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Difference between revisions of "Thom-Boardman singularities"

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Consider a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200501.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200503.png" /> are smooth manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200505.png" />, respectively (by smooth one understands: class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200506.png" />; cf. also [[Manifold|Manifold]]). In order to understand the local structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200507.png" />, it is natural to distinguish among points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200508.png" /> according to the rank of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t1200509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005010.png" /> denotes the tangent space. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005011.png" />, set:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005012.png" /></td> </tr></table>
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Out of 130 formulas, 130 were replaced by TEX code.-->
  
Using local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005014.png" />, this set is defined locally by the vanishing of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005015.png" />-minors of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005016.png" />-matrix of first-order partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005017.png" />. If one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005018.png" /> is a smooth submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005019.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005020.png" /> one can define
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Consider a smooth mapping $f : V ^ { n } \rightarrow W ^ { p }$, where $V ^ { n }$ and $W ^ { p }$ are smooth manifolds of dimension $n$ and $p$, respectively (by smooth one understands: class $\mathcal{C} ^ { \infty }$; cf. also [[Manifold|Manifold]]). In order to understand the local structure of $f$, it is natural to distinguish among points $x \in \mathbf{V}$ according to the rank of the derivative $d f _ { x } : T V _ { x } \rightarrow T W _ { f  ( x )}$, where $T ( . )$ denotes the tangent space. For $i \in \{ 0 , \dots , n \}$, set:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005021.png" /></td> </tr></table>
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\begin{equation*} \Sigma ^ { i } ( f ) = \{ x \in V : \operatorname { dim } \operatorname { Ker } d f _ { x } = i \}. \end{equation*}
  
This can be visualized as follows: at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005022.png" /> there are two vector subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005023.png" />, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005025.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005026.png" /> if and only if the intersection of these two subspaces has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005027.png" />.
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Using local coordinates on $V$ and $W$, this set is defined locally by the vanishing of the $( n - i + 1 ) \times ( n - i + 1 )$-minors of the $( n \times p )$-matrix of first-order partial derivatives of $f$. If one assumes that $\Sigma ^ { i } ( f )$ is a smooth submanifold of $V$, for $0 \leq i \leq i$ one can define
  
Again, if one assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005028.png" /> is a smooth submanifold, then one can define a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005029.png" />, etc. At the end, one has partitioned the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005030.png" /> into a collection of locally closed submanifolds, such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005031.png" /> to each submanifold is of maximal rank; in fact, if the local equations defining the various submanifolds of the collection are of maximal rank, it turns out that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005032.png" /> the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005033.png" /> to each submanifold is an immersion, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005034.png" /> the same holds except at the points of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005035.png" />, where it is a submersion.
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\begin{equation*} \Sigma ^ { i , j } ( f ) = \Sigma ^ { j } ( f | _ { \Sigma ^ { i } ( f ) } ). \end{equation*}
  
This program has been initiated by R. Thom in his seminal paper [[#References|[a9]]], inspired by earlier work of H. Whitney [[#References|[a10]]]. Thom handles completely the first-order case, by showing that for a generic mapping (i.e. for mappings in a dense subset of all smooth mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005037.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005038.png" /> is a locally closed submanifold of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005040.png" />, and that for the closure one has:
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This can be visualized as follows: at a point $x \in \Sigma ^ { i } ( f )$ there are two vector subspaces of $T V _ { X }$, namely $\operatorname {Ker} d f_x$ and $T ( \Sigma ^ { i } ( f ) ) _ { x }$. Then $x \in \Sigma ^ { i , j } ( f )$ if and only if the intersection of these two subspaces has dimension $j$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005041.png" /></td> </tr></table>
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Again, if one assumes that $\Sigma ^ { i , j } ( f )$ is a smooth submanifold, then one can define a subset $\Sigma ^ { i , j , k } ( f ) \subset \Sigma ^ { i , j } ( f )$, etc. At the end, one has partitioned the manifold $V$ into a collection of locally closed submanifolds, such that the restriction of $f$ to each submanifold is of maximal rank; in fact, if the local equations defining the various submanifolds of the collection are of maximal rank, it turns out that for $n \leq p$ the restriction of $f$ to each submanifold is an immersion, while for $n &gt; p$ the same holds except at the points of rank $p - n$, where it is a submersion.
  
