Cerf theory

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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 M, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space.

An example[edit]

Marston Morse proved that, provided is compact, any smooth function

could be approximated by a Morse function. So for many purposes, one can replace arbitrary functions on by Morse functions.

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 . At time

it has no critical points, but at time

it is a Morse function with two critical points

Jean Cerf[1] 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 an index 0 and index 1 critical point are created (as increases).

A stratification of an infinite-dimensional space[edit]

Let's return to the general case that is a compact manifold. Let denote the space of Morse functions

and the space of smooth functions

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 (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 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[edit]

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 1-parameter theorem asserts the essential property of the co-dimension one stratum.

Precisely, if is a 1-parameter family of smooth functions on with , and Morse, then there exists a smooth 1-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 1-parameter family of functions where two critical points are created (as increases), and for it is a 1-parameter family of functions where two critical points are destroyed.

Origins[edit]

The PL-Schoenflies problem for was solved by Alexander in 1924. His proof was adapted to the smooth case by Morse and Baiada. 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 1-parameter extension of the Schoenflies theorem for . The corollary 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 1-parameter extension of Smale's proof of the h-cobordism theorem (the rewriting of Smale's proof into the functional framework was done by Morse, also Milnor,[4] and also by Cerf-Gramain-Morin [5] following a suggestion of Thom).

Cerf's proof is built on the work of Thom and Mather.[6] A useful modern summary of Thom and Mather's work from the period is the book of Golubitsky and Guillemin.[7]

Applications[edit]

Beside the above-mentioned applications, Robion Kirby used Cerf Theory as a key step in justifying the Kirby calculus.

Generalization[edit]

A stratification of the complement of an infinite co-dimension subspace of the space of smooth maps was eventually developed by Sergeraert.[8]

During the seventies, the classification problem for pseudo-isotopies of non-simply connected manifolds was solved by Hatcher and Wagoner,[9] discovering algebraic -obstructions on () and () and by Igusa, discovering obstructions of a similar nature on ().[10]

References[edit]

  1. ^ French mathematician, born 1928
  2. ^ J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.
  3. ^ J.Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. No. 39 (1970) 5--173.
  4. ^ J. Milnor, Lectures on the h-cobordism theorem, Notes by L.Siebenmann and J.Sondow, Princeton Math. Notes 1965
  5. ^ Le theoreme du h-cobordisme (Smale) Notes by Jean Cerf and Andre Gramain (Ecole Normale Superieure, 1968).
  6. ^ J. Mather, Classification of stable germs by R-algebras, Publ. Math. IHES (1969)
  7. ^ M. Golubitsky, V.Guillemin. Stable Mappings and Their Singularities. Springer-Verlag Graduate Texts in Mathematics 14 (1973)
  8. ^ F. Sergeraert "Un theoreme de fonctions implicites sur certains espaces de Fréchet et quelques applications," Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 599-660.
  9. ^ Allen Hatcher and John Wagoner, Pseudo-isotopies of compact manifolds. Astérisque, No. 6. Société Mathématique de France, Paris, 1973. 275 pp.
  10. ^ K.Igusa, Stability theorem for smooth pseudoisotopies. K-Theory 2 (1988), no. 1-2, vi+355.