---
abstract: 'The following inequality $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le \operatorname{{\mbox{\rm cat$_{\rm LS}$}}}Y+\left\lceil\frac{hd(X)-r}{r+1}\right\rceil$$ holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le \left\lceil\frac{\dim X-1}{2}\right\rceil+cd(\pi_1(X))$$ for every complex $X$ with $cd(\pi_1(X))\le 2$.'
address: 'Department of Mathematics, University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA'
author:
- 'Alexander N. Dranishnikov'
title: |
    On the Lusternik-Schnirelmann category of spaces with\
    2-dimensional fundamental group
---

Introduction
============

In [@DKR] we proved that if the Lusternik-Schnirelmann category of a closed $n$-manifold, $n\ge 3$, equals 2 then the fundamental group of $M$ is free. In opposite direction we proved that if the fundamental group of an $n$-manifold is free, then $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}M\le n-2$. Then J. Strom proved that $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le \frac{2}{3}n$ for every $n$-complex, $n>4$, with free fundamental group [@St]. Yu. Rudyak suggested that the coefficient $2/3$ in Strom’s result should be improved to $1/2$. Precisely, he conjectured that the function $f$ defined as $f(n)=\max\{\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}M^n\}$ is asymptotically $\frac{1}{2}n$ where the maximum is taken over all closed $n$-manifolds with free fundamental group.

In this paper we prove Rudyak’s conjecture. Our method gives the same estimate for $n$-complexes. Moreover, it gives the same asymptotic upper bound for $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}$ of $n$-complexes with the fundamental group of cohomological dimension $\le 2$. In view of this the following generalization of Rudyak’s conjecture seems to be natural.

CONJECTURE. [*For every $k$ the function $f_k$ defined as $$f_k(n)=\max\{\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}M^n\mid cd(\pi_1(M^n)\le k\}$$ is asymptotically $\frac{1}{2}n$.*]{}

The smallest $k$ when it is unknown is 3.

The paper is organized as follows. Section 2 is an introduction to the Lusternik-Schnirelmann category based on an analogy with the dimension theory. Section 3 contains a version of a fibration theorem for $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}$. In Section 4 this fibration theorem is applied for the proof of Rudyak’s conjecture.

Kolmogorov-Ostrand’s approach to the Lusternik-Schnirelmann category
====================================================================

An open cover $\scr U=\{U_i\}$ of a topological space $X$ is called [*$X$-contractible*]{} if each $U_i$ can be contracted to a point in $X$. By the definition $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le n$ if there is an $X$-contractible cover of $X$ by $n+1$ elements.

We recall [@CLOT] that a sequence $\emptyset=O_0\subset
O_1\subset\dots\subset O_{n+1}=X$ is called [*categorical of length $n+1$*]{} if each difference $O_{i+1}\setminus O_i$ is contained in an open set that contracts to a point. It was proven in [@CLOT] that $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le n$ if and only if $X$ admits a categorical sequence of length $n+1$.

Let $\scr U$ be a family of open set in a topological space $X$. The multiplicity of $\scr U$ (or the order) at a point $x\in X$, $\Ord_x\scr U$ is the number of elements of $\scr U$ that contain $x$. The multiplicity of $\scr U$ is defined as $\Ord\scr
U=\sup_{x\in X}\Ord_x\scr U$. We recall that [*the covering dimension of a topological space $X$ does not exceed $n$, $\dim X\le
n$, if for every open cover $\scr C$ of $X$ there is an open refinement $\scr U$ with $\Ord\scr U\le n+1$.*]{}

The following proposition makes the LS-category analogous to the covering dimension.

For a paracompact topological space $X$, $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le n$ if and only if $X$ admits an $X$-contractible locally finite open cover $\scr V$ with $Ord\scr V\le n+1$.

If $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le n$ then by the definition $X$ admits an open contractible cover that consists of $n+1$ elements and therefore its multiplicity is at most $n+1$.

