B_{k}F\times F$. Thus, it follows from Theorem 7.1 that $F$ is a
$C(k,\ell)$-space. ∎

######  Theorem 8.2.

Let $f\colon H\to G$ be a homomorphism between topological groups of which the
underlying spaces are CW complexes and $k\geq 1$. Suppose that $f$ is an
$N_{k}(k)$-map and the LS category of the homotopy quotient $EH\times_{f}G$ is
estimated as $\operatorname{cat}(EH\times_{f}G)\leq k$. Then $EH\times_{f}G$
is an $H$-space.

###### Proof.

Take $F$ as in the proof of Proposition 8.1 so that the homotopy equivalence
$BF\simeq EH\times_{f}G$ and the estimate $\operatorname{cat}BF\leq k$ hold.
Then, as in [Iwa98], the inclusion $B_{k}F\to BF$ admits a homotopy section
$s\colon BF\to B_{k}F$. Since $F$ is a $C(k,k)$-space, it follows from the
existence of $s$ and Theorem 7.1 that $BF$ is an $H$-space. ∎

######  Remark 8.3.

As a consequence of Theorem 7.3, one cannot expect that $N_{k}(\ell)$-map
implies any higher homotopy associativity of the homotopy quotient $H$-space.
In order to guarantee higher homotopy associativity, we need even higher
homotopy normality.

We have only shown the existence of $H$-structure on the homotopy quotient in
the previous theorem. We do not try to answer the following natural question
in the present work.

######  Problem 8.4.

Suppose a homomorphism $f\colon H\to G$ is an $N_{k}(k)$-map and
$\operatorname{cat}(EH\times_{f}G)\leq k$. Construct a canonical $H$-structure
on $EH\times_{f}G$. If it could be done, is the natural map $G\to
EH\times_{f}G$ an $H$-map?

##  9\. Fiberwise projective spaces

Let us recall the fiberwise projective spaces of topological monoids.

######  Definition 9.1.

The fiberwise $n$-th projective space $\mathscr{B}_{n}E$ of a fiberwise
topological monoid $E\to B$ is defined to be the quotient

$\mathscr{B}_{n}E=\left(\coprod_{0\leq i\leq n}\Delta^{i}\times
E^{i}\middle)\right/{\sim}$

by the usual simplicial relation, where $E^{i}$ denotes the $i$-fold fiber
product. In particular, $\mathscr{B}E=\mathscr{B}_{\infty}E$ is called the
fiberwise classifying space. Let
$\iota_{n}\colon\mathscr{B}_{n}E\to\mathscr{B}E$ denote the canonical
inclusion. As in Sections 2, the fiberwise projective space induce the
fiberwise map

$\mathscr{B}_{n}\colon\mathscr{A}_{n}(E,E^{\prime})\to\operatorname{map}_{B}^{B}(\mathscr{B}_{n}E,\mathscr{B}_{n}E^{\prime})$

to the fiberwise mapping space of fiberwise pointed maps for fiberwise
topological monoids $E$ and $E^{\prime}$.

The following is a variant of Theorem 2.7 in the category of fiberwise spaces.

######  Theorem 9.2.

Let $E$ and $E^{\prime}$ be fiberwise topological monoids over a CW complex
$B$ and suppose that their fiberwise basepoints of $E$ and $E^{\prime}$ have
the fiberwise homotopy extension property and their projections are Hurewicz
fibrations. Then, the composite of fiberwise maps

$\mathscr{A}_{n}(E,E^{\prime})\xrightarrow{\mathscr{B}_{n}}\operatorname{map}_{B}^{B}(\mathscr{B}_{n}E,\mathscr{B}_{n}E^{\prime})\xrightarrow{(\iota_{n})_{\sharp}}\operatorname{map}_{B}^{B}(\mathscr{B}_{n}E,\mathscr{B}E^{\prime})$

is a weak homotopy equivalence in the category of spaces.

###### Proof.

In this setting, the projections $\mathscr{A}_{n}(E,E^{\prime})\to B$ and
$\operatorname{map}_{B}^{B}(\mathscr{B}_{n}E,\mathscr{B}E^{\prime})\to B$ are
Hurewicz fibrations. Thus the given composite is a weak homotopy equivalence
since it restricts to a weak homotopy equivalence on each fiber by Theorem
2.7. ∎

Although the following result is well-known, we give a proof here.

######  Proposition 9.3.

Let $G$ be a topological group. Then, the fiberwise classifying space
$EG\times_{G}BG$ is homeomorphic to $BG\times BG$ through a fiberwise map over
$BG$, where the fiberwise basepoint of $BG\times BG$ is given by the diagonal
map $BG\to BG\times BG$.

