Date: Tue, 15 Aug 2000 15:41:17 -0400
From: "Allen E. Hatcher" <hatcher@polygon.math.cornell.edu>
Subject: Query for discussion list

Is there a finite CW complex with infinite cyclic fundamental group
and some higher homotopy group not finitely generated as a module
over the fundamental group? How about with just an abelian
fundamental group?

Background: Examples with more complicated fundamental groups are
certainly known. Probably the first was found by Stallings (Am. J.
Math 1963). The examples I'm aware of are obtained by constructing a
K(pi,1) CW complex with finite n-skeleton (n>1) and nonfinitely
generated homology in dimension n+1. The n-skeleton then has pi_n
nonfinitely generated over pi_1. An example with abelian pi_1 would
have to be constructed in quite a different fashion.

A similar sort of question: Is there a finite CW complex with some
homotopy group isomorphic to the rational numbers?

Allen Hatcher