We know that a lower bound on the LS category is given by the cuplength. I am interested in knowing if there is a criteria in terms of homology or homotopy of the space (assume a compact manifold) so that the dimension of the space provides an upper bound on the category. For example this is true for simply connected spaces, follows from $Cat_{LS}(X) \leq \frac{1}{2} \big(cd(\pi_1(X)) + \dim(X)\big)$, where $cd(\pi_1(X))$ is the cohomological dimension of $\pi_1(X)$. This also suggest that the dimension may have some role in the bound.
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3$\begingroup$ If $X$ is path-connected, locally contractible and paracompact then $\operatorname{cat}(X)\leq \dim(X)$, the covering dimension. In particular for an $n$-manifold the category cannot exceed $n$. (Here I'm using the noramlization in which the category of a point is zero.) $\endgroup$– Mark GrantCommented Feb 11, 2022 at 15:24
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$\begingroup$ By the way, there must be some symbols missing from your upper bound, as I can't add a number and a space. $\endgroup$– Mark GrantCommented Feb 11, 2022 at 15:25
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$\begingroup$ @Mark Thanks, I have edited it now. $\endgroup$– ArunCommented Feb 11, 2022 at 15:39
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$\begingroup$ @ Michael Albanese: How do you prove the statement in your comment for manifolds? (For general spaces, it is false! E.g. look at acyclic non-contractible spaces. There also seem to be examples for bigger $k$, see the paragraph after exercise 1.6 in math.uni-bielefeld.de/~tcutler/pdf/…) $\endgroup$– Jens ReinholdCommented Feb 11, 2022 at 16:28
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$\begingroup$ @JensReinhold: Sorry, I got confused between the Lusternik-Schnirelman category and cup length. I have deleted my comment. $\endgroup$– Michael AlbaneseCommented Feb 11, 2022 at 16:32
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If $X$ is path-connected, locally contractible and paracompact then $\operatorname{cat}(X)\leq\dim(X)$, the covering dimension. This is Theorem 1.7 in the book Lusternik-Schnirelmann Category by Cornea, Lupton, Oprea and Tanré. (Here I'm using the normalization convention used in that book, in which the LS-category of a point is zero.) In particular the LS-category of an $n$-manifold cannot exceed $n$.
As well as the result (due to Dranishnikov) mentioned in the question, this can also be strengthened by considering the connectivity of the space: If $X$ is $r$-connected, then $\operatorname{cat}(X)\leq \dim(X)/r$. This is easily proved once you know the interpretation of LS-category in terms of the Ganea fibrations and basic obstruction theory.
answered Feb 11, 2022 at 20:43