Normal maps and submanifolds (Ex)
From Manifold Atlas
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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 $i \colon Y \subset X$ is the inclusion of a codimension $k$ oriented submanifold $X$ with normal bundle $\nu_{Y \subset X}$ and that that $f$ is [[transverse]] to $Y$. |
+ | {{beginthm|Exercise}} | ||
+ | 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 | + | # This defines well-defined maps $$\pitchfork_{Y, \xi} \colon \mathcal{N}(X,\xi)\rightarrow \mathcal{N}(Y,\xi|_Y \oplus \nu_{Y \subset X}) \quad \text{and} \quad \pitchfork_Y \colon \mathcal{N}(X)\rightarrow \mathcal{N}(Y).$$ |
− | + | # 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: $$\xymatrix{ \mathcal{N}(X) \ar[d]^{\pitchfork_Y} \ar[r] & [X, G/Cat] \ar[d]^{i^*} \\ \mathcal{N}(Y) \ar[r] & [Y, G/Cat]. } $$ | |
− | Of course we have the surgery obstruction map | + | # Of course we have the surgery obstruction map |
$$ \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$ | $$ \sigma \colon \mathcal{N}(Y, \xi_y \oplus \nu_{Y \subset X}) \to L_{n-k}(\pi_1(Y))$$ | ||
and the composite map | and the composite map | ||
$$ \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y)) $$ | $$ \sigma \circ \pitchfork_Y \colon \mathcal{N}(X, \xi) \to L_{n-k}(\pi_1(Y)) $$ | ||
− | which is called the splitting obstruction map along $Y$. | + | 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. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
<!-- == References == | <!-- == References == |
Latest revision as of 00:12, 26 August 2013
Tex syntax errorand 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
Tex syntax error
andTex syntax error
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.