# On the LS-category of homomorphisms

Alexander Dranishnikov and Nursultan Kuanyshov  Alexander Dranishnikov,
Department of Mathematics, University of Florida, 358 Little Hall,
Gainesville, FL 32611-8105, USA<EMAIL_ADDRESS>Nursultan Kuanyshov,
Department of Mathematics, University of Florida, 358 Little Hall,
Gainesville, FL 32611-8105, USA<EMAIL_ADDRESS>
###### Abstract.

We prove the equality $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)$ for
homomorphisms $\phi:\Gamma\to\Lambda$ of a torsion free finitely generated
nilpotent groups $\Gamma$ to an arbitrary group $\Lambda$. We construct an
epimorphism $\psi:G\to H$ between geometrically finite groups with
$\operatorname{cat}(\psi)>\operatorname{cd}(\psi)$.

###### Key words and phrases:

Lusternik-Schnirelmann category, group homomorphism

###### 2000 Mathematics Subject Classification:

Primary 55M30; Secondary 55M25, 57R65, 57R67

##  1\. Introduction

######  Definition 1.1.

The (reduced) Lusternik-Schnirelmann category (LS-category),
$\operatorname{cat}X$, of an ANR space $X$ is the minimal number $k$ such that
$X$ admits an open cover by $k+1$ sets $U_{0},U_{1},\dots,U_{k}$ such that
each $U_{i}$ is contractible in $X$.

The Lusternik-Schnirelmann category is an important invariant, since it gives
a lower bound on the number of critical points for a smooth real-valued
function on a closed manifold [LS]. Since it is a homotopy invariant, it can
be defined for discrete groups $\Gamma$ as
$\operatorname{cat}\Gamma=\operatorname{cat}B\Gamma$ where
$B\Gamma=K(\Gamma,1)$ is a classifying space. Eilenberg and Ganea [EG] proved
that the LS-category of a discrete group equals its cohomological dimension,
$\operatorname{cat}(\pi)=\operatorname{cd}(\pi)$. We recall that the
cohomological dimension of a group $\Gamma$ is defined as follows,

$\operatorname{cd}(\Gamma)=\max\\{k\mid H^{k}(\Gamma,M)\neq 0\\}$

where the maximum is taken over all ${\mathbb{Z}}\Gamma$-modules $M$.

######  Theorem 1.2 ([Sch, DR]).

For the cohomological dimension of a discrete group $\Gamma$,

$\operatorname{cd}(\Gamma)=\max\\{k\mid(\beta_{\Gamma})^{k}\neq 0\\}$

where $\beta_{\Gamma}\in H^{1}(\Gamma,I(\Gamma))$ is the Berstein-Schwarz
class of $\Gamma$.

The LS-category of the map $f:X\to Y$, $\operatorname{cat}f$, is the minimal
number $k$ such that X admits an open cover by $k+1$ open sets
$U_{0},U_{2},...,U_{k}$ with nullhomotopic restrictions $f|_{U_{i}}:U_{i}\to
Y$ for all $i$. The LS-category $\operatorname{cat}\phi$ of a group
homomorphism $\phi:\Gamma\to\pi$ is defined as $\operatorname{cat}f$ where the
map $f:B\Gamma\to B\pi$ induces the homomorphism $\phi$ for the fundamental
groups.

The cohomological dimension $\operatorname{cd}(\phi)$ of a group homomorphism
$\phi:\Gamma\to\pi$ was defined by Mark Grant about 10 years ago on
Mathoverflow [Gr] as maximum of $k$ such that there is a $\pi$-module $M$ with
the nonzero induced homomorphism $\phi^{*}:H^{k}(\pi,M)\to H^{k}(\Gamma,M)$.
In view of universality of the Berstein-Schwarz class [DR] for any
homomorphism $\phi:\Gamma\to\pi$

$\operatorname{cd}(\phi)=\max\\{k\mid\phi^{*}(\beta_{\pi})^{k}\neq 0\\}.$

This brings the inequality $\operatorname{cd}(\phi)\leq\operatorname{cat}\phi$
for all homomorphisms.

In view of the eqality $\operatorname{cd}(\Gamma)=\operatorname{cat}\Gamma$,
the following conjecture seems to be natural:

######  Conjecture 1.3.

For any group homomorphism $\phi:\Gamma\to\pi$ always

$\operatorname{cat}\phi=\operatorname{cd}(\phi).$

In [Sc] Jamie Scott considered this conjecture for geometrically finite groups
and he proved it for monomorphisms of any groups and for homomorphisms of free
and free abelian groups. In [Gr] Tom Goodwillie gave an example of an
epimorphism of an infinitely generated group $\phi:G\to{\mathbb{Z}}^{2}$ with
$cd(\phi)=1$ that disproves the conjecture.

In the first part of this paper we prove Conjecture 1.3 for finitely generated
torsion free nilpotent groups. In the second part we present a finitely
generated counterexample by constructing a map between aspherical manifolds
$f:M\to N$ with

$\operatorname{cat}f>\operatorname{cd}(f_{\\#}:\pi_{1}(M)\to\pi_{1}(N)).$

##  2\. Preliminaries

###  2.1. Nilpotent groups

The upper central series of a group $\Gamma$ is a chain of subgroups

${e}=Z_{0}\leqslant Z_{1}\leqslant....\leqslant Z_{n}\leqslant....$

where $Z_{1}=Z(\Gamma)$ is the center of the group, and $Z_{i+1}$ is the
preimage under the canonical epimorphism $\Gamma\to\Gamma/Z_{i}$ of the center
of $\Gamma/Z_{i}$. A group $\Gamma$ is nilpotent if $Z_{n}=\Gamma$ for some
$n$. The least such $n$ is called the nilpotency class of $\Gamma$, denoted
$\operatorname{nil}(\Gamma)$. Note that the groups with the nilpotency class
one are exactly abelian groups.

