On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(\pi_1(X)) for every complex $X$ with $cd(\pi_1(X))\le 2$.

💡 Research Summary

The paper establishes a new upper bound for the Lusternik‑Schnirelmann (LS) category of a space by exploiting the structure of a fibration that admits a section. The main theorem states that for any locally trivial fibration (f\colon X\to Y) between ANE (absolute neighbourhood extensor) spaces, if the fibre is (r)-connected and a section (s\colon Y\to X) exists, then
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