###### Proof.

The homeomorphism $EG\times_{G}BG\to BG\times BG$ is induced from the
geometric realization of the simplicial map $\\{G^{i}\times G^{i}\to
G^{i}\times G^{i}\\}_{i}$ given by the homeomorphisms

$\displaystyle(g_{1},\ldots,g_{i};x_{1},x_{2},\ldots,x_{i})$
$\displaystyle\mapsto(g_{1},\ldots,g_{i};(g_{1}\cdots g_{i})x_{1}(g_{2}\cdots
g_{i})^{-1},(g_{2}\cdots g_{i})x_{2}(g_{3}\cdots
g_{i})^{-1},\ldots,g_{i}x_{i}),$

where $EG\times_{G}BG$ and $BG\times BG$ are cosidered to be the geometric
realizations of appropriate simplicial spaces $\\{G^{i}\times G^{i}\\}_{i}$
with different simplicial structures. ∎

The following theorem provides a criterion for non-$N_{k}(\ell)$-map.

######  Theorem 9.4.

Let $f\colon H\to G$ be a homomorphism between topological groups of which the
underlying spaces are CW complexes and $k,\ell\geq 1$. Then, if $f$ is an
$N_{k}(\ell)$-map, then there exist fiberwise pointed maps

$E_{k}H\times_{H}B_{\ell}H\xrightarrow{\Phi}\mathcal{E}\xrightarrow{\Psi}EG\times_{G}BG$

for some fiberwise pointed space $\mathcal{E}$ over $B_{k}G$, where $\Phi$ is
precisely a fiberwise pointed map as a map
$E_{k}H\times_{H}B_{\ell}H\to(B_{k}f)^{\ast}\mathcal{E}$, satisfying the
following conditions:

  1. (1)

$\Phi$ restricts to a weak homotopy equivalence on each fiber,

  2. (2)

the restriction of the composite $\Psi\circ\Phi$ to the fiber over the
basepoint is homotopic to $\iota_{\ell}\circ B_{\ell}f\colon B_{\ell}H\to BG$,

  3. (3)

the composite $\Psi\circ\Phi\colon E_{k}H\times_{H}B_{\ell}H\to
E_{k}H\times_{H}BG$ is homotopic to the map induced from $f$ as a fiberwise
pointed map.

As in Theorem 6.2, we again contain the obvious consequence (2) of (3).

######  Remark 9.5.

Indeed, the converse of this theorem should hold as well. But, to verify it,
we need to generalize Theorem 9.2 for fiberwise $A_{n}$-maps from fiberwise
$A_{n}$-spaces in a way compatible with the composition of fiberwise
$A_{n}$-maps between fiberwise topological monoids. We leave this problem
here.

###### Proof.

Suppose $f\colon H\to G$ is an $N_{k}(\ell)$-map. Then we have a fiberwise
$A_{\ell}$-space $E$ over $B_{k}G$ and fiberwise $A_{\ell}$-maps $\phi\colon
E_{k}H\times_{\operatorname{conj}}H\to(B_{k}f)^{\ast}E$ and $\psi\colon E\to
E_{k}G\times_{\operatorname{conj}}G$ as in Theorem 6.2. Again by [Tsu15,
Proposition 4.1], we may suppose that $\phi$ and $\psi$ are fiberwise
$A_{\ell}$-homomorphisms. Moreover, we may suppose the composite of $\phi$ and
the fiberwise homomorphism $E_{k}H\times_{\operatorname{conj}}H\to
E_{k}H\times_{\operatorname{conj}\circ f}G$ is homotopic to $\psi$ through
fiberwise $A_{\ell}$-homomorphisms by this argument. As in [Sak10], the
projective space for an $A_{n}$-space by Stasheff [Sta63] using the
associahedra can be extended to fiberwise $A_{n}$-spaces. Let
$\mathcal{E}=\mathscr{B}^{\mathcal{K}}_{\ell}E$ denote the $\ell$-th fiberwise
projective space of $E$. Note that a fiberwise $A_{\ell}$-homomorphism induces
a fiberwise pointed map between the fiberwise projective spaces in the obvious
manner. Then we have fiberwise $A_{\ell}$-maps
$\Phi^{\prime}\colon\mathscr{B}^{\mathcal{K}}_{\ell}(E_{k}H\times_{\operatorname{conj}}H)\to(B_{k}f)^{\ast}\mathcal{E}$,
which is the homotopy inverse of the induced map of $\phi$,
$\Psi^{\prime}\colon\mathcal{E}\to\mathscr{B}^{\mathcal{K}}_{\infty}(EG\times_{\operatorname{conj}}G)$,
which is the induced map of $\psi$, and
$F^{\prime}\colon\mathscr{B}^{\mathcal{K}}_{\ell}(E_{k}H\times_{H}H)\to\mathscr{B}^{\mathcal{K}}_{\infty}(E_{k}H\times_{\operatorname{conj}\circ
f}G)$, which is the induced map of $f$. We also note that there exists a
canonical homotopy commutative diagram of fiberwise pointed spaces