The lower central series of a group $\Gamma$ is a chain of subgroups

$\Gamma=\gamma_{0}(\Gamma)\geq\gamma_{1}(\Gamma)\geq\gamma_{2}(\Gamma)\geq...$

defined as $\gamma_{i+1}(\Gamma)=[\gamma_{i}(\Gamma),\Gamma]$. It’s known that
for nilpotent groups $\Gamma$ the nilpotency class
$\operatorname{nil}(\Gamma)$ equals the least $n$ for which
$\gamma_{n}(\Gamma)=1$.

######  Proposition 2.1.

(1) Let $\phi:\Gamma\rightarrow\Gamma^{\prime}$ be an epimorphism. Then
$\phi(Z(\Gamma))\subset Z(\Gamma^{\prime})$ and
$\phi(\gamma_{i}(\Gamma))=\gamma_{i}(\Gamma^{\prime})$ for all $i$.

(2) For any finitely generated torsion free nilpotent group $\Gamma$, any
$z\in\Gamma$, and any $n\in{\mathbb{N}}$ the condition $z^{n}\in Z(\Gamma)$
implies $z\in Z(\Gamma)$.

###### Proof.

(1) Straightforward (see for example [B], Theorem 5.1.3).

(2) This is Mal’cev’s Theorem 1 in [Ma2]. We note that the proof there is not
selfcontained. This statement follows from the fact that $z$ and $z^{n}$ have
the same centralizers. The latter can be proven using Mal’cev theorem about
embedding $\Gamma$ into the group of unipotent upper triangular matrices [Ra]
and the fact that $z$ belongs to Zariski closure of $z^{n}$. ∎

######  Corollary 2.2.

For any finitely generated torsion free nilpotent group $\Gamma$ the group
$\Gamma/Z(\Gamma)$ is torsion free finitely generated nilpotent group.

###### Proof.

The torsion free part follows from (2). The rest follows from (1). ∎

We note that
$\operatorname{nil}(\Gamma/Z(\Gamma))<\operatorname{nil}(\Gamma)$.

Suppose that $G$ is a connected, simply connected nilpotent Lie group and
$\Gamma\subset G$ is a uniform lattice. Then $G$ is the universal cover for
$\Gamma$ and $N=G/\Gamma$ is an aspherical manifold, called a nilmanifold.

Mal’cev Theorem. [Ma1] Every torsion free finitely generated nilpotent group
$\Gamma$ can be realized as the fundamental group of some nilmanifold.

The corresponding simply connected Lie groups $G$ is obtained as the Mal’cev
completion of $\Gamma$.

###  2.2. Ganea-Schwarz’s approach to cat

Recall that an element of an iterated join $X_{0}*X_{1}*\cdots*X_{n}$ of
topological spaces is a formal linear combination
$t_{0}x_{0}+\cdots+t_{n}x_{n}$ of points $x_{i}\in X_{i}$ with $\sum t_{i}=1$,
$t_{i}\geq 0$, in which all terms of the form $0x_{i}$ are dropped. Given
fibrations $f_{i}:X_{i}\to Y$ for $i=0,...,n$, the fiberwise join of spaces
$X_{0},...,X_{n}$ is defined to be the space

$X_{0}*_{Y}X_{1}*_{Y}\cdots*_{Y}X_{n}=\\{\ t_{0}x_{0}+\cdots+t_{n}x_{n}\in
X_{0}*\cdots*X_{n}\ |\ f_{0}(x_{0})=\cdots=f_{n}(x_{n})\ \\}.$

The fiberwise join of fibrations $f_{0},...,f_{n}$ is the fibration

$f_{0}*_{Y}*\cdots*_{Y}f_{n}:X_{0}*_{Y}X_{1}*_{Y}\cdots*_{Y}X_{n}\longrightarrow
Y$

defined by taking a point $t_{0}x_{0}+\cdots+t_{n}x_{n}$ to $f_{i}(x_{i})$ for
any $i$ such that $t_{i}\neq 0$.

When $X_{i}=X$ and $f_{i}=f:X\to Y$ for all $i$ the fiberwise join of spaces
is denoted by $*^{n+1}_{Y}X$ and the fiberwise join of fibrations is denoted
by $*_{Y}^{n+1}f$.

For a path connected space $X$, we turn an inclusion of a point $*\to X$ into
a fibration $p^{X}_{0}:G_{0}(X)\to X$, whose fiber is known to be the loop
space $\Omega X$. The $n$-th Ganea space of $X$ is defined to be the space
$G_{n}(X)=*_{X}^{n+1}G_{0}(X)$, while the $n$-th Ganea fibration
$p^{X}_{n}:G_{n}(X)\to X$ is the fiberwise join $\ast^{n+1}_{X}p^{X}_{0}$.
Then the fiber of $p^{X}_{n}$ is $\ast^{n+1}\Omega X$.

The following theorem give the Ganea-Shwarz characterization of
$\operatorname{cat}$ [Sch],[CLOT]:

######  Theorem 2.3.

If $X$ is a connected ANR, then $\operatorname{cat}X\leq n$ if and only if the
fibration $p^{X}_{n}:G_{n}(X)\to X$ admits a section.

This characterization can be extendend to maps:

######  Theorem 2.4.

If $f:X\to Y$ is a map between connected ANRs, then $\operatorname{cat}f\leq
n$ if and only if there is a lift of $f$ with respect to the fibration
$p^{Y}_{n}:G_{n}(Y)\to Y$ admits a section.

###  2.3. Berstein-Schwarz cohomology class

The Berstein-Schwarz class of a discrete group $\pi$ is the first obstruction
$\beta_{\pi}$ to a lift of $B\pi=K(\pi,1)$ to the universal covering $E\pi$.
Note that $\beta_{\pi}\in H^{1}(\pi,I(\pi))$ where $I(\pi)$ is the
augmentation ideal of the group ring ${\mathbb{Z}}\pi$ [Be],[Sch].

