A is path-connected &

B is not empty & all path components of B are open in B.

Show that the suspension of the smsh product of A & B has trivial fundamental group.

Any ideas? (Worried)

Any help would be appreciated.

Thanks x

Printable View

- November 24th 2009, 08:00 PMTTBTopology Question...
A is path-connected &

B is not empty & all path components of B are open in B.

Show that the suspension of the smsh product of A & B has trivial fundamental group.

Any ideas? (Worried)

Any help would be appreciated.

Thanks x - November 24th 2009, 10:10 PMaliceinwonderland
OK. My idea is as follows:

Theorem (Munkres p161). A space X is locally path connected if and only if for every open set U of X, each path component of U is open in X.

B is locally path-connected by the above theorem. Let path components of B be . Then , where each is simply-connected. We see that is not necessarily simply-connected. However, if we apply the smash product of A and B, is identified for . Now each simply-connected space has a common point. It follows that the smash product is simply-connected. We see that a suspension of a simply-connected space is simply connected.

Thus we concluce that the suspension of a smash product is simply connected having a trivial fundamental group. - November 25th 2009, 03:29 AMTTB
It's ok, I've sorted it out now. :)

Thanks anyway x