On the homotopy of finite 𝐶𝑊-complexes with polycyclic fundamental group

Damian, Mihai

doi:10.1090/S0002-9947-08-04632-1

On the homotopy of finite ${CW}$-complexes with polycyclic fundamental group
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Trans. Amer. Math. Soc. 361 (2009), 1791-1809 Request permission

Abstract:

Let $X$ be a finite connected CW-complex of dimension $q$. If its fundamental group $\pi _1(X)$ is polycyclic of Hirsch number $h>q$, we show that at least one homotopy group $\pi _{i}(X)$ is not finitely generated. If $h=q$ or $h=q-1$ the same conclusion holds unless $X$ is an Eilenberg-Mac Lane space $K(\pi _1(X),1)$.

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