######  Theorem 2.5 (Universality [DR],[Sch]).

For any cohomology class $\alpha\in H^{k}(\pi,L)$ there is a homomorphism of
$\pi$-modules $I(\pi)^{k}\to L$ such that the induced homomorphism for
cohomology takes $(\beta_{\pi})^{k}\in H^{k}(\pi;I(\pi)^{k})$ to $\alpha$
where $I(\pi)^{k}=I(\pi)\otimes\dots\otimes I(\pi)$ and
$(\beta_{\pi})^{k}=\beta_{\pi}\smile\dots\smile\beta_{\pi}$.

In the paper we use notations $H^{*}(\Gamma,A)$ for cohomology of a group
$\Gamma$ with coefficient in $\Gamma$-module $A$. The cohomology groups of a
space $X$ with the fundamental group $\Gamma$ we denote as $H^{*}(X;A)$. Thus,
$H^{*}(\Gamma,A)=H^{*}(B\Gamma;A)$ where $B\Gamma=K(\Gamma,1)$.

##  3\. Reduction to epimorphisms

######  Lemma 3.1.

Let $\pi\subset\Lambda$ be a subgroup, $j:B\pi\to B\Lambda$ be a map generated
by this inclusion, and $p^{\prime}:j^{*}E\Lambda\to B\pi$ be the pull-back of
the universal covering $p_{\Lambda}:E\Lambda\to B\Lambda$. Then $p^{\prime}$
has a lift with respect to the universal covering $p_{\pi}:E\pi\to B\pi$.

###### Proof.

Clearly, for each path component $C$ of $j^{*}E\Lambda$ the restriction
$p^{\prime}|_{C}:C\to B\pi$ of a covering $p^{\prime}$ is a covering. Easy
diagram chasing shows that $C$ is simply connected. Hence, each path component
$C$ of $j^{*}E\Lambda$ is homeomorphic to $E\pi$ and the restriction
$p^{\prime}|_{C}:C\to B\pi$ is the universal covering. We may assume that
$j^{*}E\Lambda$ is a CW complex. This would imply that each path component $C$
is open. Thus, the lift $j^{*}E\Lambda\to E\pi$ can be defined independently
on each path component. ∎

Given a homomorphism $\phi:\Gamma\to\Lambda$, by $\phi^{\prime}:\Gamma\to
im(\phi)$ we denote the restriction of $\phi$ from the codomain to its range.
The following theorem is a generalization of Theorem 6.11 from [Sc].

######  Theorem 3.2.

For any group homomorphism $\phi:\Gamma\to\Lambda$,
$\operatorname{cat}\phi=\operatorname{cat}\phi^{\prime}$ .

###### Proof.

Clearly, $\operatorname{cat}\phi\leq\operatorname{cat}\phi^{\prime}$. We show
that $\operatorname{cat}\phi\geq\operatorname{cat}\phi^{\prime}$. Let
$\operatorname{cat}\phi=k$. This means that there is a lift of the
corresponding map $f:B\Gamma\to B\Lambda$ with respect to the Ganea fibration
$p_{k}:G_{k}(B\Lambda)\to B\Lambda$. Since the path fibration
$p:P_{0}(B\pi)\to B\pi$ is fiber-wise homotopy equivalent to the universal
covering $E\pi\to B\pi$ for any discrete group $\pi$, the $k$-th Ganea
fibration $G_{k}(B\pi)\to B\pi$ is fiberwise homotopy equivalent to the
iterated fiberwise join $*^{k+1}_{B\pi}E\pi$ of the universal covering. Then
$f$ admits a lift with respect to

$*^{k+1}_{B\Lambda}p_{\Lambda}:*^{k+1}_{B\Lambda}E\Lambda\to B\Lambda.$

The map $f$ factors as $f=j\circ f^{\prime}$ where $f^{\prime}:B\Gamma\to
B\pi$, $j:B\pi\to B\Lambda$ and $\pi=im(\phi)$. Hence, the map $f^{\prime}$
admits a lift to the pull-back

$\bar{f}^{\prime}:B\Gamma\to j^{*}(*^{k+1}_{B\Lambda}E\Lambda)$

By Lemma 3.1 there is a lift $s$ of
$*^{k+1}_{B\pi}p^{\prime}:j^{*}(*^{k+1}_{B\Lambda}E\Lambda)\to B\pi$ with
respect to

$*^{k+1}_{B\pi}p_{\pi}:*^{k+1}_{B\pi}E\pi\to B\pi.$

Then the composition $s\circ\bar{f}^{\prime}$ is a lift of $f^{\prime}$ with
respect to $*^{k+1}_{B\pi}p_{\pi}$. By Theorem 2.4
$\operatorname{cat}f^{\prime}\leq k$ and, hence,
$\operatorname{cat}\phi^{\prime}\leq k$. ∎

######  Theorem 3.3.

For any group homomorphism $\phi:\Gamma\to\Lambda$ and
$\phi^{\prime}:\Gamma\to im(\phi)=\pi$ as above

$\operatorname{cd}(\phi)=\operatorname{cd}(\phi^{\prime}).$

###### Proof.

Clearly, $\operatorname{cd}(\phi^{\prime})\geq\operatorname{cd}(\phi)$. We
show that $\operatorname{cd}(\phi^{\prime})\leq\operatorname{cd}(\phi)$. Let
$\operatorname{cd}(\phi^{\prime})=k$. Then $\phi^{\prime*}:H^{k}(\pi,M)\to
H^{k}(\Gamma,M)$ is not zero for some $\pi$-module $M$. Let

$\alpha:\operatorname{Coind}^{\Lambda}_{\pi}M=Hom_{\pi}({\mathbb{Z}}\Lambda,M)\to
M$

denote the canonical $\pi$-homomorphism, defined for $f:\Lambda\to M$ as
$\alpha(f)=f(1)$. Consider the commutative diagram

