proved that, provided is compact, any smooth function can be approximated by a Morse function. Thus, for many purposes, one can replace arbitrary functions on by Morse functions. As a next step, one could ask, 'if you have a one-parameter family of functions which start and end at Morse functions, can you assume the whole family is Morse?' In general, the answer is no. Consider, for example, the one-parameter family of functions on given by At time, it has no critical points, but at time, it is a Morse function with two critical points at. Cerf showed that a one-parameter family of functions between two Morse functions can be approximated by one that is Morse at all but finitely manydegenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when, at, an index 0 and index 1 critical point are created as increases.
Returning to the general case where is a compact manifold, letdenotethe space of Morse functions on, and the space of real-valued smooth functions on. Morse proved that is an open and dense subset in the topology. For the purposes of intuition, here is an analogy. Think of the Morse functions as the top-dimensional open stratum in a stratification of . Notice that in stratified spaces, the co-dimension 0 open stratum is open and dense. For notational purposes, reverse the conventions for indexing the stratifications in a stratified space, and index the open strata not by their dimension, but by their co-dimension. This is convenient since is infinite-dimensional if is not a finite set. By assumption, the open co-dimension 0 stratum of is, i.e.:. In a stratified space, frequently is disconnected. The essential property of the co-dimension 1 stratum is that any path in which starts and ends in can be approximated by a path that intersects transversely in finitely many points, and does not intersect for any. Thus Cerf theory is the study of the positive co-dimensional strata of, i.e.: for. In the case of only for is the function not Morse, and has a cubic degenerate critical point corresponding to the birth/death transition.
A single time parameter, statement of theorem
The Morse Theorem asserts that if is a Morse function, then near a critical point it is conjugate to a function of the form where. Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum. Precisely, if is a one-parameter family of smooth functions on with, and Morse, then there exists a smooth one-parameter family such that, is uniformly close to in the -topology on functions. Moreover, is Morse at all but finitely many times. At a non-Morse time the function has only one degenerate critical point, and near that point the family is conjugate to the family where. If this is a one-parameter family of functions where two critical points are created, and for it is a one-parameter family of functions where two critical points are destroyed.
Origins
The PL-Schoenflies problem for was solved by J. W. Alexander in 1924. His proof was adapted to the smooth case by Morse and Emilio Baiada. The essential property was used by Cerf in order to prove that every orientation-preserving diffeomorphism of 3-sphere| is isotopic to the identity, seen as a one-parameter extension of the Schoenflies theorem for. The corollary Exotic sphere| at the time had wide implications in differential topology. The essential property was later used by Cerf to prove the pseudo-isotopy theorem for high-dimensional simply-connected manifolds. The proof is a one-parameter extension of Stephen Smale's proof of the h-cobordism theorem. Cerf's proof is built on the work of Thom and John Mather. A useful modern summary of Thom and Mather's work from that period is the book of Marty Golubitsky and Victor Guillemin.
Applications
Beside the above-mentioned applications, Robion Kirby used Cerf Theory as a key step in justifying the Kirby calculus.
Generalization
A stratification of the complement of an infinite co-dimension subspace of the space of smooth maps was eventually developed by Francis Sergeraert. During the seventies, the classification problem for pseudo-isotopies of non-simply connected manifolds was solved by Allen Hatcher and John Wagoner, discovering algebraic -obstructions on and and by Kiyoshi Igusa, discovering obstructions of a similar nature on .
