Cerf theory

In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions
on a smooth manifold formula_2, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. The theory is named after Jean Cerf, who initiated it in the late 1960s.
An example.
Marston Morse proved that, provided formula_2 is compact, any smooth function formula_1 can be approximated by a Morse function. Thus, for many purposes, one can replace arbitrary functions on formula_2 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 formula_6 given by
At time formula_8, it has no critical points, but at time formula_9, it is a Morse function with two critical points at formula_10.
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 many degenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when, at formula_11, an index 0 and index 1 critical point are created as formula_12 increases.
A "stratification" of an infinite-dimensional space.
Returning to the general case where formula_2 is a compact manifold, let formula_14 denote the space of Morse functions on formula_2, and formula_16 the space of real-valued smooth functions on formula_2. Morse proved that formula_18 is an open and dense subset in the formula_19 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 formula_16 (we make no claim that such a stratification exists, but suppose one does). 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 formula_16 is infinite-dimensional if formula_2 is not a finite set. By assumption, the open co-dimension 0 stratum of formula_16 is formula_14, i.e.: formula_25. In a stratified space formula_26, frequently formula_27 is disconnected. The essential property of the co-dimension 1 stratum formula_28 is that any path in formula_26 which starts and ends in formula_27 can be approximated by a path that intersects formula_28 transversely in finitely many points, and does not intersect formula_32 for any formula_33.
Thus Cerf theory is the study of the positive co-dimensional strata of formula_16, i.e.: formula_35 for formula_36. In the case of
only for formula_11 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 formula_40 is a Morse function, then near a critical point formula_41 it is conjugate to a function formula_42 of the form
where formula_44.
Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum.
Precisely, if formula_45 is a one-parameter family of smooth functions on formula_2 with formula_47, and formula_48 Morse, then there exists a smooth one-parameter family formula_49 such that formula_50, formula_51 is uniformly close to formula_52 in the formula_53-topology on functions formula_54. Moreover, formula_55 is Morse at all but finitely many times. At a non-Morse time the function has only one degenerate critical point formula_41, and near that point the family formula_55 is conjugate to the family
where formula_59. If formula_60 this is a one-parameter family of functions where two critical points are created (as formula_12 increases), and for formula_62 it is a one-parameter family of functions where two critical points are destroyed.
Origins.
The PL-Schoenflies problem for formula_63 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 formula_64 is isotopic to the identity, seen as a one-parameter extension of the Schoenflies theorem for formula_63. The corollary formula_66 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 (the rewriting of Smale's proof into the functional framework was done by Morse, and also by John Milnor and by Cerf, André Gramain, and Bernard Morin following a suggestion of René Thom).
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 formula_67 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 formula_68-obstructions on formula_69 (formula_70) and formula_71 (formula_72) and by Kiyoshi Igusa, discovering obstructions of a similar nature on formula_69 (formula_74).