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

Abstract

The following inequality \[ c a t L S X ≤ c a t L S Y + ⌈ h d ( X ) − r r + 1 ⌉ \mathrm {cat}_{\mathrm {LS}} X\le \mathrm {cat}_{\mathrm {LS}} Y+\bigg \lceil \frac {hd(X)-r}{r+1}\bigg \rceil \] holds for every locally trivial fibration f : X → Y f:X\to Y between A N E ANE spaces which admits a section and has the r r -connected fiber, where h d ( X ) hd(X) is the homotopical dimension of X X . We apply this inequality to prove that \[ c a t L S X ≤ c d ( π 1 ( X ) ) + ⌈ dim ⁡ X − 1 2 ⌉ \mathrm {cat}_{\mathrm {LS}} X\le cd(\pi _1(X))+\bigg \lceil \frac {\dim X-1}{2}\bigg \rceil \] for every complex X X with c d ( π 1 ( X ) ) ≤ 2 cd(\pi _1(X))\le 2 , where c d ( π 1 ( X ) ) cd(\pi _1(X)) denotes the cohomological dimension of the fundamental group of X X .