# N-sphere and its equator

1. Sep 30, 2009

### 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 $$E$$?

2. Sep 30, 2009

### 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 wich 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.

3. Sep 30, 2009

### TFT

Thanks a lot!