Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

TL;DR

This paper generalizes existing inequalities relating cohomological dimension, connectivity, and Lusternik--Schnirelmann category, providing a unified bound based on the fundamental group and homotopy connectivity.

Contribution

It synthesizes and extends previous inequalities by establishing a new bound on category involving cohomological dimension and higher connectivity conditions.

Findings

Unified inequality for category involving cohomological dimension and connectivity.

Clarification of Dranishnikov's inequality through the Oprea--Strom inequality.

Generalization applicable to spaces with higher connectivity conditions.

Abstract

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(\pi_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(\pi)$ denotes the cohomological dimension of a group $\pi$ and ${\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } \pi_i(X)=0 \text{ for } i\leq k. \] We make a synthesis and generalization of both of these results, by demonstrating the main result: \[ {\rm cat}\leq {\rm cd}(\pi_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{k+1}\Bigr\rceil \text { if }\pi_i(X)=0 \text{ for } i=2, \ldots, k. \] The proof of the main theorem uses the Oprea--Strom inequality ${\rm cat} X\leq {\rm hd} (B\pi_1(X))+{\rm cat}^1X$, \cite{OS} where ${\rm cat}^1$ is the Clapp-Puppe ${\rm cat} \mathcal{A}$ with $\mathcal{A}$ the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

Equations

$$\operatorname{cat}X\leq\operatorname{cd}(\pi_{1}(X))+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{2}\Bigr{\rceil}.$$

$$\operatorname{cat}X\leq\Bigl{\lceil}\frac{\operatorname{hd}(X)}{k+1}\Bigr{\rceil}\text{ if }\pi_{i}(X)=0\text{ for }i\leq k.$$

$$\operatorname{cat}X\leq\operatorname{cd}(\pi_{1}(X))+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{k+1}\Bigr{\rceil}\text{ if }\pi_{i}(X)=0\text{ for }i=2,\ldots,k.$$

$$\operatorname{cat}X\leq\operatorname{hd}(B\pi_{1}(X))+\operatorname{cat}^{1}(X)\leq\operatorname{hd}(B\pi_{1}(X))+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{2}\Bigr{\rceil}.$$

$$\operatorname{cat}X\leq\operatorname{hd}(B\pi)+\operatorname{cat}^{k}(X)\leq\operatorname{hd}(B\pi)+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{k+1}\Bigr{\rceil}.$$

$$\operatorname{cat}X\leq\operatorname{cd}(\pi)+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{k+1}\Bigr{\rceil}.$$

$$\operatorname{cat}^{k}(X)\leq\Bigl{\lfloor}\frac{\operatorname{hd}(X)}{k+1}\Bigr{\rfloor}\leq\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{k+1}\Bigr{\rceil}$$

$$p:E\pi {\times }_{\pi }\tilde{Z}\to B\pi .$$

$$E\pi {\times }_{\pi }\tilde{Z}\cong Z.$$

$$\begin{array}{ccc}E\pi {\times }_{\pi }{\tilde{X}}_k& \stackrel{f}{\to }& E\pi {\times }_{\pi }\tilde{X}\\ p_0\downarrow & & \downarrow p_1\\ B\pi & =& B\pi \end{array}$$

$$\Gamma_{k}(X)=\inf\{\gamma(\mathcal{U})\bigm{|}\mathcal{U}\text{ is a $\Gamma_{k}$-cover of }E\pi\times_{\pi}\widetilde{X}\}.$$

$$\operatorname{cat}X\leq\operatorname{hd}(B\pi_{1}(X))+\operatorname{cat}^{k}(X)\leq\operatorname{hd}(B\pi_{1}(X))+\Bigl{\lceil}\frac{\dim(X)-1}{k+1}\Bigr{\rceil}.$$

$$\operatorname{cat}X\leq\operatorname{cat}Y+\Bigl{\lceil}\frac{\operatorname{hd}(X)-k}{k+1}\Bigr{\rceil}.$$

$$\operatorname{cat}X\leq\operatorname{cd}\pi+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{k+1}\Bigr{\rceil}$$

