Is There a Smooth Map from S^n to the Equator E of S^n for n ≥ 2?

[TFT]

For n\geq 2, is there a smooth map f: S^n\rightarrow E (E is the equator of S^n) which has the property that the restriction of f to E is a diffeomorphism from E to E?

 

[quasar987]

Do you know how to prove using homology theory that there is no retraction from the n-disk to its boundary? It is the same here:

Suppose such a function f exists. Call g the restriction of f to the northern hemisphere of S^n which we will regard as the n-disk D^n. Call h the restriction of g to E. By hypothese, h is a diffeo and so has degree ±1. The following diagram commute:

D^n<---E
|...|
|g...| h
|...|
E<-------|

that is, g o i = h where i is the inclusion of E in D^n. Passing to the realm of (n-1)-degree homology, the above diagram becomes the following comutative diagram

0<------Z
|...|
|...| ±1
|...|
Z<-------|

which is absurd.

 

[TFT]

Thanks a lot!