On Thu, Oct 16, 2003 at 12:43:45PM -0400, John Conway wrote:
On Thu, 16 Oct 2003, Jon Perry wrote:
Well is anthing wrong with this one then;
Let P_n be a compact, simply connected n-mainfold. It is known that P_n is homeomorphic to S_n for n>=4. As P_4 for example is homeomorphic to S_4, and P_7 is homeomorphic to S_7, then P_4*P_3=P_7 must be homeomorphic to S_7=S_4*S_3, so P_3 is homeomorphic to S_3.
Quite a lot. You don't say what "*" means, so one thing might be the assumption that Pm*Pn is again a simply-connected manifold, which is false if * means direct product.
No, actually, that's right: the fundamental group of a direct product is the product of the fundamental groups. The error is, instead, the assertion that any simply connected n manifold, n>=4, is a sphere: there are many 4-manifolds with trivial fundamental group, for instance. (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.)
Another is the assumption that we can "cancel" from a homeomorphism between A*B and A*C to deduce the existence of one between B and C. This again is false if * means direct product.
Yes, theorems like this are very unlikely to be true. In this case, for instance, a thrice perforated sphere and a once perforated torus are not homeomorphic, but their products with an interval are homeomorphic. Does anybody know a minimal counterexample with compact manifolds? Peace, Dylan