# On the LS-category of homomorphism of almost nilpotent groups

Nursultan Kuanyshov  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
epimorphisms $\phi:\Gamma\to\Lambda$ between torsion-free, finitely generated
almost nilpotent groups $\Gamma$ and $\Lambda$.

In addition, we prove the equality
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)$ for homomorphisms
$\phi:\Gamma\to\Lambda$ between torsion-free, finitely generated virtually
nilpotent groups.

###### Key words and phrases:

Lusternik-Schnirelmann category, group homomorphism

###### 2020 Mathematics Subject Classification:

Primary 55M30, 20J06; Secondary 55M10, 22E20, 22E40

##  1\. Introduction

The LS-category, denoted as $\operatorname{cat}(X)$, of a topological space
$X$ is defined as the minimum number $k$ such that $X$ admits an open cover
${U_{0},U_{1},\dots,U_{k}}$ by $k+1$ contractible sets in X. This concept
provides a lower bound on the number of critical points for smooth real-valued
functions on closed manifolds [LS], [CLOT].

Since LS-category is a homotopy invariant, it can be extended to discrete
groups $\Gamma$ as $\operatorname{cat}(\Gamma)=\operatorname{cat}(B\Gamma)$,
where $B\Gamma$ is a classifying space. Computing the LS-category of spaces,
even when considering nice spaces such as manifolds, presents significant
challenges [DS]. In the case of groups in the 1950s, Eilenberg and Ganea [EG]
established the equality between LS-category and cohomological dimension,
$\operatorname{cat}(\Gamma)=\operatorname{cd}(\Gamma)$, for discrete groups
$\Gamma$.

We recall the cohomological dimension $\operatorname{cd}(\Gamma)$ of a group
$\Gamma$ is defined as the supremum of $k$ such that $H^{k}(\Gamma,M)\neq 0$,
where $M$ is a ${\mathbb{Z}}\Gamma$-module [Br]. Dranishnikov and Rudyak [DR]
showed that $\operatorname{cd}(\Gamma)=\max\\{k\mid(\beta_{\Gamma})^{k}\neq
0\\}$, where $\beta_{\Gamma}\in H^{1}(\Gamma,I(\Gamma))$ is the Berstein-
Schwarz class of $\Gamma$ [Be].

For a map $f:X\to Y$, the LS-category of the map, denoted as
$\operatorname{cat}(f)$, is the minimum number $k$ such that $X$ can be
covered by $k+1$ open sets $U_{0},U_{1},\dots,U_{k}$ with nullhomotopic
restrictions $f|_{U_{i}}:U_{i}\to Y$ for all $i$. We define the LS-category
$\operatorname{cat}(\phi)$ for a group homomorphism $\phi:\Gamma\to\Lambda$ as
$\operatorname{cat}(f)$, where the map $f:B\Gamma\to B\Lambda$ induces the
homomorphism $\phi$ on fundamental groups.

Mark Grant [Gr] introduced the cohomological dimension
$\operatorname{cd}(\phi)$ of a group homomorphism $\phi:\Gamma\to\Lambda$ as
the maximum of $k$ such that there exists a ${\mathbb{Z}}\Lambda$-module $M$
with a non-zero induced homomorphism $\phi^{:}H^{k}(\Lambda,M)\to
H^{k}(\Gamma,M)$. In light of the universality of the Berstein-Schwarz class
[DR], we have
$\operatorname{cd}(\phi)=\max\\{k\mid\phi(\beta_{\Lambda})^{k}\neq 0\\}$.
Combining this with the cup-length lower bound for LS-category, we obtain the
inequality $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ for all group
homomorphisms.

Considering the Eilenberg-Ganea equality
$\operatorname{cd}(\Gamma)=\operatorname{cat}(\Gamma)$, a natural conjecture
arises:

######  Conjecture 1.1.

For any group homomorphism $\phi:\Gamma\to\Lambda$, it is always the case that
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)$.

