On Homology Roses and the D(2)-problem

For a commutative ring $R$ with a unit, an $R$-homology rose is a topological space whose homology groups with $R$-coefficients agree with those of a bouquet of cirlces. In this paper, we study some special properties of covering spaces and fundamental groups of $R$-homology roses, from which we obtain some result supporting the Carlsson conjecture on free $(Z_p)^r$ actions. In addition, for a group $G$ and a field $F$, we define an integer called the $F$-gap of $G$, which is an obstruction for $G$ to be realized as the fundamental group of a 2-dimensional $F$-homology rose. Furthermore, we discuss how to search candidates of the counterexamples of Wall's D(2)-problem among $F$-homology roses and $F$-acyclic spaces.