# Normal maps and submanifolds (Ex)

From Manifold Atlas

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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as 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.