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
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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