Even dimensional surgery obstruction (Ex)

Revision as of 23:39, 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 nullcobordism of the torus. Note if there is surgery obstruction that even though we know the torus is nullcobordant, the wrong normal map in the surgery programme will not find this nullcobordism. </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 nullcobordism.

References