16 Oct
2003
16 Oct
'03
10:44 a.m.
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. 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. John Conway