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 $Y\subset K(G,1)$.