Full text

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where $\operatorname{cd}(\pi)$ denotes the cohomological dimension of a group $\pi$ and $\operatorname{hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman, [G]:

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We make a synthesis and generalization of both of these results, by demonstrating the main result:

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The proof of the main theorem uses the Oprea–Strom inequality $\operatorname{cat}X\leq\operatorname{hd}(B\pi_{1}(X))+\operatorname{cat}^{1}X$, [OS] where $\operatorname{cat}^{1}$ is the Clapp-Puppe $\operatorname{cat}\mathcal{A}$ with $\mathcal{A}$ the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

1. Introduction

We work in the category of connected CW complexes and continuous maps. We use the sign $\cong$ for homotopy equivalences. All covers are assumed to be open. Given a space $X$, $\operatorname{hd}(X)$ denotes the homotopy dimension of $X$, that is, the minimum cellular dimension of all CW complexes homotopy equivalent to $X$. Given a group $\pi$, $B\pi$ denotes a classifying space for $\pi$, and $\operatorname{cd}(\pi)$ denotes the cohomological dimension of $\pi$, [B]. A classifying map for $X$ is a map $c=c_{X}:X\to B\pi_{1}(X)$ that induces an isomorphism of fundamental groups.

Below $\operatorname{cat}X$ denotes theLusternik–Schnirelmann category of a space $X$, [LS, CLOT]. A well-known inequality $\operatorname{cat}X\leq\operatorname{hd}X$, [LS, F, CLOT] can be generalized as follows [G, CLOT]:

1.1 Theorem**.**

If $\pi_{i}(X)=0$ for $i\leq k$ then $\displaystyle\operatorname{cat}X\leq\frac{\operatorname{hd}X}{k+1}.$

Concerning the case of $\pi_{1}(X)\neq 0$, the author conjectured that $\operatorname{cat}X$ can be asymptotically bounded above by $\operatorname{hd}(X)/2$, provided that $\pi_{1}(X)$ has finite cohomological dimension, i.e., $\operatorname{cd}(\pi_{1}(X))<\infty$. Later Dranishnikov [D2] proved the following fact:

1.2 Theorem**.**

\displaystyle\operatorname{cat}X\leq\operatorname{cd}(\pi_{1}(X))+\Bigl{\lceil}\frac{\operatorname{hd}(X)-1}{2}\Bigr{\rceil}.**

This theorem can be regarded as a confirmation of the conjecture.

Also, I suspected that there should be a synthesis of Theorem 1.1 and Theorem 1.2, so that the equation Theorem 1.2 can be improved by replacing (approximately) $\operatorname{hd}X$ by $\operatorname{hd}X/(k+1$ for $X$ $k$-connected. In other words, I expected to have a claim that generalizes both Theorem 1.1 and Theorem 1.2. Now I know how to make this improvement (synthesis, generalization). Let me tell you the precise statements (Theorem 1.6 and Corollary 1.7 below).

1.3 Definition** (Clapp and Puppe [CP]).**

Given a class $\mathcal{A}$ of CW complexes and a space $X$, define a subset $A$ of $X$ to be $\mathcal{A}$-categorical if the inclusion $A\to X$ factors, up to homotopy, through a space in $\mathcal{A}$. Follow Clapp and Puppe [CP], define the $\mathcal{A}$-cover of $X$ to be the cover $\{U_{0},U_{1},\ldots,U_{m}\}$ such that each $U_{i}$ is $\mathcal{A}$-categorical. Define $\operatorname{cat}_{\mathcal{A}}X$, the $\mathcal{A}$*-category of X * to be the minimal number $k$ such that there exists an $\mathcal{A}$-categorical cover $\{U_{0},U_{1},\ldots,U_{k}\}$ .

For example, $\operatorname{cat}_{\mathcal{A}}(X)=\operatorname{cat}(X)$ if $\mathcal{A}$ is the class of contractible spaces.