$\textstyle{\mathscr{B}^{\mathcal{K}}_{\ell}(E_{k}H\times_{\operatorname{conj}}H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{\prime}}$$\scriptstyle{\simeq}$$\textstyle{\mathscr{B}^{\mathcal{K}}_{\infty}(E_{k}H\times_{\operatorname{conj}\circ
f}G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{E_{k}H\times_{H}B_{\ell}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{E_{k}H\times_{H}BG}$

where $F$ is the induced map of $f$ and the vertical maps are fiberwise
pointed homotopy equivalences. This follows from the classification theorem of
fiberwise principal bundles and the fact that both
$\mathscr{B}^{\mathcal{K}}_{\ell}(E_{k}H\times_{\operatorname{conj}}H)$ and
$E_{k}H\times_{H}B_{\ell}H$ are characterized by the fiberwise variant of
Ganea’s pullback-pushuout construction. Composing the above fiberwise pointed
homotopy equivalences, we obtain the fiberwise pointed maps $\Phi$ and $\Psi$
satisfying the desired conditions. ∎

##  10\. $p$-local normality of Lie groups

By the result of James [Jam67] and Remark 6.4, the inclusion
$\operatorname{SU}(m)\to\operatorname{SU}(n)$ ($2\leq m<n$) is not
($2$-locally) an $N_{1}(1)$-map. But one might expect some normality when the
spaces are localized at a large prime $p$. In the rest of this section, we
assume $k\geq 1$ and $\ell\geq 1$ and often omit the $p$-localization symbol
like $G=G_{(p)}$.

As is well-known, any subgroup of an abelian group is normal. But the analogue
of this fact for a homomorphism between infinite loop spaces does not hold as
in the following example.

######  Example 10.1.

Let $f\colon H\to G$ be a homomorphism such that the delooping is homotopy
equivalent to the map between the Eilenberg–MacLane spaces $g\colon
K(\mathbb{Q},2n)\to K(\mathbb{Q},4n)$ classified by the square $u^{2}\in
H^{4n}(K(\mathbb{Q},2n);\mathbb{Q})$ of the generator $u\in
H^{2n}(K(\mathbb{Q},2n);\mathbb{Q})$. The homotopy fiber of $g$ is homotopy
equivalent to the rationalized $2n$-sphere $S^{2n}_{(0)}$, which is not an
$H$-space. Then, since we have $\operatorname{cat}S^{2n}_{(0)}=1$, $f$ is not
an $N_{1}(1)$-map by Theorem 8.2. Similarly, the inclusion
$\operatorname{SO}(2n)_{(0)}\to\operatorname{SO}(2n+1)_{(0)}$ is not an
$N_{1}(1)$-map for $n\geq 1$.

But we have a normality result for some class of homomorphisms as follows.

######  Theorem 10.2.

Let $f\colon H\to G$ be a homomorphism between compact connected semisimple
Lie groups. Let $2m-1$ and $2n-1$ denote the largest degrees of generators of
the rational cohomology algebra $H^{\ast}(H;\mathbb{Q})$ and
$H^{\ast}(G;\mathbb{Q})$, respectively. Suppose that $f^{\ast}\colon
H^{\ast}(G;\mathbb{Q})\to H^{\ast}(H;\mathbb{Q})$ is surjective and $p\geq
kn+\ell m$. Then the homomorphism $f$ is $p$-locally an $N_{k}(\ell)$-map.

###### Proof.

We verify the conditions in Theorem 6.2 for the trivial fiberwise topological
group $E=B_{k}G\times H$. We employ the technique to reduce the projective
spaces in [HKT19, Sections 3 and 4]. Since we assume $p\geq kn+\ell m$, we
have the $A_{k}$-equivalences

$H\simeq S^{2m_{1}-1}\times\cdots\times S^{2m_{s}-1}\quad\text{and}\quad
G\simeq S^{2n_{1}-1}\times\cdots\times S^{2n_{r}-1}$

for some sequences $2\leq m_{1}\leq\cdots\leq m_{s}\leq m$ and $2\leq
n_{1}\leq\cdots\leq n_{r}\leq n$. Then we have the homotopy equivalences

$G\simeq H\times G/H\quad\text{and}\quad G/H\simeq
S^{2q_{1}-1}\times\cdots\times S^{2q_{r-s}-1}$

for some sequence $2\leq q_{1}\leq\cdots\leq q_{r-s}\leq n$ since
$f^{\ast}\colon H^{\ast}(G;\mathbb{Q})\to H^{\ast}(H;\mathbb{Q})$ is
surjective. Note that the sequences $m_{1},\ldots,m_{s}$ and
$q_{1},\ldots,q_{r-s}$ are subsequences of the sequence $n_{1},\ldots,n_{r}$.
We can consider these equivalences are $A_{k}$-equivalences with an
appropriate $A_{k}$-form on $G/H$. Consider the space

$B_{k}(i_{1},\ldots,i_{t})=\bigcup_{k_{1}+\cdots+k_{t}}B_{k_{1}}S^{2i_{1}-1}\times\cdots\times
B_{k_{t}}S^{2i_{t}-1}.$

