Spheroidal groups, virtual cohomology and lower dimensional G-spaces

Abstract

A space is defined to be “n-spheroidal” if it has the homotopy type of an n-dimensional CW-complex X with $H_{n}(X; \mathbb {Z})$ not zero and finitely generated. A group G is called “n-spheroidal” if its classifying space K(G, 1) is n-spheroidal. Examples include fundamental groups of compact manifold K(G, 1)s. Moreover, the class of groups G, which are n-spheroidal for some n, is closed under products, free products, and group extensions. If Y is a space with $\pi _{1}(Y)$ n-spheroidal, and if $H_{k}(Y;\mathbb {F}_{p})$ is non-zero and finitely generated, and if $H_{i}(Y;\mathbb {F}_{p}) = 0$ for $i>k$, then $H_{n+k}(\overline{Y};\mathbb {F}_{p}) \ne 0$ for $\overline{Y}$ a finite sheeted covering space of Y. Hence, dim$(Y) \ge n+k$. Thus, it follows that if dim$(Y) < n$, and if $H_{k}(Y;\mathbb {F}_{p}) \ne 0$ and $H_{i}(Y;\mathbb {F}_{p}) = 0$ for $i>k>0$, then $H_{k}(Y;\mathbb {F}_{p})$ is not finitely generated. Similar results follow for $Y\subset K(G,1)$.