$\begin{CD}H^{k}(\Lambda,\operatorname{Coind}^{\Lambda}_{\pi}M)@>{j^{*}}>{}>H^{k}(\pi,\operatorname{Coind}^{\Lambda}_{\pi}M)@>{\alpha^{*}}>{}>H^{k}(\pi,M)\\\
@V{\phi^{\prime*}}V{}V@V{\phi^{\prime*}}V{}V\\\
H^{k}(\Gamma,\operatorname{Coind}^{\Lambda}_{\pi}M)@>{\alpha^{*}}>{}>H^{k}(\Gamma,M)\\\
\end{CD}$

where $\alpha^{*}$ is the coefficient homomorphism generated by $\alpha$. By
Shapiro Lemma [Br] the top row through homomorphism is an isomorphism.
Therefore, the homomorphism

$\phi^{*}=\phi^{\prime*}j^{*}:H^{k}(\Lambda,\operatorname{Coind}^{\Lambda}_{\pi}M)\to
H^{k}(\Gamma,\operatorname{Coind}^{\Lambda}_{\pi}M)$

is nonzero. Hence, $\operatorname{cd}(\phi)\geq k$. ∎

Theorem 3.2 and Theorem 3.3 imply the following:

######  Corollary 3.4.

Suppose that Conjecture 1.3 holds true for all epimorphisms
$\phi:\Gamma\to\pi$ for some class of groups. Then it holds for all
homomorphisms for groups from that class.

##  4\. Homomorphisms of nilpotent groups

######  Lemma 4.1.

Let $\Gamma$ be $\pi$ are finitely generated, torsion free nilpotent groups.
Then every epimorphism $\phi:\Gamma\rightarrow\pi$ can be realized as a
locally trivial bundle of nilmanifolds with the fiber a nilmanifold.

###### Proof.

We prove it by induction on $n=s+t$ where $s=\operatorname{nil}(\Gamma)$ and
$t=\operatorname{nil}(\pi)$ are the nilpotency classes of $\Gamma$ and $\pi$.
The base of induction is the case of abelian groups where any epimorphism
$\phi:{\mathbb{Z}}^{k+\ell}\to{\mathbb{Z}}^{k}$ is the projection onto a
factor, since it is a split surjection. Clearly, the projection $\phi$ can be
realized as a trivial fiber bundle of tori $T^{k+\ell}\to T^{k}$ with the
fiber a torus $T^{\ell}$.

Suppose that $n>2$ and the statement of the lemma holds true for $s+t<n$. We
denote by $B=Z(\Gamma)\cap\operatorname{Ker}\phi$ and by $A=Z(\Gamma)/B$,
where $Z(\Gamma)$ is the center of $\Gamma$. Note that $B$ is a direct summand
in $Z(\Gamma)$ or, equivalently, $A$ is free. For the later we claim that if
$z^{n}\in B$, then $z\in B$. Indeed, If $z^{n}\in\operatorname{Ker}\phi$ then
$z\in\operatorname{Ker}\phi$, since $\pi$ is torsion free. If $z^{n}\in
Z(\Gamma)$ then $z\in Z(\Gamma)$ by Proposition 2.1. Thus, $Z(\Gamma)\cong
B\oplus A$. Let $\bar{A}$ be the direct summand of $Z(\pi)$ that contains
$\phi(Z(\Gamma))\cong A$ as a finite index subgroup. In view of Corollary 2.2,
$\pi/\bar{A}$ is a torsion free nilpotent group.

We consider the nilpotent group $\Gamma^{\prime}=\Gamma/B$ and the epimorphism
$\phi^{\prime}:\Gamma^{\prime}\to\pi$. induced by $\phi$. In view of the
principal fiber bundle $B\Gamma\to B\Gamma^{\prime}$ with the fiber a torus,
it suffices to show that $\phi^{\prime}$ can be realized as a fiber bundle of
nilmanifolds. We consider the commutative diagram

$\begin{CD}\Gamma^{\prime}@>{}>{}>\Gamma^{\prime}/A\\\
@V{\phi^{\prime}}V{}V@V{\bar{\phi}}V{}V\\\ \pi @>{}>{}>\pi/\bar{A}.\\\
\end{CD}$

Note that $\Gamma^{\prime}/A=\Gamma/Z(\Gamma)$ and, hence, $\bar{\phi}$ is an
epimorphism with
$\operatorname{nil}(\Gamma^{\prime}/A)+\operatorname{nil}(\pi/\bar{A})<s+t.$
The homomorphism $\phi^{\prime}$ factors through the pull-back
$\Lambda=\bar{\phi}^{*}\pi$, $\phi^{\prime}=\bar{\phi}^{\prime}\circ\xi$ with
respect to $\bar{\phi}$. Since $\phi^{\prime}|_{A}:A\to\bar{A}$ is an
embedding of a finite index subgroup, the homomoprhism
$\xi:\Gamma^{\prime}\to\Lambda$ in the commutative diagram of short exact
sequences generated by $\phi^{\prime}$ and the pull-back

$\setcounter{MaxMatrixCols}{11}\begin{CD}1@>{}>{}>A@>{}>{}>\Gamma^{\prime}@>{}>{}>\Gamma/Z(\Gamma)@>{}>{}>1\\\
@V{}V{\subset}V@V{\xi}V{\subset}V@V{}V{=}V\\\ 1@>{}>{}>\bar{A}@>{}>{}>\Lambda
@>{}>{}>\Gamma/Z(\Gamma)@>{}>{}>1\\\
@V{}V{=}V@V{\bar{\phi}^{\prime}}V{}V@V{\bar{\phi}}V{}V\\\
1@>{}>{}>\bar{A}@>{}>{}>\pi @>{}>{}>\pi/\bar{A}@>{}>{}>1\\\ \end{CD}$

is an embedding of a finite index subgroup.

