Normal maps and submanifolds (Ex)
Let be a degree one normal map. For simplicity, assume that and are closed oriented -manifolds of dimension . Suppose that is the inclusion of a codimension oriented submanifold with normal bundle and that that is transverse to .
Exercise 0.1. Prove the following:
- There is a canonical degree one normal map .
- This defines well-defined maps
- If we use
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Tex syntax erroras base-points to identify and , show there is a commutative diagram:
- Of course we have the surgery obstruction map
and the composite map
which is called the splitting obstruction map along . Prove the following:
If is normally bordant to a homeomorphism then the splitting obstruction along vanishes.