Then we have the homotopy commutative diagram

$\textstyle{B_{k}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{B_{k}f}$$\textstyle{B_{k}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inclusion$\scriptstyle{\alpha}$$\textstyle{B_{k}(m_{1},\ldots,m_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inclusion$\textstyle{B_{k}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle{\bigcup_{k_{1}+k_{2}=k}B_{k_{1}}H\times
B_{k_{2}}(G/H)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\times\beta}$$\textstyle{B_{k}(n_{1},\ldots,n_{r})}$

for maps

$\alpha\colon B_{k}H\to
B_{k}(m_{1},\ldots,m_{s})\quad\text{and}\quad\beta\colon B_{k}(G/H)\to
B_{k}(q_{1},\ldots,q_{r-s})$

inducing isomorphisms on generators of mod $p$-cohomologies, where the
homotopy commutativity of the left square follows from the construction of the
left bottom arrow in [HKT19, Proposition 3.2]. Let $\gamma\colon B_{k}G\to
B_{k}(n_{1},\ldots,n_{r})$ denote the composite of the bottom arrows. We also
have the maps

$\alpha^{\prime}\colon B_{k}(m_{1},\ldots,m_{s})\to
BH\quad\text{and}\quad\gamma^{\prime}\colon B_{k}(n_{1},\ldots,n_{r})\to BG$

such that the compositions $\alpha^{\prime}\circ\alpha$ and
$\gamma^{\prime}\circ\gamma$ are homotopic to the inclusions. Consider the
fiberwise pointed spaces

$\displaystyle B_{k}(m_{1},\ldots,m_{s})\to B_{k}(m_{1},\ldots,m_{s})\times
B_{k}(m_{1},\ldots,m_{s})\to B_{k}(m_{1},\ldots,m_{s})\quad\text{and}$
$\displaystyle B_{k}(n_{1},\ldots,n_{r})\to B_{k}(n_{1},\ldots,n_{r})\times
B_{k}(m_{1},\ldots,m_{s})\to B_{k}(n_{1},\ldots,n_{r})$

with the first projection and the first inclusion. Since we suppose $p\geq
kn+\ell m$, we have

$\pi_{2i-1}(BH)=\pi_{2i-1}(BG)=0$

for $i\leq 2kn+2\ell m$. Then there exists a commutative diagram of fiberwise
pointed maps

$\textstyle{B_{k}H\times
B_{k}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\times\alpha}$$\scriptstyle{B_{k}f\times\operatorname{id}}$$\textstyle{B_{k}(m_{1},\ldots,m_{s})\times
B_{k}(m_{1},\ldots,m_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inclusion$\textstyle{EH\times_{H}BH\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{k}G\times
B_{k}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma\times\alpha}$$\textstyle{B_{k}(n_{1},\ldots,n_{s})\times
B_{k}(m_{1},\ldots,m_{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{EG\times_{G}BG}$

where the right horizontal arrows cover $\alpha^{\prime}$ and
$\gamma^{\prime}$, respectively, and the right vertical arrow is the induced
map of $f$ Thus taking the adjoints in the outer rectangle in the sense of
Theorem 9.2, the conditions in Theorem 6.2 are verified. Therefore, the
homomorphism $f$ is $p$-locally an $N_{k}(\ell)$-map. ∎

######  Remark 10.3.

In the proof of the theorem, the action $G\to\mathcal{A}_{\ell}(H,H)$ of the
resulting $N_{k}(\ell)$-map is null-homotopic since $E$ is trivial. Let us
propose to call such an $N_{k}(\ell)$-map a $Z_{k}(\ell)$-map, where $Z$ comes
from the German word zentral. Although it seems stronger condition than being
an $N_{k}(\ell)$-map, we have no example at this point for an
$N_{k}(\ell)$-map but not a $Z_{k}(\ell)$-map between Lie groups.

Let us investigate non-normality of the inclusion
$\operatorname{SU}(m)\to\operatorname{SU}(n)$.

######  Theorem 10.4.

Suppose $\max\\{kn-m,(k-1)n+2\\}<p\leq kn+(\ell-2)m$ for some $2\leq m<n$ and
$k,\ell\geq 1$, then the inclusion
$\operatorname{SU}(m)\to\operatorname{SU}(n)$ is not $p$-locally an
$N_{k}(\ell)$-map.

When $m=2$, the requirement on $p$ is loosened as follows.

######  Theorem 10.5.

Suppose $\max\\{kn-2,(k-1)n+2\\}<p\leq kn+2(\ell-1)$ for some $n\geq 3$ and
$k,\ell\geq 1$, then the inclusion
$\operatorname{SU}(2)\to\operatorname{SU}(n)$ is not $p$-locally an
$N_{k}(\ell)$-map.

