Surgery Theory - Application To Classification of Manifolds

Application To Classification of Manifolds

The origin and main application of surgery theory lies in the classification of manifolds of dimension greater than four. Loosely, the organizing questions of surgery theory are:

• Is X a manifold?
• Is f a diffeomorphism?

More formally, one must ask whether up to homotopy:

• Does a space X have the homotopy type of a smooth manifold of the same dimension?
• Is a homotopy equivalence f: M → N between two smooth manifolds homotopic to a diffeomorphism?

It turns out that the second ("uniqueness") question is a relative version of a question of the first ("existence") type; thus both questions can be treated with the same methods.

Note that surgery theory does not give a complete set of invariants to these questions. Instead, it is obstruction-theoretic: there is a primary obstruction, and a secondary obstruction called the surgery obstruction which is only defined if the primary obstruction vanishes, and which depends on the choice made in verifying that the primary obstruction vanishes.