Jamie Scott [Sc] investigated this conjecture for geometrically finite groups
and proved it for monomorphisms of any groups, as well as for homomorphisms of
free and free abelian groups. However, Tom Goodwillie [Gr] provided a
counterexample presenting an epimorphism $\phi:G\to{\mathbb{Z}}^{2}$ from an
infinitely generated group $G$ with $\operatorname{cd}(\phi)=1$, thus
disproving the conjecture.

In our joint work with Dranishnikov [DK], we reduced the conjecture from
arbitrary homomorphisms to epimorphisms and presented a counterexample with an
epimorphism between geometrically finite groups. However, we also managed to
prove the conjecture for epimorphisms between finitely generated, torsion-free
nilpotent groups. It is worth exploring whether the torsion-free restriction
can be removed from our result. While dealing with torsion, we investigated a
related question: whether it is possible to have a homomorphism of torsion
groups $\phi:\Gamma\to\Lambda$ with $\operatorname{cd}(\phi)<\infty$. In our
recent paper [Ku], we provide a negative answer to this question.

In this paper, we prove the conjecture for finitely generated, torsion-free
almost nilpotent groups. We recall an almost nilpotent group is the extension
group of the infinite cyclic group by the torsion-free nilpotent groups. More
explicitly, one can think that it is the semidirect product of the torsion-
free nilpotent group with infinite cyclic group
${\mathbb{Z}}\ltimes_{\phi}\Gamma,$ where $\phi:{\mathbb{Z}}\to Aut(\Gamma)$
is a one-parameter group.

######  Theorem 1.2.

(Theorem 3.5) Let
$f:{\mathbb{Z}}\ltimes_{\phi}\Gamma\to{\mathbb{Z}}\ltimes_{\nu}\Lambda$ be
epimorphism between torsion-free almost nilpotent groups such that
$f|_{{\mathbb{Z}}}=Id.$ Then
$\operatorname{cat}(f)=\operatorname{cd}(f)=\operatorname{cd}(\Lambda)+1.$

Furthermore, we confirm that the conjecture holds for finitely generated,
torsion-free virtually nilpotent groups.

######  Theorem 1.3.

(Theorem 4.2) Let $f:M^{m}\to N^{n}$ be a map of infranilmanifold $M$ that
induces epimorphisms $\phi:\pi_{1}(M)\to\pi_{1}(N).$ Then
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$ In particular,
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=n.$

##  2\. Preliminaries

We will begin by recalling some classical theorems in group theory and Lie
group theory. Discrete groups will be denoted by Greek letters, while Lie
groups will be denoted by Latin letters.

######  Lemma 2.1.

Let $\Gamma$ be a group and let $\Gamma^{\prime}$ be a subgroup of finite
index. Then there exists a normal subgroup $\Gamma^{\prime\prime}$ of $\Gamma$
such that $\Gamma^{\prime\prime}$ is of finite index in $\Gamma$ and
$\Gamma^{\prime\prime}\leq\Gamma^{\prime}$.

###### Proof.

Let X be the set of left cosets of $\Gamma^{\prime}.$ Consider $\phi:\Gamma\to
Sym(X)$ given by $\phi(g)(a\Gamma^{\prime})=(ga)\Gamma^{\prime}.$ Then $\phi$
is a homomorphism. Take now $\Gamma^{\prime\prime}=\ker\phi.$ Then
$\Gamma^{\prime\prime}$ is a normal subgroup of $\Gamma$ contained in
$\Gamma^{\prime}$. Finally, $\Gamma/\Gamma^{\prime\prime}$ is isomorphic to a
subgroup of $Sym(X),$ which has order $n!,$ where
$n=[\Gamma:\Gamma^{\prime}].$ Thus, $[\Gamma:\Gamma^{\prime\prime}]$ is
finite. ∎

A group $\Gamma$ is called a virtually nilpotent if it has a nilpotent
subgroup $\Gamma^{\prime}$ of finite index. By Lemma 2.1, we can always assume
that the finite index subgroup is indeed a finite index normal subgroup.

###  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 2.2.

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 $i$.