Let $\scr V$ be a contractible cover of $X$ of multiplicity $\le
n+1$. We construct a categorical sequence $O_0\subset
O_1\subset\dots\subset O_{n+1}$ of length $n+1$. We define $O_1=\{x\in X\mid \Ord_x\scr V\}=n+1$. Note that $O_1=\cup_{V_i\in\scr V} V_0\cap\dots\cap V_n$. Note that this is the disjoint union and every nonempty summand is $X$-contractible. Thus $O_1$ is $X$-contractible. Next, we define $O_2=\{x\in X\mid
\Ord_x\scr V\ge n\}$. Then $O_2\setminus O_1=cup_{V_i\in\scr
V}(V_1\cap\dots\cap V_n\setminus O_1)$ is a disjoint union of closed in $O_2$ subsets. Since this family is locally finite, we can take open (in $O_2$ and hence in $X$) disjoint neighborhoods of these summands $V_1\cap\dots\cap V_n\setminus O_1$ such that each of which is contained in an open contractible set from $\scr V$. Define $O_3=\{x\in X\mid Ord_x\ge n-1\}$ as the union of $n-1$-fold intersections and so on. $O_{n+1}$ is the union of elements of $\scr
V$ (1-fold intersections) and hence $O_{n+1}=X$. Clearly, for every $k$, $O_{k+1}\setminus O_k$ is the union of disjoint sets each of which is contained in an element of $\scr V$. Thus, the categorical sequence conditions are satisfied.

A family $\scr U$ of subsets of $X$ is called a $k$-cover, $k\in
N$ if every subfamily of $k$ elements form a cover of $X$. The following is obvious.

\[n-cover\] A family $\scr U$ of $m$-elements is an $(n+1)$-cover of $X$ if and only if $\Ord_x\scr U\ge m-n$ for all $x\in X$.

If $\Ord_x\scr U< m-n$ for some $x\in X$, then $n+1=m-(m-n)+1$ elements of $\scr U$ do not cover $x$.

If $n+1$ elements of $\scr U$ do not cover some $x$, then $\Ord_x\scr U\le m-(n+1)<m-n$.

Inspired by the work of Kolmogorov on Hilbert’s 13 problem Ostrand gave the following characterization of the covering dimension [@Os].

A metric space $X$ is of dimension $\leq n$ if and only if for each open cover $\scr C$ of $X$ and each integer $m\geq n+1$, there exist $m$ discrete families of open sets $\scr U_1,\cdots,\scr U_k$ such that their union $\cup\scr U_i$ is an $(n+1)$-cover of $X$.

Let $\scr U$ be a family of subsets in $X$ and let $A\subset X$. We denote by $\scr U|_A-\{U\cap A\mid U\in\scr U\}$.

Let $f:X\to Y$ be a map. An open cover $\scr U=\{U_0, U_1,\dots, U_n\}$ of $X$ is called [*uniformly $f$-contractible*]{} if it satisfies the property that for every $y\in Y$ there is a neighborhood $V$ such that the restriction $\scr U|_{f^{-1}(V)}$ of $\scr U$ to the preimage $f^{-1}(V)$ consists of $X$-contractible sets.

\[criterion\] Let $\{U_0',\dots,U_n'\}$ be an open cover of a normal topological space $X$. Then there is an infinite open $(n+1)$-cover of $X$, $\{U_k\}_{k=0}^{\infty}$ such that $U_k=U_k'$ for $k\le n$ and $U_k=\cup_{i=0}^nV_i$ is the disjoint union with $V_i\subset U_i$ for $k>n$.

In particular, if $\{U_0',\dots,U_n'\}$ is $X$-contractible, the cover $\{U_k\}_{k=0}^{\infty}$ is $X$-contractible. If $\{U_0',\dots,U_n'\}$ is uniformly $f$-contractible for some $f:X\to Y$, the cover $\{U_k\}_{k=0}^{\infty}$ is uniformly $f$-contractible.

By induction on $m$ we construct the family $\{U_i\}_{i=0}^{m}$. For $m=n$ we take $U_i=U_i'$.

