From: frandag@math.ksu.edu (Francis Fung) Subject: K(G, 1) uniqueness? To: rusin@math.niu.edu Date: Mon, 26 Dec 94 13:27:12 CST Hi! I was just wondering about the two spaces you mentioned in the response to LAURAHELEN's post some time ago. They were not homotopy equivalent, but they had the same cohomology rings, and the same homotopy groups, and all pi_i were 0 except the first. I was under the impression that pi_1 determined a classifying space up to homotopy equivalence. Am I missing something? Is someone not a CW-complex? Thanks a lot! Francis Fung ============================================================================== Date: Tue, 27 Dec 94 14:51:46 CST From: rusin (Dave Rusin) To: frandag@math.ksu.edu, rusin@math.niu.edu Subject: Re: K(G, 1) uniqueness? You wrote: >Hi! I was just wondering about the two spaces you mentioned in the >response to LAURAHELEN's post some time ago. They were not homotopy >equivalent, but they had the same cohomology rings, and the same >homotopy groups, and all pi_i were 0 except the first. I was under ^\.........(except pi_1) >the impression that pi_1 determined a classifying space up to homotopy >equivalence. Am I missing something? Is someone not a CW-complex? Right, pi_1 determines K in the homotopy category, but you don't have pi_1 here, you only have its cohomology ring.I have here a preprint by Ian Leary which gives the examples. I don't know where he published it. You may have met Ian; this was his dissertation under Charles Thomas about 2-3 years ago. The title is "3-groups are not determined by their integral cohomology rings". He has generalized it. The main result is that if G(e) is the group then G(1) and G(-1) are non-isomorphic groups of order 3^5 whose integral cohomology rings are isomorphic. Equivalently, the two spaces K(G(e),1) have all homotopy groups equal except the first (in fact all are zero) and have identical cohomology rings (with any trivial coefficient ring) but are not homotopy equivalent. dave