Spheroidal groups, virtual cohomology and lower dimensional G-spaces

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 $π_{1}(Y)$ $n$-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 $H_{n+k}(\overline{Y};\mathbb{F}_{p}) \neq 0$ for $\overline{Y}$ a finite sheeted covering space of $Y$. Hence, dim$(Y) \geq n+k$. Thus, it follows that if dim$(Y) < n$, and if $H_{k}(Y;\mathbb{F}_{p}) \neq 0$ 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 $Y\subset K(G,1)$.

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