Cohomological local connectedness of decomposition spaces

Abstract:

For a map $f:X \to Y$, let ${\mathcal {H}^k}\left [ f \right ]$ denote the associated $k$-dimensional cohomology sheaf. The main result is that, for a proper map between locally compact metrizable spaces, if the sheaves ${\mathcal {H}^k}\left [ f \right ]$ are locally constant and $X$ is cohomologically locally connected, then $Y$ is cohomologically locally connected. The result can be viewed as a variation on a number of similar results dating to work of Vietoris. The setting for this paper is quite general and the proof is not difficult, involving a routine analysis using the Leray-Grothendieck spectral sequence. Versions of known comparable results for homotopical local connectedness can be recovered by combining the result with standard universal coefficient theorems that translate cohomological information to homological information and with a local Hurewicz theorem.