# 0708-1300/not homeomorphic

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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 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 to to be a homotopy of to a constant map. But then is a contraction of which is a contradiction with the fact that no such contraction exists.