Difference between revisions of "0708-1300/not homeomorphic"
From Drorbn
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Assume <math>f_0 : R^n \rightarrow R^m</math> is a homeomorphism. Since <math>f_0</math> is proper we can extend it to a continuous map <math>f : S^n \rightarrow S^m</math> which in fact will be a homeomorphism. Taking inverse if necessary we may assume <math>n < m</math>. | Assume <math>f_0 : R^n \rightarrow R^m</math> is a homeomorphism. Since <math>f_0</math> is proper we can extend it to a continuous map <math>f : S^n \rightarrow S^m</math> which in fact will be a homeomorphism. Taking inverse if necessary we may assume <math>n < m</math>. | ||
| − | Let <math>F:S^n\times[0, 1] \rightarrow S^m</math> be a homotopy of <math>f</math> to a smooth map i.e. <math>F</math> is continuous, <math>F(x, 0) = f(x)</math> and <math>F(x, 1)</math> is smooth. Since <math>F(x, 1)</math> is smooth and <math>n < m</math> 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 <math>S^m</math> not in the image of <math>F(x, 1)</math>, but the complement of that point is contractible. This means that we can extend <math>F</math> to <math>F_0:S^n\times[0, 2] \rightarrow S^m</math> to be a homotopy of <math>f</math> to a constant map. But then <math>f^{－1} | + | Let <math>F:S^n\times[0, 1] \rightarrow S^m</math> be a homotopy of <math>f</math> to a smooth map i.e. <math>F</math> is continuous, <math>F(x, 0) = f(x)</math> and <math>F(x, 1)</math> is smooth. Since <math>F(x, 1)</math> is smooth and <math>n < m</math> 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 <math>S^m</math> not in the image of <math>F(x, 1)</math>, but the complement of that point is contractible. This means that we can extend <math>F</math> to <math>F_0:S^n\times[0, 2]\rightarrow S^m</math> to be a homotopy of <math>f</math> to a constant map. But then <math>f^{－1}F_0</math> is a contraction of <math>S^n</math> which is a contradiction with the fact that no such contraction exists. |
Revision as of 13:26, 18 November 2007
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 ![F:S^n\times[0, 1] \rightarrow S^m](/images/math/f/e/4/fe47981f60da67fff6d4615823847e61.png)
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 ![F_0:S^n\times[0, 2]\rightarrow S^m](/images/math/d/6/3/d635e4a94388e509c834c8f165860c91.png)
to be a homotopy of to a constant map. But then Failed to parse (lexing error): f^{－1}F_0
is a contraction ofwhich is a contradiction with the fact that no such contraction exists.
