Normal maps and submanifolds (Ex)

From Manifold Atlas
Jump to: navigation, search

Let (f, b) \colon (M, \nu_M) \to (X, \xi) be a degree one normal map. For simplicity, assume that M/var/www/vhost/ and X are closed oriented \text{Cat}-manifolds of dimension n. Suppose that i \colon Y \subset X is the inclusion of a codimension k oriented submanifold X with normal bundle \nu_{Y \subset X} and that that f is transverse to Y.

Exercise 0.1. Prove the following:

  1. There is a canonical degree one normal map (f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X}).
  2. This defines well-defined maps
    \displaystyle \pitchfork_{Y, \xi} \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X}) \quad \text{and} \quad \pitchfork_Y \colon \mathcal{N}(X)\rightarrow \mathcal{N}(Y).
  3. If we use
    Tex syntax error
    Tex syntax error
    as base-points to identify \mathcal{N}(X) \equiv [X, G/Cat] and \mathcal{N}(Y) \equiv [Y, G/Cat], show there is a commutative diagram:
    \displaystyle \xymatrix{ \mathcal{N}(X) \ar[d]^{\pitchfork_Y} \ar[r] & [X, G/Cat] \ar[d]^{i^*} \\ \mathcal{N}(Y) \ar[r] & [Y, G/Cat]. }
  4. Of course we have the surgery obstruction map
\displaystyle  \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))

and the composite map

\displaystyle  \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y))

which is called the splitting obstruction map along Y. Prove the following:

If (f, b) is normally bordant to a homeomorphism then the splitting obstruction along Y vanishes.

Personal tools