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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
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| :<math>f:M \to \mathbb R</math>
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| on a smooth manifold ''M'', their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space.
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| == An example ==
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| [[Marston Morse]] proved that, provided <math>M</math> is compact, any smooth function
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| :<math>f:M \to \Bbb R</math>
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| could be approximated by a [[Morse theory|Morse function]]. So for many purposes, one can replace arbitrary functions on <math>M</math> by Morse functions.
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| 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:
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| :<math>f_t(x)=(1/3)x^3-tx,\,</math>
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| as a 1-parameter family of functions on <math>M=\mathbb R</math>.
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| At time
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| :<math>t=-1\,</math>
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| it has no critical points, but at time
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| :<math>t=1\,</math>
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| it is a Morse function with two critical points
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| :<math>x=\pm 1.\,</math>
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| Jean Cerf<ref>[http://serge.mehl.free.fr/chrono/Cerf.html French mathematician, born 1928]</ref> 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 <math>t=0</math> an index 0 and index 1 critical point are created (as <math>t</math> increases).
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| == A ''stratification'' of an infinite-dimensional space ==
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| Let's return to the general case that <math>M</math> is a compact manifold.
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| Let <math>\operatorname{Morse}(M)</math> denote the space of Morse functions
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| :<math>f : M \to \mathbb R\,</math>
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| and <math>\operatorname{Func}(M)</math> the space of smooth functions
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| :<math>f : M \to \mathbb R.\,</math>
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| Morse proved that
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| :<math>\operatorname{Morse}(M) \subset \operatorname{Func}(M)\,</math>
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| is an open and dense subset in the <math>C^\infty</math> topology.
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| For the purposes of intuition, here is an analogy. Think of the Morse functions as the top-dimensional open stratum in a [[Topologically stratified space|stratification]] of <math>\operatorname{Func}(M)</math> (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 <math>\operatorname{Func}(M)</math> is infinite-dimensional if <math>M</math> is not a finite set. By assumption, the open co-dimension 0 stratum of <math>\operatorname{Func}(M)</math> is <math>\operatorname{Morse}(M)</math>, ie: <math>\operatorname{Func}(M)^0=\operatorname{Morse}(M)</math>. In a stratified space <math>X</math>, frequently <math>X^0</math> is disconnected. The '''essential property''' of the co-dimension 1 stratum <math>X^1</math> is that any path in <math>X</math> which starts and ends in <math>X^0</math> can be approximated by a path that intersects <math>X^1</math> transversely in finitely many points, and does not intersect <math>X^i</math> for any <math>i>1</math>.
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| Thus Cerf theory is the study of the positive co-dimensional strata of <math>\operatorname{Func}(M)</math>, i.e.: <math>\operatorname{Func}(M)^i</math> for <math>i>0</math>. In the case of
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| :<math>f_t(x)=x^3-tx,\,</math>
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| only for <math>t=0</math> is the function not Morse, and
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| :<math>f_0(x)=x^3\,</math>
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| has a cubic [[degenerate critical point]] corresponding to the birth/death transition.
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| == A single time parameter, statement of theorem ==
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| The [[Morse_theory#The_Morse_lemma|Morse Theorem]] asserts that if <math>f : M \to \mathbb R</math> is a Morse function, then near a critical point <math>p</math> it is conjugate to a function <math>g : \mathbb R^n \to \mathbb R</math> of the form
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| :<math>g(x_1,x_2,\cdots,x_n) = f(p) + \epsilon_1 x_1^2 + \epsilon_2 x_2^2 + \cdots + \epsilon_n x_n^2</math>
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| where <math>\epsilon_i \in \{\pm 1\}</math>.
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| Cerf's 1-parameter theorem asserts the '''essential property''' of the co-dimension one stratum.
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| Precisely, if <math>f_t : M \to \mathbb R</math> is a 1-parameter family of smooth functions on <math>M</math> with <math>t \in [0,1]</math>, and <math>f_0, f_1</math> Morse, then there exists a smooth 1-parameter family <math>F_t : M \to \mathbb R</math> such that <math>F_0 = f_0, F_1 = f_1</math>, <math>F</math> is uniformly close to <math>f</math> in the <math>C^k</math>-topology on functions <math>M \times [0,1] \to \mathbb R</math>. Moreover, <math>F_t</math> is Morse at all but finitely many times. At a non-Morse time the function has only one degenerate critical point <math>p</math>, and near that point the family <math>F_t</math> is conjugate to the family
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| :<math>g_t(x_1,x_2,\cdots,x_n) = f(p) + x_1^3+\epsilon_1 tx_1 + \epsilon_2 x_2^2 + \cdots + \epsilon_n x_n^2</math>
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| where <math> \epsilon_i \in \{\pm 1\}, t \in [-1,1]</math>. If <math>\epsilon_1 = -1</math> this is a 1-parameter family of functions where two critical points are created (as <math>t</math> increases), and for <math>\epsilon_1 = 1</math> it is a 1-parameter family of functions where two critical points are destroyed.
