Journal article

Homological finiteness of abelian covers

Alex Suciu, Y Yang, Gufang Zhao

Annali della Scuola normale superiore di Pisa - Classe di scienze | Stampacchia Guido | Published : 2015

Abstract

We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of r-planes in H^1(X,Q). Inside this parameter space, there is a subset Ω_A^i(X) consisting of all the covers with finite Betti numbers up to degree i. Building on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth,..

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