###### Proof.

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

###  2.2. Construction of infranilmanifold, nilmanifold, and solvmanifold.

A infranilmanifold is a closed manifold diffeomorphic to the orbit space
$G/\Gamma$ of a simply-connected nilpotent Lie group G of the action of a
discrete torsion-free subgroup $\Gamma$ of the semidirect product $G\rtimes K$
where K is a maximal subgroup of $Aut(G).$ If $\Gamma$ lies in the $G$ factor,
then the infranilmanifold is called a nilmanifold. Every infranilmanifold
$G/\Gamma$ is finitely covered by the nilmanifold $G/\Gamma\cap G.$

Let $\textfrak{g}$ be the Lie algebra of a simply-connected nilpotent Lie
group G. It is well-known that the exponential map $exp:\textfrak{g}\to G$ is
a global diffeomorphism and the quotient map $G\to G/\Gamma$ is the universal
covering map. Hence, every infranilmanifold and nilmanifold are the Eilenberg-
MacLane spaces $K(\Gamma,1).$ By the Mal’cev Theorem [Ma2] every torsion-free,
finitely generated nilpotent group $\Gamma$ can be realized as the fundamental
group of some nilmanifold. The corresponding simply-connected nilpotent Lie
group $G$ is obtained as the Mal’cev completion of $\Gamma$. Moreover,
$\Gamma$ is a lattice in G. In this paper, a lattice in $G$ is a cocompact
discrete subgroup.

A solvmanifold is a closed manifold diffeomorphic to the quotient space
$G/\Gamma$ of a simply-connected solvable Lie group G by discrete cocompact
subgroup $\Gamma$ of G. It is known that every solvmanifold $G/\Gamma$ can be
naturally fibered over a torus with a nilmanifold as fiber:
$N/\Gamma_{N}=(N\Gamma)/\Gamma\to G/\Gamma\to G/(N\Gamma)=T^{k}$ where $N$ is
the nilradical of G and $\Gamma_{N}:=\Gamma\cap N$ is a lattice [Ra]. This is
known as the Mostow fiber bundle. From the Mostow fiber bundle, we obtain that
the fundamental group of a solvmanifold $G/\Gamma$ fits into the short exact
sequence:

$1\to\Gamma_{N}\to\Gamma\to{\mathbb{Z}}^{k}\to 1.$

###  2.3. Solvable group of a special type: R type

A Lie Group $G$ is completely solvable if the corresponding Lie algebra
$\textfrak{g}$ satisfies the property that any adjoint linear operator
$\text{ad}S:\textfrak{g}\to\textfrak{g},$ $S\in\textfrak{g}$ has only real
eigenvalues. In particular, any nilpotent Lie group is completely solvable
[Ma1].

######  Theorem 2.3.

([Sa]) Let $G$ and $H$ be simply-connected completely solvable Lie groups.
Suppose G contains a lattice $\Gamma.$ For any homomorphism $\phi:\Gamma\to
H,$ there exists a unique extension to a homomorphism $\bar{\phi}:G\to H.$

###  2.4. Berstein-Schwarz cohomology class

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

######  Theorem 2.4 (Universality [DR],[Sch]).

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

###  2.5. Main Lemma

The main results of this paper rely on the following:

######  Lemma 2.5.

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

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

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

In the paper, we use the notation $H^{*}(\Gamma,A)$ for the 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\. Homomorphisms of almost nilpotent group

In this section, we prove the conjecture to special types of solvable groups,
namely to almost nilpotent groups. We recall that an almost nilpotent Lie
group is a non-nilpotent Lie group with a codimension 1 nilpotent normal
subgroup. N is nilradical of a simply-connected Lie group G if the Lie algebra
of N, $\textfrak{n}$, is the largest nilpotent ideal contained in the Lie
algebra of G, $\textfrak{g}$.