Let $\scr U=\{ U_0,\dots, U_m\}$ be the corresponding family for $m\ge n$. Consider $Y=\{x\in X\mid \Ord_x\scr
U\le m-n\}$. In view of Proposition \[n-cover\] and the assumption, it is a closed subset. We show that for every $i\le n$, the set $Y\cap U_i$ is closed in $X$. Let $x$ be a limit point of $Y\cap U_i$ that does not belong to $U_i$. Let $U_{i_0},\dots,U_{i_{m-n}}$ be the elements that contain $x$. Then $\Ord_y\scr U=m-n+1$ for all $y\in Y\cap U_i\cap U_{i_0}\cap\dots\cap
U_{i_{m-n}}$. Contradiction.

We define recursively $F_0=Y\cap U_1$ and $F_{i+1}=Y\cap U_{i+1}\setminus
(\cup_{k=0}^iU_i)$. Note that $\{F_i\}$ is a disjoint finite family of closed subsets. We fix disjoint open neighborhoods $V_i$ of $F_i$ with $V_i\subset U_i$. We define $U_{m+1}=\cup_iV_i$. In view of Proposition \[n-cover\], $U_1,\dots, U_m,U_{m+1}$ is an $(n+1)$-cover.

Clearly, if all $U_i$ are $X$-contractible, $i\le n$, then $U_{m+1}$ is $X$-contractible. If all $U_i$ are uniformly $f$-contractible, for some $f:X\to Y$, then $U_{m+1}$ is uniformly $f$-contractible.

For a normal topological space $X$, $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le n$ if and only if for any $m>n$ $X$ admits an open $(n+1)$-cover by $X$-contractible sets.

This corollary is a $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}$-analog of Ostrand’s theorem. It also can be found in [@CLOT] with further reference to [@Cu].

Fibration theorems for $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}$
===================================================================

The $\ast$-category $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*f$ of a map $f:X\to Y$ is the minimal $n$, if exists, such that there is a uniformly $f$-contractible open cover $\scr U=\{U_0,
U_1,\dots, U_n\}$ of $X$.

Note that $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*c=\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X$ for a constant map $c:X\to pt$. More generally, $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*\pi=\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X$ for the projection $\pi:X\times Y\to
Y$.

\[hur\] The inequality $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le\dim Y+\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*f$ holds true for any continuous map.

The requirements to the spaces in the theorem are that the Ostrand theorem holds true for $Y$, i.e. are fairly general (say, $Y$ is normal).

Let $\dim Y=n$ and $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*f=m$. Let $\scr U=\{U_0,\dots U_m\}$ be a uniformly $f$-contractible cover. For all $y\in Y$ denote by $V_y$ a neighborhood of $y$ from the definition of the uniform $f$-contractibility. In view of Ostrand’s characterization of dimension there is a refinement $\scr V=\scr V_0\cup\dots\cup\scr
V_{n+m}$ of the cover $\{V_y\mid y\in Y\}$ such that each family $\scr V_i$ is disjoint. Let $V_i=\cup\scr V_i$. We apply Theorem \[criterion\] to extend the family $\scr U$ to a family $\{U_0,\dots,U_{n+m}\}$. Consider the family $\scr
W=\{f^{-1}(V_i)\cap U_i\}$. Note that it is contractible in $X$. We show that it is a cover of $X$. According to the Ostrand theorem every $y\in Y$ is covered by $m+1$ elements $V_{i_0},\dots, V_{i_m}$. By Theorem \[criterion\] the family $U_{i_0},\dots, U_{i_m}$ covers the fiber $f^{-1}(y)$.

\[Corollary 9.35 [@CLOT], [@OW]\] Let $p:X\to Y$ be a closed map of $ANE$. If each fiber $p^{-1}(y)$ is contractible in $X$, then $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le\dim Y$.

In this case $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*p=0$. Indeed, since $X$ is an ANE, a contraction of $p^{-1}(y)$ to a point can be extended to a neighborhood $U$. Since the map $p$ is closed there is a neighborhood $V$ of $y$ such that $p^{-1}(V)\subset U$.

We recall that the [*homotopical dimension*]{} of a space $X$, $hd(X)$, is the minimal dimension of a $CW$-complex homotopy equivalent to $X$ [@CLOT].