1.4 Definition**.**

Let $\mathcal{A}(n)$ be the class of all $n$-dimensional CW complexes. Put $\operatorname{cat}^{n}(X):=\operatorname{cat}_{\mathcal{A}(n)}(X)$.

The following Oprea–Strom Theorem recovers and clarifies Theorem 1.2.

1.5 Theorem**.**

For every space $X$ we have

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*Proof. *See Oprea and Strom [OS, Corollary 6.2]. ∎

We prove the following generalization of Theorem 1.5:

1.6 Theorem** (Corollary 2.5).**

Let $X$ be a CW complex and $\pi=\pi_{1}(X)$. Suppose that the classifying map $c:X\to B\pi$ induces an isomorphism $c_{*}:\pi_{i}(X)\to\pi_{i}(B\pi)$ for $i\leq k$. $($In particular, $\pi_{i}(X)=0$ for $i=2,\ldots,k.)$ Then

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1.7 Corollary**.**

Let $X$ be a CW complex as in Theorem 1.6. Then

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Clearly, Theorem 1.6 and Corollary 1.7 can be regarded as an above-mentioned synthesis.

2. Proofs

First, we settle the second part of the inequality noted in Theorem 1.6.

2.1 Proposition**.**

Let $X_{k}$ be the $k$-skeleton of a CW complex $X$. Then

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*Proof. *For the first inequality, see [OS, Proposition 4.4]. The second inequality is obvious. ∎

Now we prove the first part of 1.6. The proof is based (speculated) on [OS, Sections 5,6] that, in turn, exploits clever ideas of Dranishnikov [D1, D2].

Let $X$ be a CW complex. Take $k>1$, let $X_{k}$ be the $k$-skeleton of $X$, and let $\widetilde{Z}$ be the universal covering of $Z$ for $Z=X$ or $Z=X_{k}$. Put $\pi=\pi_{1}(X)$ and let $E\pi\to B\pi$ be the universal bundle for $\pi$. Note that $\pi$ acts on $\widetilde{Z}$ via deck transformations of the covering $\widetilde{Z}\to Z$, and we can form the Borel construction

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It is worth noting that $E\pi$ is contractible, and so

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The inclusion $i=i_{k}:X_{k}\to X$ yields the commutative diagram

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where $f=f_{k}$ is induced by $i$, and $p_{0}=p$ if $Z=X_{k}$, and $p_{1}=p$ if $Z=X$. Take a base point $*$ of $B\pi$ and let $F_{0},F_{1}$ be the fibers of $p_{0},p_{1}$ over $*$, respectively. Then $f$ yields a map (inclusion) $j:F_{0}\to F_{1}$ of fibers.

2.2 Proposition**.**

If $\pi_{i}(X)=0$ for $i=2,\ldots,k$ then the inclusion $j:F_{0}\to F_{1}$ is null-homotopic.

*Proof. *It follows because $F_{0}$ is homotopy equivalent to $\widetilde{X}_{k}$, while $\widetilde{X}_{k}$ is contractible $(\pi_{i}(\widetilde{X}_{k})=0$ for $i\leq k$ and $\operatorname{hd}X_{k}\leq k$). ∎

Following [OS], define a cover $\mathcal{U}=\{U_{0},U_{1},\ldots,U_{n}\}$ of $E\pi\times_{\pi}\widetilde{X}$ to be a $\Gamma_{k}$-cover if each inclusion $U_{m}\subset E\pi\times_{\pi}\widetilde{X}$ passes through the inclusion $f$, up to homotopy over $B\pi$. In this case we define $\gamma(\mathcal{U})=n$. Now, set

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2.3 Proposition**.**

We have $\operatorname{cat}X=\operatorname{cat}(E\pi\times_{\pi}\widetilde{X})\leq\operatorname{hd}B(\pi)+\Gamma_{k}$.

