 Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by Milnor (1961). Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with, handlebody decompositions. It is a major tool in the study and classification of manifolds of dimension greater than 4.
More technically, the idea is to start with a wellunderstood manifold M and perform surgery on it to produce a manifold M' having some desired property, in such a way that the effects on the homology, homotopy groups, or other interesting invariants of the manifold are known.
The classification of exotic spheres by Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in highdimensional topology.
Contents
Surgery on a manifold
Recall that in general, if X, Y are manifolds with boundary, then the boundary of the product manifold is . The basic observation which justifies surgery is that the space can be understood either as the boundary of or as the boundary of . In symbols, .
Now, given a manifold M of dimension n = p + q and an embedding , define another ndimensional manifold M' to be
One says that the manifold M' is produced by a surgery cutting out and gluing in , or by a psurgery 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. Notice that the submanifold that was replaced in M was of the same dimension as M (it was of codimension 0).
Surgery is closely related to (but not the same as) handle attaching. Given an (n + 1)manifold with boundary and an embedding , where n = p + q, define another (n + 1)manifold with boundary L' by
The manifold L' is obtained by attaching a (p + 1)handle, with obtained from by a psurgery
A surgery on M not only produces a new manifold M', but also a cobordism W between M and M'. The trace of the surgery is the cobordism (W;M,M'), with
the (n + 1)dimensional manifold with boundary obtained from the product by attaching a (p + 1)handle .
Surgery is symmetric in the sense that the manifold M can be reobtained from M' by a (q − 1)surgery, the trace of which coincides with the trace of the original surgery, up to orientation.
In most applications, the manifold M comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endow M' with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery on normal maps: such a process changes a normal map to another normal map within the same bordism class.
Examples
1. Surgery on the circle
As per the above definition, a surgery on the circle consists of cutting out a copy of and gluing in . The pictures in Fig. 1 show that the result of doing this is either (i) S^{1} again, or (ii) two copies of S^{1}.
2. Surgery on the 2sphere
In this case there are more possibilities, since we can start by cutting out either or .
 (a) : If we remove a cylinder from the 2sphere, we are left with two disks. We have to glue back in  that is, two disks  and it's clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)
 (b) : Having cut out two disks , we glue back in the cylinder . Interestingly, there are two possible outcomes, depending on whether our glueing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is the torus , but if they are different, we obtain the Klein Bottle (Fig. 2c).
3. Surgery on the nsphere
If n = p + q, then . The psurgery on S^{n} is therefore . Examples 1 and 2 above were a special case of this.
4. Morse functions
Suppose that f is a Morse function on an (n + 1)dimensional manifold, and suppose that c is a critical value with exactly one critical point in its preimage. If the index of this critical point is p + 1, then the levelset is obtained from by a psurgery. The bordism can be identified with the trace of this surgery.
Indeed, in some coordinate chart around the critical point, the function f is of the form , with , and p + q + 1 = n + 1. Fig. 3 shows, in this local chart, the manifold M in blue and the manifold M' in red. The colored region between M and M' corresponds to the bordism W. The picture shows that W is diffeomorphic to the union
(neglecting the issue of straightening corners), where is colored in yellow, and is colored in green. The manifold M', being a boundary component of W, is therefore obtained from M by a psurgery.
Since every bordism between closed manifolds has a Morse function where different critical points have different critical values, this shows that any bordism can be decomposed into traces of surgeries (handlebody decomposition). In particular, every manifold M may be regarded as a bordism from the boundary (which may be empty) to the empty manifold, and so may be obtained from by attaching handles.
Effects on homotopy groups, and comparison to cellattachment
Intuitively, the process of surgery is the manifold analog of attaching a cell to a topological space, where the embedding ϕ takes the place of the attaching map. A simple attachment of a (q + 1)cell to an nmanifold would destroy the manifold structure for dimension reasons, so it has to be thickened by crossing with another cell.
Up to homotopy, the process of surgery on an embedding can be described as the attaching of a (p + 1)cell, giving the homotopy type of the trace, and the detaching of a qcell to obtain N. The necessity of the detaching process can be understood as an effect of Poincaré duality.
In the same way as a cell can be attached to a space to kill an element in some homotopy group of the space, a psurgery on a manifold M can often be used to kill an element . Two points are important however: Firstly, the element has to be representable by an embedding (which means embedding the corresponding sphere with a trivial normal bundle). For instance, it is not possible to perform surgery on an orientationreversing loop. Secondly, the effect of the detaching process has to be considered, since it might also have an effect on the homotopy group under consideration. Roughly speaking, this second point is only important when p is at least of the order of half the dimension of M.
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?
 Is a homotopy equivalence 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 obstructiontheoretic: 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.