Let us give a proof of the theorems. Let $H^{\ast}$ denote the mod $p$
cohomology. Then we have

$H^{\ast}(B\operatorname{SU}(n))=\mathbb{F}_{p}[c_{2},\ldots,c_{n}],$

where $c_{i}$ denotes the $i$-th Chern class of degree $2i$. For the
projective spaces, we have

$H^{\ast}(B_{k}\operatorname{SU}(n))=\mathbb{F}_{p}[c_{2},\ldots,c_{n}]/(c_{2},\ldots,c_{n})^{k+1}\oplus
S$

as in [Hem94, Iwa84], where the first summand
$\mathbb{F}_{p}[c_{2},\ldots,c_{n}]/(c_{2},\ldots,c_{n})^{k+1}$ is the image
from $H^{\ast}(B\operatorname{SU}(n))$, the second summand $S$ is mapped to
$0$ in $H^{\ast}(B_{k-1}\operatorname{SU}(n))$ and both summands are closed
under the action of Steenrod operations.

We assume that the inclusion $\operatorname{SU}(m)\to\operatorname{SU}(n)$ is
$p$-locally an $N_{k}(\ell)$-map and derives a contradiction. As in Theorem
9.4, we have a fiberwise projective space $\mathcal{E}=\mathscr{B}_{\ell}E$
for some fiberwise $A_{\ell}$-space $E$ over $B_{k}\operatorname{SU}(n)$ with
fiber $A_{\ell}$-equivalent to $\operatorname{SU}(m)$ and fiberwise pointed
maps $\Phi\colon
E_{k}\operatorname{SU}(m)\times_{\operatorname{SU}(m)}B_{\ell}\operatorname{SU}(m)\to\mathcal{E}_{B_{k}\operatorname{SU}(m)}$
and $\Psi\colon\mathcal{E}\to
E_{k}\operatorname{SU}(n)\times_{\operatorname{SU}(n)}B\operatorname{SU}(m)$
satisfying the properties in Theorem 9.4.

######  Lemma 10.6.

We have the submodule

$\mathbb{F}_{p}[c_{2}^{B},\ldots,c_{n}^{B}]/(c_{2}^{B},\ldots,c_{n}^{B})^{k+1}\otimes\mathbb{F}_{p}\\{(c_{2}^{F})^{i_{2}}\cdots(c_{m}^{F})^{i_{m}}\mid
0\leq i_{2}+\cdots+i_{m}\leq\ell-1\\}\subset H^{\ast}(\mathcal{E})$

closed under the multiplication by
$\mathbb{F}_{p}[c_{2}^{B},\ldots,c_{n}^{B}]/(c_{2}^{B},\ldots,c_{n}^{B})^{k+1}$
and Steenrod operations, where we write $c_{j}^{B}=F^{\ast}(c_{j}\times 1)$,
$c_{j}^{F}=F^{\ast}(1\times c_{j})$, and $\mathbb{F}_{p}S$ with some set $S$
denotes the mod $p$ vector space with basis $S$. When $m=2$, we also have a
bigger submodule

$\mathbb{F}_{p}[c_{2}^{B},\ldots,c_{n}^{B}]/(c_{2}^{B},\ldots,c_{n}^{B})^{k+1}\otimes\mathbb{F}_{p}\\{(c_{2}^{F})^{i}\mid
0\leq i\leq\ell\\}\subset H^{\ast}(\mathcal{E}).$

###### Proof.

Consider the cohomology Serre spectral sequence of $\mathcal{E}\to
B_{k}\operatorname{SU}(n)$. The $E_{2}$-term is given as

$E_{2}=(\mathbb{F}_{p}[c_{2}^{B},\ldots,c_{n}^{B}]/(c_{2}^{B},\ldots,c_{n}^{B})^{k+1}\oplus
S^{B})\otimes(\mathbb{F}_{p}[c_{2}^{F},\ldots,c_{\ell}^{B}]/(c_{2}^{F},\ldots,c_{m}^{F})^{\ell+1}\oplus
S^{F})$

for some modules $S^{B}$ and $S^{F}$. The differentials must be trivial on
$\mathbb{F}_{p}[c_{2}^{F},\ldots,c_{m}^{F}]/(c_{2}^{F},\ldots,c_{n}^{F})^{\ell+1}$
since so are in the Serre spectral sequence of
$E\operatorname{SU}(n)\times_{\operatorname{conj}}B\operatorname{SU}(n)$.
Although the differentials on $S^{F}$ could be non-trivial, their image is
contained in the kernel of
$H^{\ast}(\mathcal{E})=H^{\ast}(\mathscr{B}_{\ell}E)\to
H^{\ast}(\mathscr{B}_{\ell-1}E)$ since so is $S^{F}$. This implies the
statement of the lemma for $m\geq 2$. When $m=2$, we have $S^{F}=0$. Then the
desired bigger submodule exists. ∎

We concentrate on the proof of Theorem 10.4 in the following. The only
difference of the proof of Theorem 10.5 is that we employ the second submodule
of Lemma 10.6 instead of the first one.