*Proof. *The equality $\operatorname{cat}X=\operatorname{cat}(E\pi\times_{\pi}\widetilde{X})$ is explained in (2.2). The inequality follows from [OS, Prop. 5.1] because of Proposition 2.2. ∎

2.4 Theorem**.**

For any CW complex $X$ and every $k\geq 1$, we have $\Gamma_{k}(X)=\operatorname{cat}^{k}(X)$

*Proof. *For $k=1$, this is [OS, Theorem 6.1]. For $k>1$, the proof is literally the same as for $k=1$. The only change is to replace $X_{1}$ by $X_{k}$, $\operatorname{cat}^{1}$ by $\operatorname{cat}^{k}$, and $\Gamma_{1}$ by $\Gamma_{k}$, in  [OS, Theorem 6.1]. ∎

2.5 Corollary**.**

Let $X$ be a CW complex and $\pi=\pi_{1}(X)$. Suppose that the classifying map $c:X\to B\pi$ induces an isomorphism $c_{*}:\pi_{i}(X)\to\pi_{i}(B\pi)$ for $i\leq k$. Then

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*Proof. *The first inequality follows from Proposition 2.3 and Theorem 2.4, the second inequality follows from Proposition 2.1. ∎

Now we prove Corollary 1.7. First, given a group $\pi$, recall that $\operatorname{cd}(\pi)=\operatorname{hd}(B\pi)$ if either $\operatorname{cd}(\pi)\leq 3$, [EG] or $\operatorname{cd}(\pi)=1$, [Stal, Swan]. Furthermore, recall that $\operatorname{cd}(\pi)=\operatorname{cat}(B\pi)$ for all groups $\pi$, [EG, Stal, Swan]. So, for $\operatorname{cd}(\pi)\neq 2$, Corollary 1.7 follows from Corollary 2.5 directly. (Note also that if $\operatorname{cd}(\pi)=2$ then either $\operatorname{hd}(B\pi)=2$ or $\operatorname{hd}(B\pi)=3$, and it is unknown question whether there exists a group $\pi$ with $\operatorname{cd}(\pi)=2$ and $\operatorname{hd}(B\pi)=3$.)

Consider a space $X$ and the classifying map $c:X\to B\pi$ where $\pi=\pi_{1}(X)$. Note that $c_{*}:\pi_{1}(X)\to\pi_{1}(B\pi)$ is an isomorphism. Given $k\in\mathbb{N}$, assume that $c_{*}:\pi_{i}(X)\to\pi_{i}(B\pi)$ is an isomorphism for $i\leq k$.

2.6 Lemma**.**

Let $f:X\to Y$ be a locally trivial bundle with an $k$-connected fiber $F$. Suppose that $f$ admits a section. Then

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*Proof. *See [D1, Theorem 3.7]. ∎

We apply Lemma 2.6 to the Borel construction $p:E\pi\times_{\pi}{\widetilde{X}}\to B\pi$ as in (2.1), with $\operatorname{cd}(\pi)=2$. Note that the bundle $p$ is the classifying map for $X$. Furthermore, the fiber $F$ of $p$ is homotopy equivalent to $\widetilde{X}$.

For $k=1$, Corollary 1.7 is the Dranishnikov theorem Theorem 1.2. So, assume that $k>1$. Then $\pi_{2}(F)=\pi_{2}(\widetilde{X})=0$, since $\pi_{2}(\widetilde{X})=\pi_{2}(X)=\pi_{2}(B\pi)$.

We have $\operatorname{hd}(B\pi)\leq 3$ and $\pi_{i}(F)=0$ for $i=1,2$. So, because of the elementary obstruction theory, $p$ has a section. Thus, because of 2.6 and since $\operatorname{cd}(\pi)=\operatorname{cat}(B\pi)$, we conclude that

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for $\operatorname{cd}(\pi)=2$, and therefore for all $\pi$. This completes the proof of Corollary 2.5.

Acknowledgments: The work was partially supported by a grant from the Simons Foundation (#209424 to Yuli Rudyak).