######  Lemma 3.1.

Let G be the simply connected Lie group, and let N be a nilradical of G. If
$dimG/N=1,$ then the fundamental group $\Gamma$ of solvmanifold $G/\Gamma$ is
given by semidirect product
$\Gamma={\mathbb{Z}}\ltimes_{\phi}{\mathbb{Z}}_{N}.$ Further, the simply
connected Lie group G has the form $G=R\ltimes_{\bar{\phi}}N.$

###### Proof.

Since $\Gamma$ is a lattice in G, we define $\Gamma_{N}:=\Gamma\cap N.$ It is
known $\Gamma_{N}$ is a lattice in N [Ra]. We have the following commutative
diagram with exact horizontal rows

${1}$${N}$${G}$${G/N}$${1}$${1}$${\Gamma_{N}}$${\Gamma}$${\Gamma/\Gamma_{N}}$${1}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\pi}$

such that $G/N=R^{s}$ and $\Gamma/\Gamma_{N}={\mathbb{Z}}^{s}.$ Since
$dimG/N=1,$ we get $s=1.$ Since $\Gamma/\Gamma_{N}={\mathbb{Z}}$ is free, then
the lower in the diagram admits a section $s:{\mathbb{Z}}\to\Gamma.$ Hence
$\Gamma\cong{\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}.$

Since $\Gamma\subset G,$ we can uniquely extend the section
$s:{\mathbb{Z}}\to\Gamma$ to a map $\bar{s}:R=G/N\to G$ by Theorem 2.3.

We claim that $\bar{s}$ is a section to $\bar{\pi}:G\to G/N=R.$

This is because $\bar{s}\circ\bar{\pi}$ is extension map of the homomorphism
$s\circ\pi:\Gamma/\Gamma_{N}\to\Gamma/\Gamma_{N}$ by the Saito theorem
(Theorem 2.3) again. Since $s\circ\pi=Id|_{\Gamma/\Gamma_{N}}$ and $\bar{Id}$
is the extension of the identity homomorphism
$Id:\Gamma/\Gamma_{N}\to\Gamma/\Gamma_{N},$ the uniqueness of the Saito
theorem gives us $\bar{s}\circ\bar{\pi}=\bar{Id}.$

It follows that $G=R\ltimes_{\bar{\phi}}N$ and $\bar{\pi}|_{\Gamma}=\pi.$ ∎

######  Lemma 3.2.

Let $\Gamma={\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}$ and
$\Lambda={\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$ be finitely generated,
torsion-free almost nilpotent groups. Any epimorphism
$f:{\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}\to{\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$
such that $f|_{{\mathbb{Z}}}=Id:{\mathbb{Z}}\to{\mathbb{Z}}$ can be extended
uniquely to a homomorphism $\bar{f}:G\to H,$ where $G:=R\ltimes_{\bar{\phi}}N$
and $H=R\ltimes_{\bar{\nu}}N^{\prime}.$

###### Proof.

We have the following commutative diagram where the rows are short exact
sequence

${1}$${N}$${G}$${G/N=R}$${1}$${1}$${\Gamma_{N}}$${\Gamma}$${\Gamma/\Gamma_{N}={\mathbb{Z}}}$${1}$${1}$${\Lambda_{N^{\prime}}}$${\Lambda}$${\Lambda/\Lambda_{N}={\mathbb{Z}}}$${1}$${1}$${N^{\prime}}$${H}$${H/N^{\prime}=R}$${1}$$\scriptstyle{\bar{\pi_{\Gamma}}}$$\scriptstyle{f|}$$\scriptstyle{\pi_{\Gamma}}$$\scriptstyle{f}$$\scriptstyle{Id}$$\scriptstyle{\pi_{\Lambda}}$$\scriptstyle{\bar{\pi_{\Lambda}}}$

Since $\Lambda\subset H,$ the epimorphism $f:\Gamma\to\Lambda$ extends the
uniquely $\bar{f}:G\to H$ by the Saito theorem. Similarly
$Id:{\mathbb{Z}}\to{\mathbb{Z}}$ extends to a map $\bar{Id}:R\to R.$ Since
$Id=\pi_{\Lambda}\circ f\circ s_{\Gamma},$ we get
$\bar{Id}=\bar{\pi_{\Lambda}}\circ\bar{f}\circ\bar{s_{\Gamma}}$ by applying
the Saito theorem twice.