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| == Origins ==
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| The [[Piecewise linear manifold|PL]]-[[Jordan-Schönflies theorem|Schoenflies problem]] for <math>S^2 \subset \mathbb R^3</math> was solved by Alexander in 1924. His proof was adapted to the [[Smooth manifold|smooth]] case by Morse and Baiada. The '''essential property''' was used by Cerf in order to prove that every orientation-preserving [[diffeomorphism]] of [[3-sphere|<math>S^3</math>]] is isotopic to the identity,<ref>J.Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ<sub>4</sub>=0), Lecture Notes in Mathematics, No. 53. Springer-Verlag, Berlin-New York 1968.</ref> seen as a 1-parameter extension of the Schoenflies theorem for <math>S^2 \subset \mathbb R^3</math>. The corollary [[Exotic sphere|<math>\Gamma_4 = 0</math>]] at the time had wide implications in differential topology. The '''essential property''' was later used by Cerf to prove the [[Pseudoisotopy theorem|pseudo-isotopy theorem]]<ref>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.</ref> for high-dimensional simply-connected manifolds. The proof is a 1-parameter extension of Smale's proof of the [[h-cobordism|h-cobordism theorem]] (the rewriting of Smale's proof into the functional framework was done by Morse, also Milnor,<ref>J. Milnor, Lectures on the h-cobordism theorem, Notes by L.Siebenmann and J.Sondow, Princeton Math. Notes 1965</ref> and also by [http://www.maths.ed.ac.uk/~aar/surgery/cerf-gramain.pdf Cerf-Gramain-Morin] <ref>Le theoreme du h-cobordisme (Smale) Notes by Jean Cerf and Andre Gramain (Ecole Normale Superieure, 1968).</ref> following a suggestion of Thom).
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| Cerf's proof is built on the work of Thom and Mather.<ref>J. Mather, Classification of stable germs by R-algebras, Publ. Math. IHES (1969)</ref> A useful modern summary of Thom and Mather's work from the period is the book of Golubitsky and Guillemin.<ref>[[Marty Golubitsky|M. Golubitsky]], V.Guillemin. Stable Mappings and Their Singularities. Springer-Verlag Graduate Texts in Mathematics 14 (1973)</ref>
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| == Applications ==
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| Beside the above mentioned applications, Robion Kirby used Cerf Theory as a key step in justifying the [[Kirby calculus]].
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| == Generalization ==
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| A stratification of the complement of an infinite co-dimension subspace of the space of smooth maps <math>\{ f : M \to \mathbb R \}</math> was eventually developed by Sergeraert.<ref>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.</ref>
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| During the seventies, the classification problem for pseudo-isotopies of non-simply connected manifolds was solved by [[Allen Hatcher|Hatcher]] and Wagoner,<ref>Allen Hatcher and John Wagoner, Pseudo-isotopies of compact manifolds. Astérisque, No. 6. Société Mathématique de France, Paris, 1973. 275 pp.</ref> discovering [[K-theory|algebraic <math>K_i</math>]]-obstructions on <math>\pi_1 M</math> (<math>i=2</math>) and <math>\pi_2 M</math> (<math>i=1</math>) and by [[Kiyoshi Igusa|Igusa]], discovering obstructions of a similar nature on <math>\pi_1 M</math> (<math>i=3</math>).<ref>K.Igusa, Stability theorem for smooth pseudoisotopies. K-Theory 2 (1988), no. 1-2, vi+355.</ref>
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| == References ==
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| <references/>
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| [[Category:Differential topology]]
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| [[Category:Singularity theory]]
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Friends call him Royal Seyler. I am a manufacturing and distribution officer. The thing I adore most flower arranging and now I have time to consider on new things. He currently lives in Arizona and his mothers and fathers live nearby.
Take a look at my web site: extended auto warranty (please click the next document)