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

Abstract:

The following inequality \[ \mathrm {cat}_{\mathrm {LS}} X\le \mathrm {cat}_{\mathrm {LS}} Y+\bigg \lceil \frac {hd(X)-r}{r+1}\bigg \rceil \] holds for every locally trivial fibration $f:X\to Y$ between $ANE$ spaces 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 \[ \mathrm {cat}_{\mathrm {LS}} X\le cd(\pi _1(X))+\bigg \lceil \frac {\dim X-1}{2}\bigg \rceil \] for every complex $X$ with $cd(\pi _1(X))\le 2$, where $cd(\pi _1(X))$ denotes the cohomological dimension of the fundamental group of $X$.