By the inequalities

$\displaystyle n+p-1$ $\displaystyle\geq n+((k-1)n+3)-1=kn+2,$
$\displaystyle(m+1)+p-1$ $\displaystyle\leq
m+1+(kn+(\ell-2)m)-1=kn+(\ell-1)m,$

we can find integers $0\leq\ell^{\prime}\leq\ell-1$, $m+1\leq i\leq n$ and
$2\leq j<m$ or $j=0$ satisfying

$i+p-1=kn+\ell^{\prime}m+j\quad\text{and}\quad\ell^{\prime}m+j\leq(\ell-1)m.$

By this equality, one can see the inequalities

$\displaystyle
k\leq\frac{p}{3}\quad\text{and}\quad\ell^{\prime}\leq\frac{p}{2}.$

Since the map $\Psi\colon\mathcal{E}\to
E\operatorname{SU}(n)\times_{\operatorname{SU}(n)}B\operatorname{SU}(n)\cong
B\operatorname{SU}(n)\times B\operatorname{SU}(n)$ is fiberwise pointed and
the composite $\Psi\circ\Phi\colon
E\operatorname{SU}(m)\times_{\operatorname{SU}(m)}B\operatorname{SU}(m)\to
E\operatorname{SU}(n)\times_{\operatorname{SU}(n)}B\operatorname{SU}(n)$ is
homotopic to the induced map of the homomorphism
$f\colon\operatorname{SU}(m)\to\operatorname{SU}(n)$, we have

(2) $\displaystyle\Psi^{\ast}(1\times c_{i})=c_{i}^{B}+(\text{a polynomial}),$

where the polynomial consists of terms divisible by at least one of
$c_{m+1}^{B},\ldots,c_{i-1}^{B}$. Let us compute
$\mathcal{P}^{1}(\Psi^{\ast}(1\times c_{i}))$ the image of Steenrod operation
$\mathcal{P}^{1}$.

######  Lemma 10.7.

The coefficient of $c_{j}c_{m}^{\ell^{\prime}}c_{n}^{k}$ ($c_{0}=1$ when
$j=0$) in $\mathcal{P}^{1}c_{i}\in H^{2i+2p-2}(B\operatorname{SU}(n))$ is
nonzero.

###### Proof.

Recall the mod $p$ Wu formula [Sha77]

$\displaystyle\mathcal{P}^{1}c_{i}=\sum_{2i_{2}+\cdots+ni_{n}=i+p-1}$
$\displaystyle(-1)^{i_{2}+\cdots+i_{n}-1}\frac{(i_{2}+\cdots+i_{n}-1)!}{i_{2}!\cdots
i_{n}!}$
$\displaystyle\cdot\left(i+p-1-\frac{\sum_{j=2}^{i-1}(i+p-1-j)i_{j}}{i_{2}+\cdots+i_{n}-1}\right)c_{2}^{i_{2}}\cdots
c_{n}^{i_{n}}.$

Let $\ell^{\prime\prime}=\ell^{\prime}$ if $j=0$ and
$\ell^{\prime\prime}=\ell^{\prime}+1$ otherwise. Since we have

$\displaystyle
i+p-1-\frac{\sum_{j=2}^{i-1}(i+p-1-j)i_{j}}{i_{2}+\cdots+i_{n}-1}$
$\displaystyle=kn+\ell^{\prime}m+j-\frac{(kn+\ell^{\prime}m+j)(\ell^{\prime}+1)-j-\ell^{\prime}m}{k+\ell^{\prime\prime}-1}$
$\displaystyle=\frac{k((k-1)n+\ell^{\prime}m+j)}{k+\ell^{\prime\prime}-1},$

we can compute the coefficient of $c_{j}c_{m}^{\ell^{\prime}}c_{n}^{k}$ as

$\displaystyle(-1)^{k+\ell^{\prime\prime}-1}\frac{(k+\ell^{\prime\prime}-1)!}{k!\ell^{\prime}!}\cdot\frac{k((k-1)n+\ell^{\prime}m+j)}{k+\ell^{\prime\prime}-1}$
$\displaystyle=(-1)^{k+\ell^{\prime\prime}-1}\frac{(k+\ell^{\prime\prime}-2)!}{(k-1)!\ell^{\prime}!}((k-1)n+\ell^{\prime}m+j).$

Now we have

$\displaystyle
k+\ell^{\prime\prime}-2<\frac{p}{3}+\frac{p}{2}+1-2<p\quad\text{and}\quad
0<(k-1)n+\ell^{\prime}m+j=i+p-1-n<p$

and hence the coefficient of $c_{j}c_{m}^{\ell^{\prime}}c_{n}^{k}$ is nonzero.
∎

This lemma implies that the coefficient of
$(c_{n}^{B})^{k}c_{j}^{F}(c_{m}^{F})^{\ell^{\prime}}$ in
$\mathcal{P}^{1}\Psi^{\ast}(1\times c_{i})$ is nonzero. But it cannot happen
as follows.

######  Lemma 10.8.

The coefficient of $(c_{n}^{B})^{k}c_{j}^{F}(c_{m}^{F})^{\ell^{\prime}}$
($c^{F}_{0}=1$ when $j=0$) in $\mathcal{P}^{1}\Psi^{\ast}(1\times c_{i})$ is
zero.

