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
The theory is named after Jean Cerf, who initiated it in the late 1960s.
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?'
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
Think of the Morse functions as the top-dimensional open stratum in a stratification of
(we make no claim that such a stratification exists, but suppose one does).
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.
By assumption, the open co-dimension 0 stratum of
The essential property of the co-dimension 1 stratum
Thus Cerf theory is the study of the positive co-dimensional strata of
is the function not Morse, and has a cubic degenerate critical point corresponding to the birth/death transition.
Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum.
Morse, then there exists a smooth one-parameter family
At a non-Morse time the function has only one degenerate critical point
this is a one-parameter family of functions where two critical points are created (as
it is a one-parameter family of functions where two critical points are destroyed.
His proof was adapted to the smooth case by Morse and Emilio Baiada.
[1] The essential property was used by Cerf in order to prove that every orientation-preserving diffeomorphism of
is isotopic to the identity,[2] seen as a one-parameter extension of the Schoenflies theorem for
at the time had wide implications in differential topology.
The essential property was later used by Cerf to prove the pseudo-isotopy theorem[3] 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[4] and by Cerf, André Gramain, and Bernard Morin[5] following a suggestion of René Thom).
Cerf's proof is built on the work of Thom and John Mather.
[6] A useful modern summary of Thom and Mather's work from that period is the book of Marty Golubitsky and Victor Guillemin.
[7] Beside the above-mentioned applications, Robion Kirby used Cerf Theory as a key step in justifying the Kirby calculus.
A stratification of the complement of an infinite co-dimension subspace of the space of smooth maps
was eventually developed by Francis Sergeraert.
[8] During the seventies, the classification problem for pseudo-isotopies of non-simply connected manifolds was solved by Allen Hatcher and John Wagoner,[9] discovering algebraic
) and by Kiyoshi Igusa, discovering obstructions of a similar nature on