By induction assumption applied to $\bar{\phi}$ there is a fiber bundle
between nilmanifolds

$\bar{f}:B(\Gamma/Z(\Gamma))\to B(\pi/\bar{A})$

with the fiber a nilmanifold $F$. Consider the pull-back diagram

$\begin{CD}B\Lambda @>{p^{\prime}}>{}>B(\Gamma/Z(\Gamma))\\\
@V{\bar{f}^{\prime}}V{}V@V{\bar{f}}V{}V\\\ B\pi @>{p}>{}>B(\pi/\bar{A}).\\\
\end{CD}$

Let $q:M\to B\Lambda$ be a covering that corresponds to the subgroup
$\xi(\Gamma^{\prime})\subset\Lambda=\pi_{1}(B\Lambda)$. Since
$\xi(\Gamma^{\prime})$ is of finite index, $M$ is a closed aspherical manifold
with $\pi_{1}(M)=\Gamma^{\prime}$. Thus, the composition
$f^{\prime}=\bar{f}^{\prime}\circ q:M\to B\pi$ realizes the homomorphism
$\phi^{\prime}$ as a fiber bundle. The fiber $F^{\prime}$ of $f^{\prime}$ is
homeomorphic to the total space of a fiber bundle $F^{\prime}\to F$ with a
finite fiber. The homotopy exact sequence of the fibration $F^{\prime}\to
B\Gamma^{\prime}\stackrel{{\scriptstyle f^{\prime}}}{{\to}}B\pi$ and the fact
that $\phi^{\prime}$ is surjective imply that $F^{\prime}$ is connected.
Hence, $F^{\prime}$ is a nilmanifold. ∎

######  Lemma 4.2.

For every locally trivial bundle of closed aspherical manifolds
$f:M^{m}\rightarrow N^{n}$ with compact connected fiber $F$ the induced
homomorphism

$f^{*}:H^{n}(N;{\mathbb{Z}}\pi)\rightarrow H^{n}(M;{\mathbb{Z}}\pi)$

is nonzero where $\pi=\pi_{1}(N)$.

###### Proof.

Consider the pull-back diagram:

${\widetilde{M}}$${f^{*}{\widetilde{N}}}$${\widetilde{N}}$${M}$${N}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{M}}$$\scriptstyle{p_{1}}$$\scriptstyle{\bar{f}}$$\scriptstyle{p_{N}}$$\scriptstyle{f}$

where $p_{M}$ and $p_{N}$ are he universal covering maps. Since
$f^{*}(\widetilde{N})$ is a covering of $M$ and $\widetilde{M}$ is the
universal covering, $p_{M}$ factors through covering maps $p_{2}$ and $p_{1}$.

We recall [Br, Lemma 7.4] that for every left $\pi$-module $M$ there is a
natural isomorphism of right modules

$\Theta:Hom_{\pi}(M,{\mathbb{Z}}\pi)\to Hom_{c}(M,{\mathbb{Z}})$

defined by the formula $F\mapsto f_{1}$ where $F:M\to{\mathbb{Z}}\pi$,
$F(m)=\sum_{\gamma\in\pi}f_{\gamma}(m)\gamma$. We consider the following
commutative diagrams of cochain groups:

${Hom_{\Gamma}(C_{*}(\widetilde{M}),{\mathbb{Z}}\pi)}$${Hom_{\pi}(C_{*}(X),{\mathbb{Z}}\pi)}$${Hom_{c}(C_{*}(X),{\mathbb{Z}})}$${Hom_{\pi}(C_{*}(\widetilde{N}),{\mathbb{Z}}\pi)}$${Hom_{c}(C_{*}(\widetilde{N}),{\mathbb{Z}})}$$\scriptstyle{\Psi}$$\scriptstyle{\Theta}$$\scriptstyle{p_{2}^{*}}$$\scriptstyle{\Theta}$$\scriptstyle{(\bar{f})^{\ast}}$$\scriptstyle{(\bar{f}_{c})^{\ast}}$

where $X=f^{*}\widetilde{N}$, $\Psi=\Theta\circ\Phi$, and

$\Phi:Hom_{\Gamma}(C_{*}(\widetilde{M}),{\mathbb{Z}}\pi)\to
Hom_{\pi}(C_{*}(X),{\mathbb{Z}}\pi)$

is defined as follows. For each simplex $\sigma$ in $X$ we fix a lift
$\tilde{\sigma}$ in $\widetilde{M}$. We define

$\Phi(F)(\sigma)=F(\widetilde{\sigma})$

for $F\in Hom_{\Gamma}(C_{*}(\widetilde{M}),{\mathbb{Z}}\pi)$ and simplex
$\sigma$ in $X$ .

Claim 1. $\Phi(F)\in Hom_{\pi}(C_{*}(X),{\mathbb{Z}}\pi)$.

Proof. Since the fiber $F$ is connected, it follows that $f$ induces an
epimorphism of the fundamental groups. Let $g\in\pi$ and let
$f_{*}(\bar{g})=g$. Then $\widetilde{g\sigma}$ and $\bar{g}\tilde{\sigma}$
both cover $g\sigma$. Hence there is $\gamma\in Kerf_{*}$ such that
$\gamma(\widetilde{g\sigma})=\bar{g}\tilde{\sigma}$. Then $\Phi$ is
$\pi$-equivariant:

$\Phi(F)(g\sigma)=F(\widetilde{g\sigma})=F(\gamma\widetilde{g\sigma})=F(\bar{g}\tilde{\sigma})=gF(\tilde{\sigma})=g\Phi(F)(\sigma).$

Claim 2. $\Phi\circ p_{2}^{*}=1$.

Proof. $\Phi\circ
p_{2}^{*}(F^{\prime})(\sigma)=\Phi(p_{2}^{*}(F^{\prime}))(\sigma)=(p_{2}^{*}F^{\prime})(\tilde{\sigma})=F^{\prime}(p_{2}(\tilde{\sigma}))=F^{\prime}(\sigma)$.

