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]