# 0708-1300/not homeomorphic

From Drorbn

Please, read the following carefully. It can contain some mistake.

Assume is a homeomorphism. Since is proper we can extend it to a continuous map which in fact will be
a homeomorphism. Taking inverse if necessary we may assume . Let
**Failed to parse (lexing error): F : Sn × [0, 1] --> Sm**

be a homotopy of to a smooth map i.e. is continuous, and is smooth. Since is smooth and all of its image points are singular values and by Sard's theorem constitute a set of measure zero. Then there is a point in not in the image of , but the complement of that point is contractible. This means that we can extend toFailed to parse (lexing error): \overline{F} : Sn × [0, 2] --> S^mto be a homotopy of to a constant map. But thenFailed to parse (lexing error): f^{－1}\circ\overline{F}is a contraction of which is a contradiction with the fact that no such contraction exists.