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| In [[differential topology]], the '''transversality theorem''', also known as the [[René Thom|Thom]] Transversality Theorem, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that [[transversality (mathematics)|transversality]] is a [[generic property]]: any smooth map <math>f:X\rightarrow Y</math>, may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold <math>Z \subseteq Y</math>. Together with the [[Pontryagin-Thom construction]], it is the technical heart of [[cobordism theory]], and the starting point for [[surgery theory]]. The finite dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite dimensional parametrization using the infinite dimensional version of the transversality theorem.
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| == Finite dimensional version ==
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| === Previous definitions ===
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| Let <math>f:X\rightarrow Y</math> be a smooth map between manifolds, and let <math>Z</math> be a submanifold of <math>Y</math>. We say that <math>f</math> is transverse to <math>Z</math>, denoted as <math>f \pitchfork Z</math>, if and only if for every <math>x\in f^{-1}\left(Z\right)</math> we have <math>Im\left( df_x \right) + T_{f\left(x\right)} Z = T_{f\left(x\right)} Y</math>.
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| An important result about transversality states that if a smooth map <math>f</math> is transverse to <math>Z</math>, then <math>f^{-1}\left(Z\right)</math> is a regular submanifold of <math>X</math>.
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| If <math>X</math> is a [[Manifold#Manifold_with_boundary|manifold with boundary]], then we can define the restriction of the map <math>f</math> to the boundary, as <math>\partial f:\partial X \rightarrow Y</math>. The map <math>\partial f</math> is smooth, and it allow us to state an extension of the previous result: if both <math>f \pitchfork Z</math> and <math>\partial f \pitchfork Z</math>, then <math>f^{-1}\left(Z\right)</math> is a regular submanifold of <math>X</math> with boundary, and <math>\partial f^{-1}\left( Z \right) = f^{-1}\left( Z \right) \cap \partial X</math>.
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| === Parametric transversality theorem ===
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| Consider the map <math>F:X\times S \rightarrow Y</math> and define <math>f_s\left(x\right) = F\left(x,s\right)</math>. This generates a family of mappings <math>f_s:X\rightarrow Y</math>. We require that the family vary smoothly by assuming <math>S</math> to be a manifold and <math>F</math> to be smooth.
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| The statement of the ''parametric transversality theorem'' is:
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| Suppose that <math>F:X \times S \rightarrow Y</math> is a smooth map of manifolds, where only <math>X</math> has boundary, and let <math>Z</math> be any submanifold of <math>Y</math> without boundary. If both <math>F</math> and <math>\partial F</math> are transverse to <math>Z</math>, then for almost every <math>s\in S</math>, both <math>f_s</math> and <math>\partial f_s</math> are transverse to <math>Z</math>.
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| === More general transversality theorems ===
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| The parametric transversality theorem above is sufficient for many elementary applications (see the book by Guillemin and Pollack).
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| There are more powerful statements (collectively known as ''transversality theorems'') that imply the parametric transversality theorem and are needed for more advanced applications.
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| Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or, in some cases, only a dense <math>G_\delta</math>) subset of the set of mappings. To make such a statement precise, it is necessary to define the space of mappings under consideration, and what is the topology in it. There are several possibilities; see the book by Hirsch.
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| What is usually understood by ''Thom's transversality theorem'' is a more powerful statement about [[Jet (mathematics)|jet]] transversality. See the books by Hirsch and by Golubitsky and Guillemin. The original reference is Thom, Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59–71.
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| [[John Mather (mathematician)|John Mather]] proved in the 1970s an even more general result called the ''[[Jet_(mathematics)#Multijets|multijet]] transversality theorem''. See the book by Golubitsky and Guillemin.
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| == Infinite dimensional version ==
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| The infinite dimensional version of the transversality theorem takes into account that the manifolds may be modeled in Banach spaces.
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| === Formal statement ===
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| Suppose that <math>F:X \times S \rightarrow Y</math> is a <math>C^k</math> map of <math>C^\infty</math>-Banach manifolds. Assume that
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| i) <math>X</math>, <math>S</math> and <math>Y</math> are nonempty, metrizable <math>C^\infty</math>-Banach manifolds with chart spaces over a field <math>\mathbb{K}</math>.
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| ii) The <math>C^k</math>-map <math>F:X \times S \rightarrow Y</math> with <math>k\geq 1</math> has <math>y</math> as a regular value.
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| iii) For each parameter <math>s\in S</math>, the map <math>f_s\left(x\right) = F\left(x,s\right)</math> is a [[Fredholm operator|Fredholm map]], where <math>\mathop{\mathrm{ind}} Df_s\left(x\right)<k</math> for every <math>x\in f_{s}^{-1}\left( \left\{y\right\} \right)</math>.
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| iv) The convergence <math>s_n \rightarrow s</math> on <math>S</math> as <math>n \rightarrow \infty</math> and <math>F\left(x_n, s_n \right) = y</math> for all <math>n</math> implies the existence of a convergent subsequence <math>x_n \rightarrow x</math> as <math>n \rightarrow \infty</math> with <math>x\in X</math>.
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| If Assumptions i-iv hold, then there exists an open, dense subset <math>S_0</math> of <math>S</math> such that <math>y</math> is a regular value of <math>f_s</math> for each parameter <math>s\in S_0</math>.
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| Now, fix an element <math>s\in S_0</math>. If there exists a number <math>n\geq 0</math> with <math>\mathrm{ind} Df_s\left( x \right) = n</math> for all solutions <math>x\in X </math> of <math>f_s\left(x \right) = y</math>, then the solution set <math>f_s^{-1}\left( \left\{y \right\} \right)</math> consists of an <math>n</math>-dimensional <math>C^k</math>-Banach manifold or the solution set is empty.
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| Note that if <math>\mathrm{ind} Df_s\left( x \right) = 0</math> for all the solutions of <math>f_s\left(x \right) = y</math>, then there exists an open dense subset <math>S_0</math> of <math>S</math> such that there are at most finitely many solutions for each fixed parameter <math>s\in S_0</math>. In addition, all these solutions are regular.
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| ==References==
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| * René Thom, ''Quelques propriétés globales des variétés differentiables.'' Comm. Math. Helv. 28 (1954), pp. 17–86.
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| * René Thom, "Un lemme sur les applications différentiables." Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59–71.
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| * [[Marty Golubitsky|Martin Golubitsky]] and Victor Guillemin. "Stable mappings and their singularities". Springer-Verlag, 1974. ISBN 978-0-387-90073-5.
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| * Guillemin, Victor and Pollack, Alan (1974). ''Differential Topology''. Prentice-Hall. ISBN 0-13-212605-2.
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| * Morris W. Hirsch. "Differential topology". Springer, 1976. ISBN 0-387-90148-5
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| * V. I. Arnold. "Geometrical methods in the theory of ordinary differential equations". Springer, 1988. ISBN 0-387-96649-8
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| * Zeidler, Eberhard (1997) ''Nonlinear Functional Analysis and Its Applications: Part 4: Applications to Mathematical Physics''. Springer. ISBN 978-0-387-96499-7.
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| [[Category:Theorems in differential topology]]
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| [[Category:Differential geometry]]
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My name's Gabriella Newman but everybody calls me Gabriella. I'm from Austria. I'm studying at the college (3rd year) and I play the Bass Guitar for 9 years. Usually I choose songs from the famous films ;).
I have two brothers. I love Shortwave listening, watching TV (Breaking Bad) and Sculling or Rowing.
Look into my homepage - dieta-malyshevoyru