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Mark Steinberger


Surgery

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Joking aside, surgery theory is a set of tools for deciding when two manifolds are homeomorphic, diffeomorphic, or h-cobordant.

For any of these conditions to hold, it is a necessary condition that the two manifolds be homotopy equivalent, so a surgery problem starts with a homotopy equivalence between them.

One may then ask whether the homotopy equivalence in question could be homotopic to a homeomorphism or diffeomorphism. A necessary condition for this to occur is that the homotopy equivalence be simple.

Simple or not, there is a first obstruction to further study called a normal invariant, which measures whether the homotopy equivalence is normally cobordant to the identity map on the target manifold. (Actually, there are two different normal invariants. One is used to find homeomorphisms, and the other to find diffeomorphisms. The only difference is that the normal cobordism in the latter case is a smooth manifold, but the smooth obstruction is harder to compute than the topological one.)

If the normal invariant vanishes, one then may calculate surgery obstructions to being able to perform certain kinds of coring and capping operations on the normal cobordism to make it homotopy equivalent (or, if one calculates the obstructions in a different group, simple homotopy equivalent) to the target manifold crossed with the unit interval.

If the original homotopy equivalence was simple and if the simple homotopy surgery obstruction vanishes, then the homotopy equivalence is homotopic to a homeomorphism (or diffeomorphism, if the normal cobordism is smooth).

Simple or not, if the surgery obstruction for sugery up to ordinary homotopy vanishes, then the result of the coring and pasting operations on the normal cobordism is an h-cobordism between the two manifolds.

Filling in the details of the above would make a good graduate course, which is available to interested students.

Surgery is one of the main ingredients for obtaining results on the nonlinear similarity problem.

 

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