Normal maps and submanifolds (Ex)

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<wikitex>;
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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 $k$ submanifold oriented submanifold $X$ with normal bundle $\nu_{Y \subset X}$ and that that $f$ is [[transverse]] to $Y$. Prove the following:
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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 $i \colon Y \subset X$ is the inclusion of a codimension $k$ oriented submanifold $X$ with normal bundle $\nu_{Y \subset X}$ and that that $f$ is [[transverse]] to $Y$.
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# There is a canonical degree one normal map $(f|N, b') \colon (N, \nu) \to (Y, \xi|Y \oplus \nu_{Y \subset X}$.
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{{beginthm|Exercise}}
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# ...
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Prove the following:
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# There is a canonical degree one normal map $(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$.
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# This defines well-defined maps $$\pitchfork_{Y, \xi} \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X}) \quad \text{and} \quad \pitchfork_Y \colon \mathcal{N}(X)\rightarrow \mathcal{N}(Y).$$
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# If we use ${\rm Id}_X$ and ${\rm Id}_Y$ 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: $$\xymatrix{ \mathcal{N}(X) \ar[d]^{\pitchfork_Y} \ar[r] & [X, G/Cat] \ar[d]^{i^*} \\ \mathcal{N}(Y) \ar[r] & [Y, G/Cat]. } $$
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# Of course we have the surgery obstruction map
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$$ \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$
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and the composite map
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$$ \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y)) $$
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which is called the splitting obstruction map along $Y$. Prove the following:
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If $(f, b)$ is normally bordant to a homeomorphism then the splitting obstruction along $Y$ vanishes.
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{{endthm}}
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== References ==
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<!-- == References ==
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{{#RefList:}}
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{{#RefList:}} -->
[[Category:Exercises]]
[[Category:Exercises]]
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[[Category:Exercises without solution]]

Latest revision as of 00:12, 26 August 2013

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 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
     and 
    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.

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