Cerf theory

Marston Morse proved that, provided ${\displaystyle M}$ is compact, any smooth function ${\displaystyle f\colon M\to \mathbb {R} }$ can be approximated by a Morse function. Thus, for many purposes, one can replace arbitrary functions on ${\displaystyle M}$ 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 ${\displaystyle M=\mathbb {R} }$ given by

${\displaystyle f_{t}(x)=(1/3)x^{3}-tx.}$

At time ${\displaystyle t=-1}$, it has no critical points, but at time ${\displaystyle t=1}$, it is a Morse function with two critical points at ${\displaystyle x=\pm 1}$.

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 ${\displaystyle t=0}$, an index 0 and index 1 critical point are created as ${\displaystyle t}$ increases.

Returning to the general case where ${\displaystyle M}$ is a compact manifold, let ${\displaystyle \operatorname {Morse} (M)}$ denote the space of Morse functions on ${\displaystyle M}$, and ${\displaystyle \operatorname {Func} (M)}$ the space of real-valued smooth functions on ${\displaystyle M}$. Morse proved that ${\displaystyle \operatorname {Morse} (M)\subset \operatorname {Func} (M)}$ is an open and dense subset in the ${\displaystyle C^{\infty }}$ 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 ${\displaystyle \operatorname {Func} (M)}$ (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 ${\displaystyle \operatorname {Func} (M)}$ is infinite-dimensional if ${\displaystyle M}$ is not a finite set. By assumption, the open co-dimension 0 stratum of ${\displaystyle \operatorname {Func} (M)}$ is ${\displaystyle \operatorname {Morse} (M)}$, i.e.: ${\displaystyle \operatorname {Func} (M)^{0}=\operatorname {Morse} (M)}$. In a stratified space ${\displaystyle X}$, frequently ${\displaystyle X^{0}}$ is disconnected. The essential property of the co-dimension 1 stratum ${\displaystyle X^{1}}$ is that any path in ${\displaystyle X}$ which starts and ends in ${\displaystyle X^{0}}$ can be approximated by a path that intersects ${\displaystyle X^{1}}$ transversely in finitely many points, and does not intersect ${\displaystyle X^{i}}$ for any ${\displaystyle i>1}$.

Thus Cerf theory is the study of the positive co-dimensional strata of ${\displaystyle \operatorname {Func} (M)}$, i.e.: ${\displaystyle \operatorname {Func} (M)^{i}}$ for ${\displaystyle i>0}$. In the case of

${\displaystyle f_{t}(x)=x^{3}-tx}$,

only for ${\displaystyle t=0}$ is the function not Morse, and

${\displaystyle f_{0}(x)=x^{3}}$

has a cubic degenerate critical point corresponding to the birth/death transition.

The Morse Theorem asserts that if ${\displaystyle f\colon M\to \mathbb {R} }$ is a Morse function, then near a critical point ${\displaystyle p}$ it is conjugate to a function ${\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} }$ of the form

${\displaystyle g(x_{1},x_{2},\dotsc ,x_{n})=f(p)+\epsilon _{1}x_{1}^{2}+\epsilon _{2}x_{2}^{2}+\dotsb +\epsilon _{n}x_{n}^{2}}$

where ${\displaystyle \epsilon _{i}\in \{\pm 1\}}$.

Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum.

Precisely, if ${\displaystyle f_{t}\colon M\to \mathbb {R} }$ is a one-parameter family of smooth functions on ${\displaystyle M}$ with ${\displaystyle t\in [0,1]}$, and ${\displaystyle f_{0},f_{1}}$ Morse, then there exists a smooth one-parameter family ${\displaystyle F_{t}\colon M\to \mathbb {R} }$ such that ${\displaystyle F_{0}=f_{0},F_{1}=f_{1}}$, ${\displaystyle F}$ is uniformly close to ${\displaystyle f}$ in the ${\displaystyle C^{k}}$-topology on functions ${\displaystyle M\times [0,1]\to \mathbb {R} }$. Moreover, ${\displaystyle F_{t}}$ is Morse at all but finitely many times. At a non-Morse time the function has only one degenerate critical point ${\displaystyle p}$, and near that point the family ${\displaystyle F_{t}}$ is conjugate to the family

${\displaystyle g_{t}(x_{1},x_{2},\dotsc ,x_{n})=f(p)+x_{1}^{3}+\epsilon _{1}tx_{1}+\epsilon _{2}x_{2}^{2}+\dotsb +\epsilon _{n}x_{n}^{2}}$

where ${\displaystyle \epsilon _{i}\in \{\pm 1\},t\in [-1,1]}$. If ${\displaystyle \epsilon _{1}=-1}$ this is a one-parameter family of functions where two critical points are created (as ${\displaystyle t}$ increases), and for ${\displaystyle \epsilon _{1}=1}$ it is a one-parameter family of functions where two critical points are destroyed.

The PL-Schoenflies problem for ${\displaystyle S^{2}\subset \mathbb {R} ^{3}}$ was solved by J. W. Alexander in 1924. 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 ${\displaystyle S^{3}}$ is isotopic to the identity,[2] seen as a one-parameter extension of the Schoenflies theorem for ${\displaystyle S^{2}\subset \mathbb {R} ^{3}}$. The corollary ${\displaystyle \Gamma _{4}=0}$ 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 ${\displaystyle \{f\colon M\to \mathbb {R} \}}$ 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 ${\displaystyle K_{i}}$-obstructions on ${\displaystyle \pi _{1}M}$ (${\displaystyle i=2}$) and ${\displaystyle \pi _{2}M}$ (${\displaystyle i=1}$) and by Kiyoshi Igusa, discovering obstructions of a similar nature on ${\displaystyle \pi _{1}M}$ (${\displaystyle i=3}$).[10]