On Thu, 16 Oct 2003, Dylan Thurston wrote:
On Thu, Oct 16, 2003 at 12:43:45PM -0400, John Conway wrote:
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.
Oops! What a blunder! [The reason I made it is that I was thinking of circle x circle; but of course the circle is simply connected.]
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.
I actually noticed this, but thought the above nonsense was an easier "kill"! Sorry, JHC