The elementary obstruction theory implies the following.

\[section\] Let $p:E\to X$ be a fibration with $n-1$-connected fiber where $n=hd(X)$. Then $p$ admits a section.

We recall that $\lceil x\rceil$ denotes the smallest integer $n$ such that $x\le n$.

\[r-connectedhur\] Suppose that a locally trivial fibration $f:X\to Y$ with an $r$-connected fiber $F$ admits a section. Then $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}^*f\le\left\lceil\frac{hd(X)-r}{r+1}\right\rceil.$$ Moreover, $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le \operatorname{{\mbox{\rm cat$_{\rm LS}$}}}Y+
\left\lceil\frac{hd(X)-r}{r+1}\right\rceil.$$

Let $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}Y=m$ and $hd(X)=n$.

Let $s:Y\to X$ be a section. We apply the fiber-wise Serre construction to $f$ with this $s(y)$ as the base points. Then we apply the fiber-wise Ganea construction to obtain a map $\xi_k:E_k\to X$. Namely, $$E_0=\{\phi\in X^I\mid |f(\phi(I))|=1,\ \phi(0)=sf\phi(I)\}$$ with $\xi_0(\phi)=\phi(1)$ and $$E_{k+1}=\{\phi\ast\psi\in E_0\ast E_k\mid
\xi_0(\phi)=\xi_k(\psi)\}$$ with the natural projection $\xi_{k+1}:E_{k+1}\to X$. Note that $\xi_k$ is a Hurewicz fibration with the fiber the join product $\ast^{k+1}\Omega F$ of $k+1$ copies of the loop space $\Omega F$. Thus, it is $(k+(k+1)r-1)$-connected. By Proposition \[section\] there is a section $\sigma:X\to E_k$ whenever $k(r+1)+r\ge n$. The smallest such $k$ is equal to $\lceil\frac{gcd(X)-r}{r+1}\rceil$.

For each $x\in X$ the element$\sigma(x)$ of $\ast_k\Omega F$ can be presented as the $(k+1)$-tuple $$\sigma(x)=((\phi_0,t_0),\dots,(\phi_k,t_k))\mid \sum t_i=1, t_i\ge
0).$$ We use the notation $\sigma(x)_i=t_i$. Clearly, $\sigma(x)_i$ is a continuous function.

A section $\sigma:X\to E_k$ defines a cover $\scr U=\{U_0,\dots,
U_k\}$ of $X$ as follows: $$U_i=\{x\in X\mid\sigma(x)_i>0\}.$$ Let $\{U_0,\dots, U_{m+k}\}$ be an extension of $\scr U$ to an $(k+1)$-cover of $X$ from Theorem \[criterion\].

Let $\scr V=\{V_0,\dots V_{m+k}\}$ be an open $Y$-contractible $(m+1)$-cover of $Y$. We show that the sets $W_i=f^{-1}(V_i)\cap U_i$ are contractible in $X$ for all $i$. By the construction of $U_i$ for $i\le n$ for every $x\in U_i$ there is a canonical path connecting $x$ with $sf(x)$. We use these paths to contract $f^{-1}(V_i)\cap U_i$ to $s(V_i)$ in $X$. Then we apply a contraction of $s(V_i)$ to a point in $s(Y)$. If $i>k$, we apply the last condition of the first part of Theorem \[criterion\].

Similarly as in the proof of Theorem \[hur\] we show that $\{W_i\}_{i=0}^{m+k}$ is a cover of $X$. Since $\scr V$ is an $(m+1)$-cover, by Proposition \[n-cover\] every $y\in Y$ is covered by at least $k+1$ elements $V_{i_0},\dots, V_{i_k}$ of $\scr
V$. By the construction $U_{i_0},\dots, U_{i_k}$ is a cover of $X$. Hence $W_{i_0},\dots, W_{i_k}$ covers $f^{-1}(y)$.

