Even dimensional surgery obstruction (Ex)
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This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction. | This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction. | ||
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# Show that any closed, orientable $m$-manifold $M^m$, possesses a degree 1 map $f:M^m\to S^m$. | # Show that any closed, orientable $m$-manifold $M^m$, possesses a degree 1 map $f:M^m\to S^m$. | ||
# For $M=\Sigma_g$, find all degree 1 normal maps $(\bar{f},f)$ that cover $f:\Sigma_g\to S^2$. | # For $M=\Sigma_g$, find all degree 1 normal maps $(\bar{f},f)$ that cover $f:\Sigma_g\to S^2$. | ||
− | # For each of the $(\bar{f},f)$, calculate the surgery obstruction $\sigma_*(\bar{f},f)\in L_2(\mathbb{Z})$. If this vanishes, write down an explicit surgery on $(\bar{f},f)$ that describes a cobordism between $\Sigma_g$ and | + | # For each of the $(\bar{f},f)$, calculate the surgery obstruction $\sigma_*(\bar{f},f)\in L_2(\mathbb{Z})$. If this vanishes, write down an explicit surgery on $(\bar{f},f)$ that describes a cobordism between $\Sigma_g$ and $S^2$. |
− | Note if there is surgery obstruction that even though we know $\Sigma_g$ is cobordant to | + | Note if there is surgery obstruction that even though we know $\Sigma_g$ is cobordant to $S^2$, the wrong normal map in the surgery programme will not find this cobordism. |
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Revision as of 00:30, 20 March 2012
This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction.
- Show that any closed, orientable -manifold , possesses a degree 1 map .
- For , find all degree 1 normal maps that cover .
- For each of the , calculate the surgery obstruction . If this vanishes, write down an explicit surgery on that describes a cobordism between and .
Note if there is surgery obstruction that even though we know is cobordant to , the wrong normal map in the surgery programme will not find this cobordism.