Even dimensional surgery obstruction (Ex)

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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 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.

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