Even dimensional surgery obstruction (Ex)

Revision as of 23:41, 19 March 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 $m=2$ and $M$ the torus, find all degree 1 normal maps $(\bar{f},f)$ that cover $f$. # 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 of the torus to the 2-sphere. Note if there is surgery obstruction that even though we know the torus is cobordant to the 2-sphere, the wrong normal map in the surgery programme will not find this cobordism. </wikitex> == References == {{#RefList:}} [[Category:Exercises]]m$-manifold $M^m$, possesses a degree 1 map $f:M^m\to S^m$.
2. For $m=2$ and $M$ the torus, find all degree 1 normal maps $(\bar{f},f)$ that cover $f$.
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 of the torus to the 2-sphere.

Note if there is surgery obstruction that even though we know the torus is cobordant to the 2-sphere, the wrong normal map in the surgery programme will not find this cobordism.

References