Since $\widetilde{N}$ is contractible, the bottom row in the diagram above
gives us an isomorphism of the cohomology groups
$\Theta^{*}:H^{*}(N;{\mathbb{Z}}\pi)\to
H^{*}_{c}(\widetilde{N};{\mathbb{Z}})$. The commutative diagram on the cochain
level produces the commutative diagram for cohomology:

${H^{n}(M;{\mathbb{Z}}\pi)}$${H_{c}^{n}(X;{\mathbb{Z}})}$${H^{n}(N;{\mathbb{Z}}\pi)}$${H_{c}^{n}(\widetilde{N};{\mathbb{Z}})={\mathbb{Z}}.}$$\scriptstyle{\Psi^{*}}$$\scriptstyle{\Theta^{*}}$$\scriptstyle{f_{c}^{*}}$$\scriptstyle{(\bar{f})^{*}}$

Since $\widetilde{N}$ is contractible, $X=f^{*}(\widetilde{N})$ is a trivial
fiber bundle with the fiber $F$. Since $X\cong\widetilde{N}\times F$ admits a
proper retraction onto $\widetilde{N}$, we obtain that $(\bar{f_{c}})^{*}$ is
a monomorphism. Hence, the homomorphism
$f^{*}:{\mathbb{Z}}=H^{n}(N;{\mathbb{Z}}\pi)\to H^{n}(M;{\mathbb{Z}}\pi)$ is
not trivial. ∎

######  Theorem 4.3.

For a homomorphism $\phi:\Gamma\to\pi$ of finitely generated torsion free
nilpotent groups $\operatorname{cat}\phi=\operatorname{cd}(\phi)$.

###### Proof.

In view of Theorem 3.2 and Theorem 3.3, it suffices to prove the theorem when
$\phi$ is surjective. By Lemma 4.1 there is a fiber bundle of nilmanifolds
$M^{m}=B\Gamma\to B\pi=N^{n}$ with a compact fiber. By Lemma 4.2,
$\operatorname{cd}(\phi)=n$. The inequalities
$\operatorname{cd}(\phi)\leq\operatorname{cat}\phi\leq\dim N=n$ complete the
proof. ∎

##  5\. Counterexample

Answering a question of M. Grant, T. Goodwillie gave an example of an
epimorphism $\phi:G\to\Gamma$ satisfying the inequality
$\operatorname{cd}(\phi)<\min\\{\operatorname{cd}(G),\operatorname{cd}(\Gamma)\\}.$
We present here a slightly modified version of his example. Let
$\Gamma={\mathbb{Z}}^{2}$ and $\phi:G\to\Gamma$ be the epimorphism defined by
the extension of $\Gamma$ by the abelian group
$I(\Gamma)^{2}=I(\Gamma)\otimes_{\mathbb{Z}}I(\Gamma)$ that corresponds to the
square of the Berstein-Schwarz cohomology class $(\beta_{\Gamma})^{2}\in
H^{2}(\Gamma,I(\Gamma)^{2})$. Then $\phi^{*}(\beta_{\Gamma})^{2}=0$. Hence,
$\operatorname{cd}(\phi)<2$. If $\operatorname{cat}\phi\leq 1$, then the
induced map $\hat{\phi}:BG\to B\Gamma$ can be lifted to the Ganea’s space
$G_{1}(B\Gamma)$ which is homotopy equivalent to 1-dimensional complex
$\Sigma\Gamma$, the suspension of $\Gamma$. [CLOT]. Then the epimorphism
$\phi$ factors through a free group $F$ via epimorphisms $G\to F\to\Gamma$.
Hence $rankF\geq 2$. Therefore, $G$ contains a free group of rank $\geq 2$. We
note that $G$ is amenable as an abelian-by-abelian extension and hence $G$
cannot contain $F$. Therefore, $\operatorname{cat}\phi\geq 2$. Thus, $\phi$ is
an infinitely generated counterexample to Conjecture 1.3.

######  Remark 5.1.

We note there are simpler examples of epimorphisms $\phi:G\to\Gamma$
satisfying

$\operatorname{cd}(\phi)<\min\\{\operatorname{cd}(G),\operatorname{cd}(\Gamma)\\}.$

Namely, the homomorphism $\phi:F_{n}\ast{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{n}$
defined by the abelianization of the first factor and by the projection onto
${\mathbb{Z}}^{n}\to{\mathbb{Z}}\subset{\mathbb{Z}}^{n}$ has
$\operatorname{cd}(\phi)=1$ and
$\operatorname{cd}(F^{n}\ast{\mathbb{Z}}^{n})=\operatorname{cd}{\mathbb{Z}}^{n}=n$.
Here $F_{n}$ denotes the free group on $n$ generators.

Let ${\mathbb{S}}$ denote the sphere spectrum. We recall that every stably
parallelizable manifold is ${\mathbb{S}}$-orientable [Sw]. The
${\mathbb{S}}$-cohomology groups of $X$ are exactly the stable cohomotopy
groups $\pi^{*}_{s}(X)$. Then Lemma 3.5 of [Ru] in the case of the spectrum
${\mathbb{S}}$ can be stated as follows:

######  Lemma 5.2.

Suppose that $f:W\to M$ is a map of degree one between closed stably
paralleliable manifolds. Then the induced map
$f^{*}:\pi^{*}_{s}(M)\to\pi_{s}^{*}(W)$ is injective.

We note that the natural map $[X,S^{n}]\to\pi^{n}_{s}(X)$ is a bijection when
$\dim X\leq 2n-2$ [Hu].

