1
$\begingroup$

Borel conjectued aspherical closed manifolds are topologically rigid.(i.e.a homotopy equivalence between two aspherical manifolds is homotopic to a homeomorphism).

now,soppuse M is a K(G,1) space, it is well known that if G is a finite group,then M should be of infinite dimension.

I am not so comfortable with spaces of infinite dimesion. are there some good (natural) prespective in viewing such spaces.

$\endgroup$
2
  • 3
    $\begingroup$ This question seems rather unfocused. In general, for a finite group G, K(G,1) is an infinite-dimensional CW complex. There are various notions of infinite dimensional manifold; are you interested in analogues of the Borel Conjecture for infinite dimensional manifolds? $\endgroup$
    – Dan Ramras
    Nov 30 '10 at 3:07
  • $\begingroup$ "reasonable" classes of infinite-dimensional manifolds are homotopy-equivalent if and only if they're diffeomorphic. i.e. from the perspective of homotopy-theory the smooth structure isn't important. These are old theorems due to people like Henderson and West (likely others) from the late 60's and early 70's. Richard Palais is here, he likely knows the history of this subject better than most. So from the point of view of these theorems, up to homotopy-equivalence such manifolds are just spaces having the homotopy-type of countable CW-complexes... $\endgroup$ Nov 30 '10 at 3:53
1
$\begingroup$

The technique of this question, together with Greg Kuperberg's observation that it can be extended to any finite group, gives a construction of BG = K(G, 1) as an infinite dimensional manifold.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.