Even dimensional surgery obstruction (Ex)
(Difference between revisions)
Patrickorson (Talk | contribs) |
Patrickorson (Talk | contribs) |
||
Line 6: | Line 6: | ||
# 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. | # 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 | + | 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> | </wikitex> |
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.
- Show that any closed, orientable -manifold , possesses a degree 1 map .
- For and the torus, 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 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.