Morse surgery

surgery

A transformation of smooth manifolds to which the level manifold of a smooth function is subjected on passage through a non-degenerate critical point; it is the most important construction in the topology of manifolds.

Let be a smooth -dimensional manifold (without boundary) in which a -dimensional sphere is (smoothly) imbedded. Suppose that the normal bundle of in is trivial, that is, a closed tubular neighbourhood of in decomposes into a direct product , where is a disc of dimension . After having chosen such a decomposition, remove the interior of from . A manifold is obtained whose boundary decomposes into a product of spheres. But the manifold has precisely the same boundary. Identifying the boundaries of and by a diffeomorphism preserving the product structure of , a manifold without boundary is again obtained. It is called the result of surgery of along .

In the realization of a surgery it is necessary to give a decomposition of the neighbourhood of into a direct product, that is, a trivialization of the normal bundle of in ; in this connection different trivializations (riggings) may give essentially distinct (even homotopically) manifolds .

The number is called the index of the surgery, and the pair its type. If is obtained from by a surgery of type , then is obtained from by a surgery of type . For , is a disjoint union of (which may be empty) and . The construction of a surgery may also be carried out for piecewise-linear and topological manifolds.

Example. For and the result of surgery is a disjoint union of spheres, and for a torus. For and the product is obtained. The case and is more complicated: if is imbedded in in the standard way (as a great circle), then, depending on the choice of the trivialization of the normal bundle, all lens spaces (cf. Lens space) are obtained; if, however, a knotting of is allowed, then a still larger set of three-dimensional manifolds is obtained.

If is the boundary of an -dimensional manifold , then will be the boundary of the manifold obtained from by glueing a handle of the index of . In particular, if is a smooth function on and if are numbers such that is compact and contains a unique non-degenerate critical point , then is obtained from by a surgery of index , where is the Morse index of . In a more general form, any surgery of a manifold of index defines a bordism (cobordism) (obtained from the product by glueing a handle of index to its "right-hand boundary" ), and on the triple there is a Morse function with a unique critical point of index ; moreover, any bordism on which there is such a Morse function is obtained in this way. Hence (and from the existence of Morse functions on triples) it follows that two manifolds are bordant if one can be obtained from the other by a sequence of surgeries.

With the known precautions on the treatment of orientations, the result of a surgery on an oriented manifold is again an oriented manifold. In general, for any structure series (see -structure) it is possible to define the idea of -surgery; in this connection, two manifolds are -bordant if they are connected by a finite sequence of -surgeries.

The important role of surgery in the topology of manifolds is explained by the fact that it allows one to "delicately" (without infringing on the various properties of manifolds) kill "superfluous" homotopy groups (the operation usually used to this end in homotopy theory, i.e. the operation of "glueing" cells, instantaneously leads out of the class of manifolds). In practice, all theorems on the classification of structures on manifolds are based on the question: Given a mapping of a closed manifold into a CW-complex , when does there exist a bordism and a mapping such that and is a homotopy equivalence. The natural route to the solution of this problem is to annihilate the kernels of the homomorphisms by a sequence of surgeries (where are the homotopy groups). If this succeeds, the resulting mapping will be a homotopy equivalence. The study of the corresponding obstructions (lying in the so-called Wall groups, see [4] and Wall group) is one of the most important stimuli in the development of algebraic -theory.

References

[1]	M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)
[2]	S.P. Novikov, "Homotopically equivalent smooth manifolds I" Transl. Amer. Math. Soc. , 48 (1965) pp. 271–396 Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 365–474
[3]	M.A. Kervaire, J.W. Milnor, "Groups of homotopy spheres: I" Ann. of Math. , 77 (1963) pp. 504–537
[4]	A.S. Mishchenko, "Hermitian -theory. The theory of characteristic classes and methods of functional analysis" Russian Math. Surveys , 31 : 2 (1976) pp. 71–138 Uspekhi Mat. Nauk , 31 : 2 (1976) pp. 69–134

Comments

There are clear relations between surgery and Morse theory, as indicated above, which is why in the Russian literature the term Morse surgery is frequently used. In the Western literature one simply speaks of surgery. The construction was invented by J.W. Milnor ([a4]).

References

[a1]	W.B. Browder, "Surgery on simply-connected manifolds" , Springer (1972)
[a2]	C.T.C. Wall, "Surgery on compact manifolds" , Acad. Press (1970)
[a3]	C.T.-C. Wall, "Surgery on non-simply connected manifolds" Ann. of Math. , 84 (1966) pp. 217–276
[a4]	J.W. Milnor, "A procedure for killing the homotopy groups of differentiable manifolds" , Differential geometry , Proc. Symp. Pure Math. , 3 , Amer. Math. Soc. (1961) pp. 39–55
[a5]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963)
[a6]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3