###  5.1. Bolotov’s example

Answering a question of Gromov [Gro1], D. Bolotov constructed [Bo] a closed
4-manifold $M$ with the fundamental group
$\pi={\mathbb{Z}}\ast{\mathbb{Z}}^{3}$ whose cohomological dimension
$\operatorname{cd}(\pi)=3$ such that a classifying map $u_{M}:M\to B\pi$
cannot be deformed to the 2-skeleton $B\pi^{(2)}$. His manifold is defined as
the pull-back $M=g^{*}(S^{3}\times S^{1})$ of the $S^{1}$-bundle $h\times
1:S^{3}\times S^{1}\to S^{2}\times S^{1}$, where $h$ is the Hopf fiber bundle,
with respect to the collapsing map $g:N=(S^{2}\times S^{1})\\#T^{3}\to
S^{2}\times S^{1}$. Here $T^{3}$ is a 3-dimensional torus. The pull-back
bundle is denoted by $p:M\to N$.

######  Proposition 5.3.

Bolotov’s manifold has the following properties:

(a) The map $p$ induces an isomorphism of the fundamental groups.

(b) The homomorphism $u_{M}^{*}:H^{3}(B\pi;A)\to H^{3}(M;A)$ is trivial for
any $\pi$-module $A$.

(c) The through map $M\stackrel{{\scriptstyle
u_{M}}}{{\rightarrow}}B\pi=S^{1}\vee T^{3}\stackrel{{\scriptstyle
q_{2}}}{{\to}}T^{3}\stackrel{{\scriptstyle q_{1}}}{{\rightarrow}}S^{3}$ is
essential where $q_{2}$ collapses $S^{1}$ to the wedge point, and $q_{1}$ is a
map of degree one.

(d) The manifold $M$ is stably parallelizable.

###### Proof.

(a) This is straightforward ([Bo]). Thus, the classifying map $u_{M}$ factors
through $p:M\to N$, $u_{M}=q_{3}q_{4}p$ where $q_{4}:N\to(S^{2}\times
S^{1})\vee T^{3}$ collapses the connected sum to the wedge and
$q_{3}:(S^{2}\times S^{1})\vee T^{3}\to S^{1}\vee T^{3}$ projects $S^{2}\times
S^{1}$ onto the factor $S^{1}$.

(b) Let $a\in H^{3}(B\pi;A)$. By the Poincare Duality for local coefficients
there is 1-dimensional class $\bar{\beta}$ in $M$ such that
$u_{M}^{*}(a)\cup\bar{\beta}\neq 0$. In view of universality of the Berstein-
Schwarz class $\beta_{\pi}$ and the fact that $u_{M}$ induces an isomorphism
of 1-cohomology, we may assume that $\bar{\beta}=u_{M}^{*}(\beta_{\pi})$. Then
$0\neq u_{M}^{*}(a)\cup\bar{\beta}=u_{M}^{*}(a\cup\beta_{\pi})=0$. The last
equality is due to dimensional reason.

(c) This was proven in [Bo]. Here we present a simplified proof. Let
$q=q_{1}q_{2}q_{3}q_{4}:N\to S^{3}$:

$N=(S^{2}\times S^{1})\\#T^{3}\stackrel{{\scriptstyle
q_{4}}}{{\to}}(S^{2}\times S^{1})\vee T^{3}\stackrel{{\scriptstyle
q_{3}}}{{\to}}S^{1}\vee T^{3}\stackrel{{\scriptstyle
q_{2}}}{{\to}}T^{3}\stackrel{{\scriptstyle q_{1}}}{{\to}}S^{3}.$

Let $a$ be a generator in $H^{3}(K({\mathbb{Z}},3))$ and let $a_{0}$ be its
restriction to $S^{3}=K({\mathbb{Z}},3)^{(4)}$. Part (b) implies that
$p^{*}(\alpha_{0})=0$ where $\alpha_{0}=q^{*}(a_{0})$. The exact sequence of
pair $(C_{p},N)$ implies $(j^{*})^{-1}(\alpha_{0})\neq\emptyset$ where $j:N\to
C_{p}$ is the inclusion of the codomain of $p$ into the mapping cone. It
suffices to show that $q:N\to S^{3}$ does not extend to $C_{p}$. Every
extension $\psi^{\prime}$ of $q$ to the 4-skeleton $C_{p}^{(4)}$ can be
extended to a map $\psi:C_{p}\to K({\mathbb{Z}},3)$ defining an element
$\alpha\in(j^{*})^{-1}(\alpha_{0})$. We show that such $\psi$ cannot be
deformed to $S^{3}\subset K({\mathbb{Z}},3)$. Here we assume that
$K({\mathbb{Z}},3)^{(5)}=S^{3}\cup_{\nu}D^{5}$.

For a cohomology class $x$ we denote by $\bar{x}$ its mod 2 reduction. Since
$q$ is a map of degree one, $\bar{\alpha}_{0}\neq 0$. We recall that the
obstruction to retraction of $S^{3}\cup_{\nu}D^{5}$ to $S^{3}$ is the Steenrod
square $Sq^{2}\bar{a}$. By the naturality of primary obstructions the
obstruction to deform the map $\psi:C_{p}\to K({\mathbb{Z}},3)$ to $S^{3}$ is
$\psi^{*}(Sq^{2}\bar{a})=Sq^{2}\bar{\alpha}$. We show that
$Sq^{2}\bar{\alpha}\neq 0$. Let $E$ be the disk bundle associated with the
circle budle $p$. For the use of cohomology we can identify $N$ with $E$. Then
$M=\partial E$ and the pair $(E,M)$ can be identified with the mapping cone
$C_{p}$. In view of the Thom isomorphism there is $x\in
H^{1}(N;{\mathbb{Z}}_{2})$ such that $x\cup u=\bar{\alpha}$ where $u$ is the
mod 2 Thom class. Since $j^{*}(\bar{\alpha})=\bar{\alpha}_{0}$ the naturality
of the cup product

$\begin{CD}H^{3}(N,\partial N;{\mathbb{Z}}_{2})\times H^{2}(N,\partial
N;{\mathbb{Z}}_{2})@>{\cup}>{}>H^{5}(N,\partial N;{\mathbb{Z}}_{2})\\\
@V{j^{*}\times 1}V{}V@V{1}V{}V\\\ H^{3}(N;{\mathbb{Z}}_{2})\times
H^{2}(N,\partial N;{\mathbb{Z}}_{2})@>{\cup}>{}>H^{5}(N,\partial
N;{\mathbb{Z}}_{2})\\\ \end{CD}$

implies that $\bar{\alpha}_{0}\cup u=\bar{\alpha}\cup u$. Then

$0\neq\bar{\alpha}_{0}\cup u=\bar{\alpha}\cup u=x\cup u\cup u=x\cup
Sq^{2}u=Sq^{2}(x\cup u)=Sq^{2}(\bar{\alpha}).$

Here we use the equality $Sg^{2}u=u\cup u$ and the Cartan formula for Steenrod
squares.