By construction, $\bar{f}$ restricted to discrete group $\Gamma$ brings the
epimorphism $f:\Gamma\to\Lambda.$

∎

######  Lemma 3.3.

Let $\Gamma={\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}$ and
$\Lambda={\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$ be finitely generated,
torsion-free almost nilpotent groups. Then every epimorphism
$f:{\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}\to{\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$
such that $f|_{{\mathbb{Z}}}=Id:{\mathbb{Z}}\to{\mathbb{Z}}$ can be realized
as a locally trivial bundle of solvmanifolds with the fiber a nilmanifold.

###### Proof.

By Lemma 3.2, the kernel of epimorphism
$f:{\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}\to{\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$
such that $f|_{{\mathbb{Z}}}=Id:{\mathbb{Z}}\to{\mathbb{Z}}$ equals the kernel
of epimorphism $f|:\Gamma_{N}\to\Lambda_{N^{\prime}}.$ Similarly,
$ker(\bar{f})=ker(\bar{f}|_{N}).$ Let us denote $K:=ker(\bar{f}|_{N})$ and
$\pi:=ker(f|_{\Gamma_{N}}).$ Note that $\pi$ is a torsion-free, finitely
generated nilpotent group since it is a subgroup of finitely generated,
torsion-free nilpotent group $\Gamma_{N}.$ By Mal’cev Theorem [Ma2], $\pi$ can
be realized the fundamental group of a nilmanifold $K/\pi,$ where $K$ is a
simply connected, nilpotent Lie group, i.e. the Mal’cev completion of the
torsion-free nilpotent group $\pi$. Since $G,H$, and $K$ are simply connected
completely solvable Lie groups, we can apply the Saito theorem to obtain the
following commutative diagram

${K/\pi}$${G/\Gamma}$${H/\Lambda}$${1}$${K}$${G}$${H}$${1}$${1}$${\pi}$${\Gamma}$${\Lambda}$${1.}$$\scriptstyle{\hat{f}}$$\scriptstyle{p_{\pi}}$$\scriptstyle{\bar{f}}$$\scriptstyle{p_{\Gamma}}$$\scriptstyle{p_{\Lambda}}$$\scriptstyle{f}$

Here $G=R\ltimes_{\bar{\phi}}N$ and $H=R\ltimes_{\bar{\nu}}N^{\prime}$ are
connected, simply connected almost nilpotent Lie groups whereas $N$ and
$N^{\prime}$ are their nilradicals.

We claim that the solvmanifold $G/\Gamma$ is a total space of a locally
trivial bundle of solvmanifold $H/\Lambda$ with a fiber nilmanifold $K/\pi.$

Since G is the principal $K$-bundle over $H$ [Co] and the projection
$p_{\Lambda}:H\to H/\Lambda$ is the universal covering map, we can pick a
sufficiently small neighborhood $U$ of a point $x\in H/\Lambda$ such that
$p^{-1}_{\Lambda}(U)$ is evenly covered by
$\\{\bar{U_{\lambda}}\\}_{\lambda\in\Lambda}$. In addition, we require that
for each $\lambda$, $\bar{U_{\lambda}}\times K$ is a local trivialization of
the fiber bundle $K\to G\to H$. Let

$X:=\underset{\lambda\in\Lambda}{\coprod}\bar{U_{\lambda}}\times K.$

Note that the action of the group $\Gamma$ on $X$ translates the summands. The
orbit space $X/\Gamma$ is homeomorphic to

$(X/\pi)/\Lambda=\left(\left(\underset{\lambda\in\Lambda}{\coprod}\bar{U}_{\lambda}\times
K\right)/\pi\right)/\Lambda=\left(\underset{\lambda\in\Lambda}{\coprod}\bar{U}_{\lambda}\times
K/\pi\right)/\Lambda\cong U\times K/\pi.$

Hence, the preimage $\hat{f}^{-1}(U)$, is homeomorphic to $U\times K/\pi.$

Since the point $x\in H/\Lambda$ is arbitrary, we get the solvmanifold
$G/\Gamma$ is the locally trivial fiber bundle over $H/\Lambda$ with fiber
$K/\pi.$