This is done by writing the mapping locally as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005043.png" /> open, then associating to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005044.png" /> the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005045.png" /> of the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005046.png" />. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005047.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005048.png" />, a condition defining a [[Schubert variety|Schubert variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005049.png" /> in the [[Grassmann manifold|Grassmann manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005051.png" />-planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005052.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005053.png" /> is seen locally as the pull-back of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005054.png" /> by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005055.png" />.
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This program has been initiated by R. Thom in his seminal paper [[#References|[a9]]], inspired by earlier work of H. Whitney [[#References|[a10]]]. Thom handles completely the first-order case, by showing that for a generic mapping (i.e. for mappings in a dense subset of all smooth mappings from $V$ to $W$), $\Sigma ^ { i } ( f )$ is a locally closed submanifold of codimension $i ( p - n + i )$ of $V$, and that for the closure one has:
  
This approach is exemplary, because it presents the singular locus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005056.png" /> as the pull-back of a universal situation, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005057.png" />; it is then straightforward to show that for most mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005058.png" />, the induced local mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005059.png" /> are transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005060.png" />, and hence that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005061.png" /> is a locally closed smooth submanifold of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005063.png" />. Moreover, this approach can serve as basis for the computation of the cohomology class that is Poincaré dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005064.png" />, which can be interpreted as the first obstruction to having a homotopy from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005065.png" /> to a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005066.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005067.png" /> (see [[#References|[a9]]], p. 80, or [[#References|[a5]]], Prop. 1.3; the dual classes for second-order singularities have been computed in [[#References|[a7]]] and [[#References|[a6]]]).
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\begin{equation*} \overline { \Sigma } \square ^ { i } ( f ) = \bigcup _ { h \geq i } \Sigma ^ { i } ( f ). \end{equation*}
  
The complete proof that the process of decomposition of the source of a generic smooth mapping can be carried out successfully has been given by J.M. Boardman [[#References|[a1]]]. See [[Singularities of differentiable mappings|Singularities of differentiable mappings]] for the notions of jet space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005070.png" />-jet extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005071.png" /> of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005072.png" />, used below.
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This is done by writing the mapping locally as $f : U \rightarrow {\bf R} ^ { n }$, $U \subset \mathbf{R} ^ { n }$ open, then associating to each $x \in U$ the graph $\Gamma _ { x } \subset \mathbf{R} ^ { n } \times \mathbf{R} ^ { p }$ of the derivative $d f _ { x } : \mathbf{R} ^ { n } \rightarrow \mathbf{R} ^ { p },$. Clearly, $x \in \Sigma ^ { i } ( f )$ if and only if $\operatorname { dim } ( \Gamma _ { X } \cap ( \mathbf{R} ^ { n } \times \{ 0 \} ) ) = i$, a condition defining a [[Schubert variety|Schubert variety]] $F_{i}$ in the [[Grassmann manifold|Grassmann manifold]] $G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$ of $n$-planes in $\mathbf{R} ^ { n } \times \mathbf{R} ^ { p }$. Thus, $\Sigma ^ { i } ( f )$ is seen locally as the pull-back of $F _ { i } \subset G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$ by the mapping $x \mapsto \Gamma _ { x }$.
  
