Cohomological Dimension, Connectivity, and Lusternik--Schnirelmann category

Dranishnikov~\cite{D2} proved that $$\mathrm{c}\mathrm{a}\mathrm{t}X\le \mathrm{c}\mathrm{d}({\pi }_1(X))+\lceil \frac{\mathrm{h}\mathrm{d}(X)-1}{2}\rceil \mathrm{.}$$ where ${\rm cd}(\pi)$denotesthecohomologicaldimensionofagroup$\pi$and${\rm hd}(X)$denotesthehomotopydimensionof$X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: $$\mathrm{c}\mathrm{a}\mathrm{t}X\le \lceil \frac{\mathrm{h}\mathrm{d}(X)}{k+1}\rceil \text{ if }{\pi }_i(X)=0\text{ for }i\le k\mathrm{.}$$ We make a synthesis and generalization of both of these results, by demonstrating the main result: $$\mathrm{c}\mathrm{a}\mathrm{t}\le \mathrm{c}\mathrm{d}({\pi }_1(X))+\lceil \frac{\mathrm{h}\mathrm{d}(X)-1}{k+1}\rceil \text{ if }{\pi }_i(X)=0\text{ for }i=2,\dots ,k\mathrm{.}$$ The proof of the main theorem uses the Oprea–Strom inequality ${\rm cat} X\leq {\rm hd} (B\pi_1(X))+{\rm cat}^1X$,\text{cite}OSwhere${\rm cat}^1$istheClapp-Puppe${\rm cat} $\mathcal{A}$$with$$\mathcal{A}$$ the class of 1-dimensional CW complexes. The inequality clarified the Dranishnikov inequality.