∎

######  Corollary 3.4.

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

######  Theorem 3.5.

Let
$f:{\mathbb{Z}}\ltimes_{\phi}\Gamma_{N}\to{\mathbb{Z}}\ltimes_{\nu}\Lambda_{N^{\prime}}$
be an epimorphism between torsion-free almost nilpotent groups such that
$f|_{{\mathbb{Z}}}=Id.$ Then

$\operatorname{cat}(f)=\operatorname{cd}(f)=\operatorname{cd}(\Lambda_{N^{\prime}})+1.$

###### Proof.

It is clear by dimensional reason that

$\operatorname{cd}(f)\leq\operatorname{cat}(f)\leq\operatorname{cd}(\Lambda_{N^{\prime}})+1.$

We prove that
$\operatorname{cd}(f)=\operatorname{cd}(\Lambda_{N^{\prime}})+1.$ By Lemma
2.5, it suffices to show that $f$ can be realized as a fiber bundle over a
closed aspherical manifold with compact fiber. Indeed, by Lemma 3.3 the map
$f$ can be realized as a locally trivial fiber bundle over solvmanifold
$(R\rtimes_{\bar{\nu}}N^{\prime})/({\mathbb{Z}}\rtimes_{\nu}\Lambda_{N^{\prime}})$
with fiber nilmanifold $K/\pi$. This finishes the proof of the theorem. ∎

##  4\. Homomorphism of virtually nilpotent groups

Let $f:M\to N$ be a map between aspherical manifolds with the fundamental
groups $\Gamma$ and $\Lambda$ respectively. We denote by
$\phi:\Gamma\to\Lambda$ the induced homomorphism
$\phi:=f_{*}:\pi_{1}(M)\to\pi_{1}(N)$.

######  Lemma 4.1.

There are no maps from a nilmanifold to a infranilmanifold that induces an
epimorphisms of the fundamental groups.

###### Proof.

Suppose there is a map $f$ from nilmanifold $N$ to infranilmanifold $M$ that
induces an epimorphism $\phi:\pi_{1}(N)\to\pi_{1}(M).$ Since $\pi_{1}(N)$ is a
nilpotent group, the image $\phi(\pi_{1}(N))$ is a nilpotent group by
Proposition 2.2. Since $\phi$ is surjective, $\pi_{1}(M)$ must be nilpotent.
This is a contradiction since $\pi_{1}(M)$ is a virtually nilpotent group. ∎

######  Theorem 4.2.

Let $f:M^{m}\to N^{n}$ be a map between infranilmanifolds that induces an
epimorphism $\phi:\pi_{1}(M)\to\pi_{1}(N).$ Then
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=n.$

###### Proof.

Using the well-known inequalities on cat [CLOT] we obtain:

$\operatorname{cat}(\phi)\leq\min\\{\operatorname{cat}(M),\operatorname{cat}{N}\\}\leq\dim
N=n.$

Since $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi),$ to obtain the
equality $\operatorname{cd}(\phi)=\operatorname{cat}(\phi),$ it suffices to
show that $\operatorname{cd}(\phi)=n.$ We consider two steps:

Step 1. Let $B\Lambda$ be a nilmanifold.

Since $\Gamma$ is a virtually nilpotent group, there exists nilpotent subgroup
$\Gamma^{\prime}$ such that the index $|\Gamma:\Gamma^{\prime}|$ is finite. By
Lemma 2.1, we can assume $\Gamma^{\prime}$ to be normal. We have the following
commutative diagrams:

${\Gamma^{\prime}}$${\Gamma}$${\Lambda}$$\scriptstyle{i_{*}}$$\scriptstyle{h_{*}}$$\scriptstyle{\phi}$
${B\Gamma^{\prime}}$${B\Gamma}$${B\Lambda}$$\scriptstyle{i}$$\scriptstyle{h}$$\scriptstyle{f}$

where $h:=f|_{B\Gamma^{\prime}}.$

Since the map $h:B\Gamma^{\prime}\to\Lambda$ induces an epimorphism
$h_{*}:\Gamma^{\prime}\to\Lambda,$ by Corollary 3.4 there is a fiber bundle of
nilmanifolds $B\Gamma^{\prime}\to B\Lambda$ with a compact fiber. By Lemma
2.5, $\operatorname{cd}(h_{*})=n$.

