In
mathematics, at the junction of
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
and
differential topology, Cerf theory is the study of families of smooth real-valued functions
:
on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. The theory is named after
Jean Cerf, who initiated it in the late 1960s.
An example
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known a ...
proved that, provided
is compact, any
smooth function can be approximated by a
Morse function
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
. Thus, for many purposes, one can replace arbitrary functions on
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
given by
:
At time
, it has no critical points, but at time
, it is a Morse function with two critical points at
.
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
, an index 0 and index 1 critical point are created as
increases.
A ''stratification'' of an infinite-dimensional space
Returning to the general case where
is a compact manifold, let
denote the space of Morse functions on
, and
the space of real-valued smooth functions on
. 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
Stratification may refer to:
Mathematics
* Stratification (mathematics), any consistent assignment of numbers to predicate symbols
* Data stratification in statistics
Earth sciences
* Stable and unstable stratification
* Stratification, or st ...
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
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 one-parameter theorem asserts the essential property of the co-dimension one stratum.
Precisely, if
is a one-parameter family of smooth functions on
with