For smooth manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005074.png" />, and integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005075.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005076.png" />, one defines the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005077.png" /> of the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005078.png" />-jets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005079.png" />; it can be proved that these are locally closed smooth submanifolds, and that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005080.png" /> is a mapping whose jet extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005081.png" /> are transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005083.png" />, then, setting
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This approach is exemplary, because it presents the singular locus $\Sigma ^ { i } ( f )$ as the pull-back of a universal situation, namely $F _ { i } \subset G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$; it is then straightforward to show that for most mappings $f$, the induced local mappings $U \rightarrow G _ { n } ( {\bf R} ^ { n } \times {\bf R} ^ { p } )$ are transversal to $F_{i}$, and hence that $\Sigma ^ { i } ( f )$ is a locally closed smooth submanifold of codimension $i ( p - n + i )$ of $V$. Moreover, this approach can serve as basis for the computation of the cohomology class that is Poincaré dual to $\overline { \Sigma } \square ^ { i } ( f )$, which can be interpreted as the first obstruction to having a homotopy from $f$ to a mapping $g : V \rightarrow W$ for which $\Sigma ^ { i } ( g ) = \emptyset$ (see [[#References|[a9]]], p. 80, or [[#References|[a5]]], Prop. 1.3; the dual classes for second-order singularities have been computed in [[#References|[a7]]] and [[#References|[a6]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005084.png" /></td> </tr></table>
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The complete proof that the process of decomposition of the source of a generic smooth mapping can be carried out successfully has been given by J.M. Boardman [[#References|[a1]]]. See [[Singularities of differentiable mappings|Singularities of differentiable mappings]] for the notions of jet space $J ^ { r } ( V , W )$ and $r$-jet extension $j ^ { r } ( f )$ of a mapping $f : V \rightarrow W$, used below.
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For smooth manifolds $V ^ { n }$ and $W ^ { p }$, and integers $i _1 , \ldots , i _ { r }$, with $n \geq i _ { 1 } \geq \ldots \geq i _ { r } \geq 0$, one defines the subsets $\sum ^ { i _ { 1 }  , \dots , i _ { r }}$ of the space of $r$-jets $J ^ { r } ( V , W )$; it can be proved that these are locally closed smooth submanifolds, and that if $f : V \rightarrow W$ is a mapping whose jet extensions $j ^ { s } ( f ) : V \rightarrow J ^ { s } ( V , W )$ are transversal to $\sum ^ { i _ { 1 }  , \dots , i _ { s }}$, $s = 1 , \dots , r$, then, setting
 +
 
 +
\begin{equation*} \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f ) = j ^ { r } ( f ) ^ { - 1 } ( \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( V , W ) ), \end{equation*}
  
 
one obtains:
 
one obtains:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005085.png" /></td> </tr></table>
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\begin{equation*} \Sigma ^ { i _ { 1 } } ( f ) = \{ x \in V : \operatorname { dim } \operatorname { Ker } ( d f _ { x } ) = i _ { 1 } \}, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005086.png" /></td> </tr></table>
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\begin{equation*} \dots \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f ) = \Sigma ^ { i _ { r } } ( f | _ { \Sigma ^ { i _ { 1 } } , \ldots , i _ { r - 1 } ( f ) } ). \end{equation*}
  
The codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005087.png" /> equals
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The codimension of $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$ equals
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005088.png" /></td> </tr></table>
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\begin{equation*} ( p - n + i _ { 1 } ) \cdot \mu _ { i _ { 1 } , \dots , i _ { r } } - ( i _ { 1 } - i _ { 2 } ) \cdot \mu _ { i _ { 2 } , \dots , i _ { r } } \dots  \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005089.png" /></td> </tr></table>
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\begin{equation*} \ldots - ( i _ { r  - 1} - i _ { r } ) \cdot \mu _ { i _ { r } }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005090.png" /> is the number of sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005091.png" /> satisfying
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where $\mu _ { i _ { 1 } , \ldots , i _ { s } }$ is the number of sequences $( j _ { 1 } , \dots , j _ { s } )$ satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005092.png" /></td> </tr></table>
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\begin{equation*} \left\{ \begin{array} { l } { j _ { 1 } \geq \ldots \geq j _ { s }; } \\ { i _ { s } \geq j _ { s } \geq 0 \quad \forall s , 1 \leq s \leq r, } \\ { j _ { 1 } &gt; 0 .} \end{array} \right. \end{equation*}
  
Moreover, local equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005093.png" /> can be given explicitly, in terms of the ideal generated by the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005094.png" />, in some local coordinates, and its Jacobian extensions, an operation which adds to an ideal of functions certain minors of the matrix of their first-order derivatives.
+
Moreover, local equations for $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$ can be given explicitly, in terms of the ideal generated by the components of $f$, in some local coordinates, and its Jacobian extensions, an operation which adds to an ideal of functions certain minors of the matrix of their first-order derivatives.
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005095.png" /> are called Thom–Boardman singularities.
+
The $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( V , W )$ are called Thom–Boardman singularities.
  
 
An alternative, more concise approach to Thom–Boardman singularities has been given later by J.N. Mather [[#References|[a2]]], and an algebraic approach can be found in [[#References|[a4]]].
 
An alternative, more concise approach to Thom–Boardman singularities has been given later by J.N. Mather [[#References|[a2]]], and an algebraic approach can be found in [[#References|[a4]]].
  
