The Lusternik-Schniremann-category and the fundamental group

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Dranishnikov, Alexander
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We prove that $$ \cat X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$$ for every CW complex $X$ where $cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$.
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Mathematics - Geometric Topology, 55M30
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