The surgery approach
In the classical approach, as developed by Browder, Novikov, Sullivan and Wall, surgery is done on normal maps of degree one. Using surgery, the question "Is the normal map of degree one cobordant to a homotopy equivalence?" can be translated (in dimensions greater than four) to an algebraic statement about some element in an Lgroup of the group ring Z[π_{1}(X)]. More precisely, the question has a positive answer if and only if the surgery obstruction is zero, where n is the dimension of M.
For example, consider the case where the dimension n=4k is a multiple of four, and π_{1}(X) = 0. It is known that L_{4k}(Z) is isomorphic to Z; under this isomorphism the surgery obstruction of f maps, up to a scalar factor, to the difference of the signatures σ(X) − σ(M) 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 multistep 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 PontryaginThom 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. If normal maps of degree one to X exist, their bordism classes (called normal invariants) are classified by the set of homotopy classes [X,G / O]. Each of these normal invariants has a surgery obstruction; X has the homotopy type of a smooth manifold if and only if one of these obstructions is zero. Stated differently, this means that there is a choice of normal invariant with zero image under the surgery obstruction map
Structure sets and surgery exact sequence
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 from some manifold to X, where two maps are identified under a bordismtype relation. A necessary (but not in general sufficient) condition for the structure set of a space X to be nonempty is that X be an ndimensional Poincaré complex, i.e. that the homology and cohomology groups be related by isomorphisms of an ndimensional manifold, for some integer n. Depending on the precise definition and the category of manifolds (smooth, PL, or topological), there are various versions of structure sets. Since, by the scobordism 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.
The structure set and the surgery obstruction map are brought together in the surgery exact sequence. 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. Examples are the classification of exotic spheres, and the proofs of the Borel conjecture for negatively curved manifolds and manifolds with hyperbolic fundamental group.
In the topological category, the surgery exact sequence is the long exact sequence induced by a fibration sequence of spectra. This implies that all the sets involved in the sequence are in fact abelian groups. On the spectrum level, the surgery obstruction map is an assembly map whose fiber is the block structure space of the corresponding manifold.
See also
 scobordism theorem
 hcobordism theorem
 Whitehead torsion
 Dehn surgery
 Manifold decomposition
 orientation character
References
 Browder, William (1972), Surgery on simplyconnected manifolds, Berlin, New York: SpringerVerlag, MR0358813
 Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan, eds. (2000), Surveys on surgery theory. Vol. 1, Annals of Mathematics Studies, 145, Princeton University Press, ISBN 9780691049373; 9780691049380, MR1746325, http://www.maths.ed.ac.uk/~aar/books/wall1.pdf
 Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan, eds. (2001), Surveys on surgery theory. Vol. 2, Annals of Mathematics Studies, 149, Princeton University Press, ISBN 9780691088143; 9780691088150, MR1818769, http://www.maths.ed.ac.uk/~aar/books/wall2.pdf
 Kervaire, Michel A.; Milnor, John W. (1963), "Groups of homotopy spheres: I", Annals of Mathematics (Annals of Mathematics) 77 (3): 504–537, doi:10.2307/1970128, JSTOR 1970128, MR0148075
 Milnor, John Willard (1961), "A procedure for killing homotopy groups of differentiable manifolds.", Proc. Sympos. Pure Math., Vol. III, Providence, R.I.: American Mathematical Society, pp. 39–55, MR0130696
 Milnor, John Willard (1965), Lectures on the hcobordism theorem, Notes by L. Siebenmann and J. Sondow, Princeton University Press, MR0190942
 Postnikov, M.M.; Rudyak, Yu.B. (2001), "Morse surgery", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/m/m065000.htm
 Ranicki, Andrew (1980), "The algebraic theory of surgery. I. Foundations", Proc. London Math. Soc 40 (3): 87–192, doi:10.1112/plms/s340.1.87, http://plms.oxfordjournals.org/cgi/reprint/s340/1/87.pdf, retrieved 20090517 PDF
 Ranicki, Andrew (1980), "The algebraic theory of surgery. II. Applications to topology", Proc. London Math. Soc 40: 193–283, doi:10.1112/plms/s340.2.193, http://plms.oxfordjournals.org/cgi/reprint/s340/2/193.pdf, retrieved 20090517 PDF
 Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 9780198509240, MR2061749, http://www.maths.ed.ac.uk/~aar/books/index.htm
 Wall, C. T. C. (1999) [1970], Ranicki, Andrew, ed., Surgery on compact manifolds, Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 9780821809426, MR1687388, http://www.maths.ed.ac.uk/~aar/books/scm.pdf
External links
Categories: Geometric topology
 Algebraic topology
 Quadratic forms
 Surgery theory
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