Even dimensional surgery obstruction (Ex)

Latest revision as of 09:16, 1 April 2012

This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction.

1. Show that any closed, orientable $<wikitex>; This question is intended to illustrate the importance of the normal map in defining the even-dimensional surgery obstruction. # 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 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 $S^2$, the wrong normal map in the surgery programme will not find this cobordism. </wikitex> <!-- == References == {{#RefList:}} --> [[Category:Exercises]] [[Category:Exercises without solution]]m$-manifold $M^m$, possesses a degree 1 map $f:M^m\to S^m$.
2. For $M=\Sigma_g$, find all degree 1 normal maps $(\bar{f},f)$ that cover $f:\Sigma_g\to S^2$.
3. 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 $S^2$, the wrong normal map in the surgery programme will not find this cobordism.