In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions
Contents
- An example
- A stratification of an infinite dimensional space
- A single time parameter statement of theorem
- Origins
- Applications
- Generalization
- References
on a smooth manifold M, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space.
An example
Marston Morse proved that, provided
could be approximated by a Morse function. So for many purposes, one can replace arbitrary functions on
As a next step, one could ask, 'if you have a 1-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 family:
as a 1-parameter family of functions on
it has no critical points, but at time
it is a Morse function with two critical points
Jean Cerf showed that a 1-parameter family of functions between two Morse functions could 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
A stratification of an infinite-dimensional space
Let's return to the general case that
and
Morse proved that
is an open and dense subset in the
For the purposes of intuition, here is an analogy. Think of the Morse functions as the top-dimensional open stratum in a stratification of
Thus Cerf theory is the study of the positive co-dimensional strata of
only for
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
where
Cerf's 1-parameter theorem asserts the essential property of the co-dimension one stratum.
Precisely, if
where
Origins
The PL-Schoenflies problem for
Cerf's proof is built on the work of Thom and Mather. A useful modern summary of Thom and Mather's work from the period is the book of Golubitsky and 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
During the seventies, the classification problem for pseudo-isotopies of non-simply connected manifolds was solved by Hatcher and Wagoner, discovering algebraic