###### Proof.

Applying $\mathcal{P}^{1}$ to the equality (2), the term
$(c_{n}^{B})^{k}c_{j}^{F}(c_{m}^{F})^{\ell^{\prime}}$ comes from
$(\mathcal{P}^{1}c_{s}^{B})c_{j}^{F}(c_{m}^{F})^{\ell^{\prime}}$ by $s\neq n$.
Here we have $m<s<n$. Since $\mathcal{P}^{1}c_{s}^{B}$ must contain the term
$(c_{n}^{B})^{k}$, $s$ is computed as $s=kn-(p-1)$. But this contradicts to
our assumption $s>m>kn-p$ in the theorem. Thus the coefficient of
$(c_{n}^{B})^{k}c_{j}^{F}(c_{m}^{F})^{\ell^{\prime}}$ is zero. ∎

This contradicts to the previous result, completing the proof of Theorem 10.4.

Let us examine the $p$-local normality of the inclusion
$\operatorname{SU}(2)\to\operatorname{SU}(3)$ for small primes $p$. By Theorem
10.5, the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$ is not
$p$-locally an $N_{k}(\ell)$-map if $3k\leq p\leq 3k+2\ell-2$. Also, by
Theorem 10.2, the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$ is
$p$-locally an $N_{k}(\ell)$-map if $p\geq 3k+2\ell$.

When $p=2$, the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$ is not
$2$-locally an $N_{k}(\ell)$-map for any $k\geq 1$ and $\ell\geq 1$ as
mentioned before.

When $p=3$, the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$ is
$3$-locally an $N_{k}(\ell)$-map for any $k\geq 1$ and $\ell\geq 1$. But note
that the quotient $\operatorname{SU}(3)/\operatorname{SU}(2)=S^{5}$ admits a
$3$-local $H$-space structure (compare Theorem 8.2).

When $p=5$, the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$ is
$5$-locally an $N_{1}(1)$-map. But it is not $5$-locally an $N_{1}(\ell)$-map
if $\ell\geq 2$ holds. We leave the remaining case in the present work.

######  Problem 10.9.

Determine whether the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$
is $5$-locally an $N_{2}(1)$-map or not. More generally, find methods to
determine when the inclusion $\operatorname{SU}(m)\to\operatorname{SU}(n)$ is
$p$-locally an $N_{k}(1)$-map.

For the inclusion $\operatorname{SU}(2)\to\operatorname{SU}(3)$,
$E_{2}\operatorname{SU}(2)\times_{\operatorname{conj}}\operatorname{SU}(2)$ is
not $5$-locally fiberwise based equivalent to the trivial bundle though its
restriction
$E_{1}\operatorname{SU}(2)\times_{\operatorname{conj}}\operatorname{SU}(2)$
over $B_{1}\operatorname{SU}(2)$ is $5$-locally trivial.

We close this paper giving the corresponding results for
$\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$, which is $p$-locally
homotopy equivalent to $\operatorname{Sp}(m)\to\operatorname{Sp}(n)$ for odd
$p$. The proof proceeds as Theorems 10.4 and 10.5 using the mod $p$ Wu formula

$\displaystyle\mathcal{P}^{1}p_{i}=\sum_{i_{1}+\cdots+ni_{n}=i+\frac{p-1}{2}}$
$\displaystyle(-1)^{i_{1}+\cdots+i_{n}-1+i+\frac{p-1}{2}}\frac{(i_{1}+\cdots+i_{n}-1)!}{i_{1}!\cdots
i_{n}!}$
$\displaystyle\cdot\left(2i+p-1-\frac{\sum_{j=1}^{i-1}(2i+p-1-2j)i_{j}}{i_{1}+\cdots+i_{n}-1}\right)p_{1}^{i_{1}}\cdots
p_{n}^{i_{n}}$

for the Pontryagin classes $p_{1},\ldots,p_{n}$.

######  Theorem 10.10.

Suppose $\max\\{2kn-2m,2(k-1)n+2\\}<p\leq 2kn+2(\ell-2)m-1$ for some $1\leq
m<n$ and $k,\ell\geq 1$, then the inclusion
$\operatorname{SO}(2m+1)\to\operatorname{SO}(2n+1)$ is not $p$-locally an
$N_{k}(\ell)$-map.

######  Theorem 10.11.

Suppose $\max\\{2kn-4,2(k-1)n+2\\}<p\leq 2kn+2\ell-3$ for some $n\geq 2$ and
$k,\ell\geq 1$, then the inclusion
$\operatorname{SO}(3)\to\operatorname{SO}(2n+1)$ is not $p$-locally an
$N_{k}(\ell)$-map.