(d) The manifold $M$ is stably parallelizable as the total space of an
orientable $S^{1}$-bundle over a stably parallelizable manifold $N$. ∎

We recall that for every closed manifold $M$ there is a hyperbolization
$f:W\to M$ which a map of degree one of a closed aspherical manifold which is
surjective on the fundamental groups. Moreover, the map $f$ induces an
isomorphism between stable tangent bundles of $M$ and $W$ [DJ],[CD],[Gro2].

######  Theorem 5.4.

There is a map of a closed aspherical 4-manifold $g:W\to T^{3}$ onto a 3-torus
that induces an epimorphism of the fundamental groups
$g_{\\#}:\pi_{1}(W)\to{\mathbb{Z}}^{3}$ such that
$\operatorname{cat}g_{\\#}=3$ and $\operatorname{cd}(g_{\\#})<3$.

###### Proof.

Let $M$ be the Bolotov’s example and let $f:W\to M$ be the hyperbolization of
$M$. Let $g=q_{2}\circ u_{M}\circ f$. Since Bolotov’s example is stably
parallelizable and the hyperbolization is a tangential map, by Lemma 5.2 we
obtain that $f^{*}:\pi_{s}^{3}(M)\to\pi_{s}^{3}(W)$ is injective. By
dimensional reason $[M,S^{3}]=\pi_{s}^{3}(M)$ and $[W,S^{3}]=\pi_{s}^{3}(W)$.
Thus, since by Proposition 5.3 (c) the map $q_{1}q_{2}u_{M}:M\to S^{3}$ is
essential, the map $q_{1}\circ g:W\to S^{3}$ is essential as well. Hence, the
map $g$ cannot be deformed to the 2-skeleton. Therefore,
$\operatorname{cat}g>2$. Clearly,
$\operatorname{cat}g=\operatorname{cat}g_{\\#}=3$.

By Proposition 5.3 (a) the homomorphism $(q_{1}\circ u_{M})^{*}$ is trivial on
3-dimensional cohomology. Hence, so is $g^{*}$. This means that
$\operatorname{cd}(g_{\\#})<3$. ∎

## Acknowledgments

The first author was supported by the Simons Foundation Grant.

## References

  * [B] H. Bechtell, The theory of groups, Addison-Wesley, 1971.
  * [Be] I. Berstein, On the Lusternik-Schnirelmann category of Grassmannians. Math. Proc. Camb. Philos. Soc. 79 (1976) 129-134.
  * [Bo] D. Bolotov, Gromov’s macroscopic dimension conjecture, AGT 6 (2006), 1669-1676.
  * [Br] K. Brown, Cohomology of Groups. _Graduate Texts in Mathematics_ , 87 Springer, New York Heidelberg Berlin, 1994.
  * [CD] R. M. Charney, M. W. Davis, Strict hyperbolization, Topology 34 (1995), no.2, 329-350.
  * [CLOT] O. Cornea, G. Lupton, J. Oprea, D. Tanre,  Lusternik-Schnirelmann Category, AMS, 2003.
  * [DJ] M. W. Davis and T. Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991), no.2, 347-388.
  * [DR] A. Dranishnikov, Yu. Rudyak, On the Berstein-Svarc theorem in dimension 2. Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 407-413.
  * [EG] S. Eilenberg, T. Ganea, On the Lusternik-Schnirelmann Category of Abstract Groups. Annals of Mathematics, 65, (1957), 517-518.
  * [Gr] M. Grant,  https://mathoverflow.net/questions/89178/cohomological-dimension-of-a-homomorphism
  * [Gro1] M. Gromov  Positive curvature, macroscopic dimension, spectral gaps and higher signatures,  Functional analysis on the eve of the 21st century. Vol. II, Birkhauser, Boston, MA, 1996.
  * [Gro2] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, (1987) 75–263.
  * [Hu] S.-T. Hu, Homotopy theory, Acad. Press, 1959.
  * [LS] L. Lusternik, L. Schnirelmann, “Sur le probleme de trois geodesiques fermees sur les surfaces de genre 0”, Comptes Rendus de l’Academie des Sciences de Paris, 189: (1929) 269-271.
  * [Ma1] A. I. Mal’tsev, On a class of homogeneous spaces, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, vol. 13 (1949), no 3, 201-212.
  * [Ma2] A. I. Mal’tsev, Nilpotent groups without torsion, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, vol. 13 (1949), no 1, 9–32.
  * [Ra] M. S. Raghunathan,  Discrete subgroups of Lie groups, Springer, 1972.
  * [Ru] Yu. Rudyak, On category weight and its applications. Topology 38 (1999) no. 1, 37–55.
  * [Sch] A. Schwarz, The genus of a fibered space. Trudy Moscov. Mat. Obsc. 10, 11 (1961 and 1962), 217-272, 99-126.
  * [Sc] J. Scott, On the Topological Complexity of Maps, preprint arXiv:2011.10646 2020.
  * [Sw] R. Switzer, Algebraic ¹opologyÐHomotopy and Homology. Springer, Berlin, 1975.