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| | + | # There is a canonical degree one normal map $(f|_N, b') \colon (N, \nu) \to (Y, \xi|_Y \oplus \nu_{Y \subset X})$. |
# ... | # ... | ||
Revision as of 10:27, 24 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 .
- ...