proof of Brouwer fixed point theorem
Proof of the Brouwer fixed point theorem:
Assume that there does exist a map from with no fixed point. Then let
be the following map: Start at , draw the ray going through and then let be
the first intersection
of that line with the sphere. This map is continuous
and well defined only
because fixes no point. Also, it is not hard to see that it must be the identity
on the boundary
sphere. Thus we have a map , which is the identity on
, that is, a retraction
. Now, if is the inclusion
map
, . Applying the reduced homology functor
, we find that
, where indicates the induced map on homology
.
But, it is a well-known fact that (since is contractible), and that
. Thus we have an isomorphism
of a non-zero group onto itself
factoring through a trivial group, which is clearly impossible. Thus we have a contradiction
,
and no such map exists.
Title | proof of Brouwer fixed point theorem |
---|---|
Canonical name | ProofOfBrouwerFixedPointTheorem |
Date of creation | 2013-03-22 13:11:24 |
Last modified on | 2013-03-22 13:11:24 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 6 |
Author | bwebste (988) |
Entry type | Proof |
Classification | msc 47H10 |
Classification | msc 54H25 |
Classification | msc 55M20 |