Normal maps and submanifolds (Ex)
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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: | 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 $(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$. | # 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 $\mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X})$. | + | # 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> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Exercises]] | [[Category:Exercises]] |
Revision as of 17:16, 27 February 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.