The Lusternik-Schnirelmann category of complexes with low dimensional fundamental groups
========================================================================================

For every complex $X$ with $cd(\pi_1(X))\le 2$ the following inequality holds true: $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le cd(\pi_1(X)) +\left\lceil\frac{hd(X)-1}{2}\right\rceil.$$

Let $\pi=\pi_1(X)$. We consider Borel’s construction $$\begin{CD}
\tilde X @<<< \tilde X\times E\pi @>>> E\pi\\
@VVV @VVV @VVV\\
X @<g<< \tilde X\times_{\pi}E\pi @>f>> B\pi.\\
\end{CD}$$ We claim that there is a section $s:B\pi\to \tilde X\times_{\pi}E\pi$ of $f$. By the condition $cd\pi\le 2$ we may assume that $B\pi$ is a complex of dimension $\le 3$. Since a fiber of $f$ is simply connected, there is a lift of the 2-skeleton. The condition $cd\pi\le 2$ implies $H^3(B\pi,E)=0$ for every $\pi$-module. Thus, we have no obstruction for the lift of the 3-skeleton.

We apply Theorem \[r-connectedhur\] to obtain the inequality $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le \operatorname{{\mbox{\rm cat$_{\rm LS}$}}}(B\pi) +\left\lceil\frac{hd(\tilde X\times_{\pi}E\pi)-1}{2}\right\rceil.$$ Since $g$ is a fibration with the homotopy trivial fiber, the space $\tilde X\times_{\pi}E\pi$ is homotopy equivalent to $X$. Thus, $hd(\tilde X\times_{\pi}E\pi)=hd(X)$. Note that $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}B\pi=cd\pi$.

For every complex $X$ with free fundamental group: $$\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le 1 +\left\lceil\frac{\dim X-1}{2}\right\rceil.$$

Note that this estimate is sharp on $X=S^1\times \C P^n$.

For every 3-dimensional complex $X$ with free fundamental group: $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le 2.$

This Corollary can be also derived from the fact that in the case of free fundamental group every 2-complex is homotopy equivalent to the wedge of circles and 2-spheres [@KR].

It is unclear whether the estimate $ \operatorname{{\mbox{\rm cat$_{\rm LS}$}}}X\le 2+\lceil\frac{\dim
X-1}{2}\rceil$ is sharp for complexes with $cd(\pi_1(X))=2$. It is sharp when the answer to the following question is affirmative.

Does there exists a 4-complex $K$ with free fundamental group and with $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}(K\times S^1)=4$?

Indeed, for $X=K\times S^1$ we would have the equality $4=2+\lceil\frac{5-1}{2}\rceil$. Note that $cd(\pi_(X))=2$. There is a connection between this question and the Problem 1.5 from [@DKR] which asks whether $\operatorname{{\mbox{\rm cat$_{\rm LS}$}}}M^4\le 2$ for closed $4$-manifolds with free fundamental group.

[\[CLOT\]a]{}

Cornea, O.; Lupton, G.; Oprea, J.; Tanré, D.: Lusternik-Schnirelmann category. [*Mathematical Surveys and Monographs*]{}, **103**. American Mathematical Society, Providence, RI, 2003.

Cuvilliez, M.: LS-catégorie et $k$-monomorphisme. Thèse, Université Catholique de Louvain, 1998.

Dranishnikov, A; Katz, M.; Rudyak, Y.: Small values of Lusternik-Schnirelmann category and systolic categories for manifolds. [Preprint]{}, (2007). `arXiv:0706.1625math.AT`

Katz, M.; Rudyak, Y.: Lusternik-Schnirelmann category and systolic category of low dimensional manifolds. [*Communications on Pure and Applied Mathematics*]{}, **59** (2006), 1433-1456. See `arXiv:math.DG/0410456`

Oprea, J. and Walsh, J. Quotient maps, group actions and Lusternik- Schnirelmann category, Topology Appl., 117 (2002), 285-305.

Ostrand, Ph. : Dimension of metric spaces and Hilbert’s problem $13$, Bull. Amer. Math. Soc. 71 1965, 619-622.

Strom, J.: Lusternik-Schnirelmann category of spaces with free fundamental group. [*Preprint*]{}, (2007).
