Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961).

, could be described as removing an imbedded sphere of dimension p from M.[2] Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.

A relatively easy argument using Morse theory shows that a manifold can be obtained from another one by a sequence of spherical modifications if and only if those two belong to the same cobordism class.

[1] The classification of exotic spheres by Michel Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

that are at distance one-or-less from a given fixed point (the center of the disk); for example, then,

and from the equation from our basic observation before, the gluing is justified then One says that the manifold M′ is produced by a surgery cutting out

, or by a p-surgery if one wants to specify the number p. Strictly speaking, M′ is a manifold with corners, but there is a canonical way to smooth them out.

One then wants the surgery process to endow M′ with the same kind of additional structure.

As per the above definition, a surgery on the circle consists of cutting out a copy of

is therefore Intuitively, the process of surgery is the manifold analog of attaching a cell to a topological space, where the embedding

-cell to an n-manifold would destroy the manifold structure for dimension reasons, so it has to be thickened by crossing with another cell.

(which means embedding the corresponding sphere with a trivial normal bundle).

For instance, it is not possible to perform surgery on an orientation-reversing loop.

Roughly speaking, this second point is only important when p is at least of the order of half the dimension of M. The origin and main application of surgery theory lies in the classification of manifolds of dimension greater than four.

Note that surgery theory does not give a complete set of invariants to these questions.

In the classical approach, as developed by William Browder, Sergei Novikov, Dennis Sullivan, and C. T. C. Wall, surgery is done on normal maps of degree one.

can be translated (in dimensions greater than four) to an algebraic statement about some element in an L-group of the group ring

More precisely, the question has a positive answer if and only if the surgery obstruction

; under this isomorphism the surgery obstruction of f is proportional to the difference of the signatures

of X and M. Hence a normal map of degree one is cobordant to a homotopy equivalence if and only if the signatures of domain and codomain agree.

Coming back to the "existence" question from above, we see that a space X has the homotopy type of a smooth manifold if and only if it receives a normal map of degree one whose surgery obstruction vanishes.

This leads to a multi-step obstruction process: In order to speak of normal maps, X must satisfy an appropriate version of Poincaré duality which turns it into a Poincaré complex.

Supposing that X is a Poincaré complex, the Pontryagin–Thom construction shows that a normal map of degree one to X exists if and only if the Spivak normal fibration of X has a reduction to a stable vector bundle.

Stated differently, this means that there is a choice of normal invariant with zero image under the surgery obstruction map The concept of structure set is the unifying framework for both questions of existence and uniqueness.

Roughly speaking, the structure set of a space X consists of homotopy equivalences M → X from some manifold to X, where two maps are identified under a bordism-type relation.

A necessary (but not in general sufficient) condition for the structure set of a space X to be non-empty is that X be an n-dimensional Poincaré complex, i.e. that the homology and cohomology groups be related by isomorphisms

Since, by the s-cobordism theorem, certain bordisms between manifolds are isomorphic (in the respective category) to cylinders, the concept of structure set allows a classification even up to diffeomorphism.

This sequence allows to determine the structure set of a Poincaré complex once the surgery obstruction map (and a relative version of it) are understood.

In important cases, the smooth or topological structure set can be computed by means of the surgery exact sequence.

This implies that all the sets involved in the sequence are in fact abelian groups.