Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

Abstract

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where $\mathrm{cd}(\pi )$ denotes the cohomological dimension of a group $\pi$ and $\mathrm{hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } π_i(X)=0 \text{ for } i\leq k. \] We make a synthesis and generalization of both of these results, by demonstrating the main result: \[ {\rm cat}\leq {\rm cd}(π_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{k+1}\Bigr\rceil \text { if }π_i(X)=0 \text{ for } i=2, \ldots, k. \] The proof of the main theorem uses the Oprea--Strom inequality $\mathrm{cat}X\le \mathrm{hd}(B{\pi }_1(X))+{\mathrm{cat}}^{1}X$, \cite{OS} where ${\mathrm{cat}}^1$ is the Clapp-Puppe $\mathrm{cat}\mathcal{A}$ with $\mathcal{A}$ the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.

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