On Thu, Oct 16, 2003 at 12:27:44PM -0700, Eugene Salamin wrote:
--- Dylan Thurston <dpt@exoskeleton.math.harvard.edu> wrote:
(There is some confusion in terminology, between the concepts of "trivial fundamental group" and "homotopy equivalent to S^n". Among topologists, "simply connected" always means the former, as far as I know. The two conditions are equivalent for 3-manifolds.)
What does "homotopy equivalent to S^n" mean?
There are a number of ways to say it. Here's one: A compact n-dimensional manifold M is "simply connected" if every map of a circle into M can be contracted to a point. It is "homotopy equivalent to S^n" if in addition, every map of a k-sphere, 1<=k<=n, can be contracted to a point. Alternatively (theorem), M is homotopy equivalent to S^n if M is simply connected and the homology groups of M, 1<=k<=n, vanish. (There is a more general notion of homotopy equivalence, which is a little harder to state.) Peace, Dylan