${H^{n}(B\Gamma^{\prime};{\mathbb{Z}}\Lambda)}$${H^{n}(B\Gamma;{\mathbb{Z}}\Lambda)}$${H^{n}(B\Lambda;{\mathbb{Z}}\Lambda)}$$\scriptstyle{i^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{h^{*}}$

We claim that that $\operatorname{cd}(\phi)=n$ since
$\operatorname{cd}(h_{*})=n.$

Let us pick a element $a\in H^{n}(B\Lambda;{\mathbb{Z}}\Lambda)$ with
$h^{*}(a)\neq 0$. Suppose the contrary that $\operatorname{cd}(\phi)<n,$ then
$f^{*}(a)=0.$ This is a contradiction, since

$0=i^{*}(f^{*}(a))=h^{*}(a)\neq 0.$

Thus, we obtain $\operatorname{cd}(\phi)=n.$

Step 2. Let $B\Lambda$ be a pure infranilmanifold. In view of Lemma 4.1, we
consider $B\Gamma$ is also a pure infranilmanifold.

Since $\Lambda$ is virtually nilpotent group, there exists a nilpotent
subgroup $\Lambda^{\prime}$ such that the index $|\Lambda:\Lambda^{\prime}|$
is finite. Therefore, the induce and the co-induced modules coinside, i.e.,

$Ind_{\Lambda^{\prime}}^{\Lambda}M=Coind_{\Lambda^{\prime}}^{\Lambda}M$

where $M$ is a $Z\Lambda^{\prime}$-module. By the Shapiro Lemma
[Br][proposition 6.2, p. 73], we have the following isomorphisms

$H^{*}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda^{\prime})\cong
H^{*}(B\Lambda;Coind_{\Lambda^{\prime}}^{\Lambda}\Lambda^{\prime})=H^{*}(B\Lambda;Ind_{\Lambda^{\prime}}^{\Lambda}{\mathbb{Z}}\Lambda^{\prime})=H^{*}(B\Lambda;{\mathbb{Z}}\Lambda),$

since
$Ind_{\Lambda^{\prime}}^{\Lambda}{\mathbb{Z}}\Lambda^{\prime}={\mathbb{Z}}\Lambda\otimes_{{\mathbb{Z}}\Lambda^{\prime}}{\mathbb{Z}}\Lambda^{\prime}\cong{\mathbb{Z}}\Lambda.$

Let

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

denote the canonical ${\mathbb{Z}}\Lambda^{\prime}$-homomorphism, defined for
$g:{\mathbb{Z}}\Lambda\to{\mathbb{Z}}\Lambda^{\prime}$ as $\alpha(g)=g(1)$.

We define the coefficient homomorphism $\beta$ as composition of coefficient
homomorphisms

$\beta:{\mathbb{Z}}\Lambda\cong{\mathbb{Z}}\Lambda\otimes_{{\mathbb{Z}}\Lambda^{\prime}}{\mathbb{Z}}\Lambda^{\prime}\cong
Hom_{{\mathbb{Z}}\Lambda^{\prime}}({\mathbb{Z}}\Lambda,{\mathbb{Z}}\Lambda^{\prime})\overset{\alpha}{\to}{\mathbb{Z}}\Lambda^{\prime}.$

One can define $\beta$ explicitly: Given $\gamma\in\Lambda,$
$\beta(\gamma)=\gamma$ if $\gamma\in\Lambda^{\prime}$ and zero otherwise.

