Even dimensional surgery obstruction (Ex)
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.