Normal maps and submanifolds (Ex)

Let $(f, b) \colon (M, \nu_M) \to (X, \xi)$$<wikitex>; Let $(f, b) \colon (M, \nu_M) \to (X, \xi)$ be a [[degree one normal map]]. For simplicity, assume that $M$ and $X$ are closed oriented $\text{Cat}$-manifolds of dimension $n$. Suppose that $Y \subset X$ is a codimension $k$ oriented submanifold $X$ with normal bundle $\nu_{Y \subset X}$ and that that $f$ is [[transverse]] to $Y$. Prove the following: # There is a canonical degree one normal map $(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$. # This defines a well-defined map $\pitchfork_Y \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X})$. Of course we have the surgery obstruction map $$ \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$ and the composite map $$ \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y)) $$ which is called the splitting obstruction map along $Y$. In addition prove the following: # If $(f, b)$ is normally bordant to a homeomorphism then the splitting obstruction along $Y$ vanishes. </wikitex> == References == {{#RefList:}} [[Category:Exercises]](f, b) \colon (M, \nu_M) \to (X, \xi)$ be a degree one normal map. For simplicity, assume that $M$$M$ and $X$$X$ are closed oriented $\text{Cat}$$\text{Cat}$-manifolds of dimension $n$$n$. Suppose that $Y \subset X$$Y \subset X$ is a codimension $k$$k$ oriented submanifold $X$$X$ with normal bundle $\nu_{Y \subset X}$$\nu_{Y \subset X}$ and that that $f$$f$ is transverse to $Y$$Y$. 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})$$(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$.
2. This defines a well-defined map $\pitchfork_Y \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X})$$\pitchfork_Y \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X})$.

Of course we have the surgery obstruction map

$\displaystyle \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$ $$\displaystyle \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$

and the composite map

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

which is called the splitting obstruction map along $Y$$Y$. In addition prove the following:

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