We get the following commutative diagram

${H^{*}(B\Lambda;{\mathbb{Z}}\Lambda)}$${H^{*}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda)}$${H^{*}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda^{\prime})}$${H^{*}(B\Gamma;{\mathbb{Z}}\Lambda)}$${H^{*}(B\Gamma^{\prime};{\mathbb{Z}}\Lambda)}$${H^{*}(f^{*}(B\Lambda^{\prime});{\mathbb{Z}}\Lambda^{\prime})}$$\scriptstyle{j^{*}}$$\scriptstyle{f^{*}}$$\scriptstyle{f^{\prime*}}$$\scriptstyle{\beta_{*}}$$\scriptstyle{f^{\prime*}}$$\scriptstyle{j^{*}}$$\scriptstyle{\beta_{*}}$

where $f^{*}(B\Lambda^{\prime})$ is pull-back of maps $f:B\Gamma\to B\Lambda$
and $i:B\Lambda^{\prime}\to B\Lambda.$

Since the map $f:B\Gamma\to B\Lambda$ induces an epimorphism of the
fundamental groups of manifolds, the pull-back space
$f^{*}(B\Lambda^{\prime})$ is path-connected. Furthermore,
$f^{*}(B\Lambda^{\prime})$ is an infranilmanifold since $B\Lambda^{\prime}\to
B\Lambda$ is a regular covering. Hence, we get that

$f^{\prime*}:H^{n}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda^{\prime})\to
H^{n}(f^{*}(B\Lambda^{\prime});{\mathbb{Z}}\Lambda^{\prime})$

is nonzero homomorphism by Step 1, so $\operatorname{cd}(f^{\prime})=n.$

We claim

$f^{*}:H^{n}(B\Lambda;{\mathbb{Z}}\Lambda)\to
H^{n}(B\Gamma;{\mathbb{Z}}\Lambda)$

is a nonzero homomorphism. Suppose the contrary and let us pick the element
$b\in H^{n}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda^{\prime})$ with
$f^{\prime*}(b)\neq 0.$ Since

$\beta_{*}j^{*}:H^{*}(B\Lambda;{\mathbb{Z}}\Lambda)\to
H^{*}(B\Lambda^{\prime};{\mathbb{Z}}\Lambda^{\prime})$

is an isomorphism by the Shapiro Lemma, there exists a nonzero element $a\in
H^{n}(B\Lambda;{\mathbb{Z}}\Lambda)$ such that $\beta_{*}j^{*}(a)=b.$ Then
$f^{\prime*}j^{*}(a)=j^{*}f^{*}(a)=j^{*}(0)=0$ since $\operatorname{cd}(f)<n.$
This is a contradiction, as we have the commutative diagram

$0\neq
f^{\prime*}(b)=f^{\prime*}\beta_{*}j^{*}(a)=\beta_{*}f^{\prime*}j^{*}(b)=\beta_{*}j^{*}f^{*}(b)=\beta_{*}j^{*}(0)=0.$

Hence, we prove the claim, which implies $\operatorname{cd}(\phi)=n.$ ∎

##  5\. Acknowledgement

The author thanks his advisor Alexander Dranishnikov for extremely helpful
discussions and his support.

## 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.
  * [Br] K. Brown, Cohomology of Groups. _Graduate Texts in Mathematics_ , 87 Springer, New York Heidelberg Berlin, 1994.
  * [CLOT] O. Cornea, G. Lupton, J. Oprea, D. Tanre,  Lusternik-Schnirelmann Category, AMS, 2003.
  * [Co] R. Cohen, The topology of fiber bundles lecture notes. Standford University (1998).
  * [DK] A. Dranishnikov, N. Kuanyshov, On the LS category of homomorphism. Math. Z. to appear, preprint arXiv:2203.03734 [math.AT]
  * [DS] A. Dranishnikov, R. Sadykov, The Lusternik–Schnirelmann category of a connected sum, Fundamenta Mathematicae 251 (2020), no. 3, 313-328.
  * [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
  * [Ku] N. Kuanyshov, On the LS category of homomorphisms of groups with torsion, preprint arXiv:2302.10998 [math.AT]
  * [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.
  * [Sa] M. Saito, Sur certains groupes de Lie resolubles, I, II, Sci. Pap. Coll. Gen. Ed. Univ. Tokyo 7 (1957), 1-11, 157-168.
  * [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, Topology and its Applications 314 (2022), Paper No. 108094, 25 pp.