Normal maps and submanifolds (Ex)
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# If $(f, b)$ is normally bordant to a homeomorphism then the splitting obstruction along $Y$ vanishes. | # If $(f, b)$ is normally bordant to a homeomorphism then the splitting obstruction along $Y$ vanishes. | ||
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[[Category:Exercises]] | [[Category:Exercises]] | ||
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Revision as of 14:53, 1 April 2012
Let be a degree one normal map. For simplicity, assume that and are closed oriented -manifolds of dimension . Suppose that is a codimension oriented submanifold with normal bundle and that that is transverse to . Prove the following:
- There is a canonical degree one normal map .
- This defines a well-defined map .
Of course we have the surgery obstruction map
and the composite map
which is called the splitting obstruction map along . In addition prove the following:
- If is normally bordant to a homeomorphism then the splitting obstruction along vanishes.