## References

  * [BV73] J. M. Boardman and R. M. Vogt.  Homotopy invariant algebraic structures on topological spaces.  Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin-New York, 1973.
  * [CJ98] Michael Crabb and Ioan James.  Fibrewise homotopy theory.  Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998.
  * [FS10] Emmanuel D. Farjoun and Yoav Segev.  Crossed modules as homotopy normal maps.  Topology Appl., 157(2):359–368, 2010.
  * [Fur85] Yasukuni Furukawa.  Homotopy-normality of Lie groups.  Quart. J. Math. Oxford Ser. (2), 36(141):53–56, 1985.
  * [Fur87] Yasukuni Furukawa.  Homotopy-normality of Lie groups. II.  Quart. J. Math. Oxford Ser. (2), 38(150):185–188, 1987.
  * [Fur95] Yasukuni Furukawa.  Homotopy-normality of Lie groups. III.  Hiroshima Math. J., 25(1):83–96, 1995.
  * [Hem94] Yutaka Hemmi.  On exterior $A_{n}$-spaces and modified projective spaces.  Hiroshima Math. J., 24(3):583–605, 1994.
  * [HKT19] Sho Hasui, Daisuke Kishimoto, and Mitsunobu Tsutaya.  Higher homotopy commutativity in localized Lie groups and gauge groups.  Homology Homotopy Appl., 21(1):107–128, 2019.
  * [Iwa84] Norio Iwase.  On the $K$-ring structure of $X$-projective $n$-space.  Mem. Fac. Sci. Kyushu Univ. Ser. A, 38(2):285–297, 1984.
  * [Iwa98] Norio Iwase.  Ganea’s conjecture on Lusternik-Schnirelmann category.  Bull. London Math. Soc., 30(6):623–634, 1998.
  * [Jam67] I. M. James.  On the homotopy theory of the classical groups.  An. Acad. Brasil. Ci., 39:39–44, 1967.
  * [Jam71] I. M. James.  Products between homotopy groups.  Compositio Math., 23:329–345, 1971.
  * [Jam76] I. M. James.  The topology of Stiefel manifolds.  London Mathematical Society Lecture Note Series, No. 24. Cambridge University Press, Cambridge-New York-Melbourne, 1976.
  * [Kac82] Hideyuki Kachi.  On the iterated Samelson product.  Math. J. Okayama Univ., 24(1):37–44, 1982.
  * [KK10] Daisuke Kishimoto and Akira Kono.  Splitting of gauge groups.  Trans. Amer. Math. Soc., 362(12):6715–6731, 2010.
  * [KN03] Akira Kono and Osamu Nishimura.  Homotopy normality of Lie groups and the adjoint action.  J. Math. Kyoto Univ., 43(3):641–650, 2003.
  * [KT18] Daisuke Kishimoto and Mitsunobu Tsutaya.  Samelson products in $p$-regular ${\rm SO}(2n)$ and its homotopy normality.  Glasg. Math. J., 60(1):165–174, 2018.
  * [KY98] Kenji Kudou and Nobuaki Yagita.  Modulo odd prime homotopy normality for $H$-spaces.  J. Math. Kyoto Univ., 38(4):643–651, 1998.
  * [KY01] Kenji Kudou and Nobuaki Yagita.  Note on homotopy normality and the $n$-connected fiber space.  Kyushu J. Math., 55(1):119–129, 2001.
  * [May75] J. Peter May.  Classifying spaces and fibrations.  Mem. Amer. Math. Soc., 1(1, 155):xiii+98, 1975.
  * [McC64] G. S. McCarty, Jr.  Products between homotopy groups and the $J$-morphism.  Quart. J. Math. Oxford Ser. (2), 15:362–370, 1964.
  * [Nis06] Osamu Nishimura.  A note on homotopy normality of $H$-spaces.  J. Math. Kyoto Univ., 46(4):913–921, 2006.
  * [Pre12] Matan Prezma.  Homotopy normal maps.  Algebr. Geom. Topol., 12(2):1211–1238, 2012.
  * [Sak10] Michihiro Sakai.  $A_{\infty}$-spaces and L-S category in the category of fibrewise spaces.  Topology Appl., 157(13):2131–2135, 2010.
  * [Sha77] P. Brian Shay.  ${\rm mod}$ $p$ Wu formulas for the Steenrod algebra and the Dyer-Lashof algebra.  Proc. Amer. Math. Soc., 63(2):339–347, 1977.
  * [Sta63] James Dillon Stasheff.  Homotopy associativity of $H$-spaces. I, II.  Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid., 108:293–312, 1963.
  * [Sta70] James Stasheff.  $H$-spaces from a homotopy point of view.  Lecture Notes in Mathematics, Vol. 161. Springer-Verlag, Berlin-New York, 1970.
  * [Tsu12] Mitsunobu Tsutaya.  Finiteness of $A_{n}$-equivalence types of gauge groups.  J. Lond. Math. Soc. (2), 85(1):142–164, 2012.
  * [Tsu15] Mitsunobu Tsutaya.  Homotopy pullback of $A_{n}$-spaces and its applications to $A_{n}$-types of gauge groups.  Topology Appl., 187:1–25, 2015.
  * [Tsu16] Mitsunobu Tsutaya.  Mapping spaces from projective spaces.  Homology Homotopy Appl., 18(1):173–203, 2016.