antipodal map on is homotopic to the identity if and only if is odd
Regard as a subspace of and define by . Since is a unit vector field, for any . Hence , so is into . Finally observe that and . Thus is a homotopy between the antipodal map and the identity map. ∎
The antipodal map is homotopic to the identity if and only if is odd.
Now suppose is odd, say . Regard has a subspace of . So each point of has coordinates with . Define a map by , pairwise swapping coordinates and negating the even coordinates. By construction, for any , we have that and . Hence is a unit vector field. Applying the lemma, we conclude that the antipodal map is homotopic to the identity. ∎
- 1 Hatcher, A. Algebraic topology, Cambridge University Press, 2002.
- 2 Munkres, J. Elements of algebraic topology, Addison-Wesley, 1984.
|Title||antipodal map on is homotopic to the identity if and only if is odd|
|Date of creation||2013-03-22 15:47:33|
|Last modified on||2013-03-22 15:47:33|
|Last modified by||mps (409)|