Thom's transversality theorem [[#References|[a8]]] implies that the set of mappings that are transversal to all possible Thom–Boardman singularities, that one may call generic mappings, is dense in the space of all mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005096.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005097.png" />. So now one may ask how useful are Thom–Boardman singularities in the understanding of generic mappings.
+
Thom's transversality theorem [[#References|[a8]]] implies that the set of mappings that are transversal to all possible Thom–Boardman singularities, that one may call generic mappings, is dense in the space of all mappings from $V$ to $W$. So now one may ask how useful are Thom–Boardman singularities in the understanding of generic mappings.
  
In some cases, they allow a full classification. This is so, for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005099.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050102.png" />, by a result of B. Morin [[#References|[a3]]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050104.png" /> one finds the catastrophes of the fold, the cusp, the swallowtail, and the butterfly, respectively (see [[Thom catastrophes|Thom catastrophes]]).
+
In some cases, they allow a full classification. This is so, for example, if $n \leq p$ and $i _ { 1 } = \ldots = i _ { r } = 1$, or $n &gt; p$ and $i_ 1 = n - p$, $i _ { 2 } = \ldots = i _ { r } = 1$, by a result of B. Morin [[#References|[a3]]]; for $n = p$ and $r = 1,2,3,4$ one finds the catastrophes of the fold, the cusp, the swallowtail, and the butterfly, respectively (see [[Thom catastrophes|Thom catastrophes]]).
  
In general, Thom–Boardman singularities allow a very useful first approach to the understanding of the structure of a mapping; however, they are not fine enough to provide an, even coarse, classification. Indeed, as pointed out by I.R. Porteous [[#References|[a6]]], a generic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050105.png" /> can present the singularities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050107.png" />, both of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050108.png" />, and some isolated points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050109.png" />, called parabolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050111.png" />-points by Porteous, can be in the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050112.png" />; the structure of such a mapping is definitely different at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050113.png" />-parabolic and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050114.png" />-non-parabolic points. Similar phenomena occur in other dimensions.
+
In general, Thom–Boardman singularities allow a very useful first approach to the understanding of the structure of a mapping; however, they are not fine enough to provide an, even coarse, classification. Indeed, as pointed out by I.R. Porteous [[#References|[a6]]], a generic mapping $f : \mathbf{R} ^ { 5 } \rightarrow \mathbf{R} ^ { 5 }$ can present the singularities $\Sigma ^ { 2 }$ and $\Sigma ^ { 1,1,1,1 }$, both of dimension $1$, and some isolated points of $\Sigma ^ { 2 }$, called parabolic $\Sigma ^ { 2 }$-points by Porteous, can be in the closure of $\Sigma ^ { 1,1,1,1 }$; the structure of such a mapping is definitely different at $\Sigma ^ { 2 }$-parabolic and $\Sigma ^ { 2 }$-non-parabolic points. Similar phenomena occur in other dimensions.
  
 
In fact, Thom–Boardman singularities provide a partition of the source of a generic mapping into locally closed submanifolds, but the closure of a submanifold is not necessarily a union of similar submanifolds.
 
In fact, Thom–Boardman singularities provide a partition of the source of a generic mapping into locally closed submanifolds, but the closure of a submanifold is not necessarily a union of similar submanifolds.
  
When studying the equations of Thom–Boardman singularities, an interesting device shows up: the intrinsic derivative, first studied by Porteous (see [[#References|[a5]]]). In general, derivatives of order higher than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050115.png" /> are not intrinsic, in the sense that are affected by higher derivatives of coordinate changes, not only the linear part of them. However, it turns out that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050116.png" />, then a suitable combination of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050117.png" /> derivatives, restricted to appropriate subspaces, is intrinsic. The simplest case is that of the second intrinsic derivative; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050118.png" />, then the bilinear mapping induced by the second derivative,
+
When studying the equations of Thom–Boardman singularities, an interesting device shows up: the intrinsic derivative, first studied by Porteous (see [[#References|[a5]]]). In general, derivatives of order higher than $1$ are not intrinsic, in the sense that are affected by higher derivatives of coordinate changes, not only the linear part of them. However, it turns out that if $x \in \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$, then a suitable combination of the first $r + 1$ derivatives, restricted to appropriate subspaces, is intrinsic. The simplest case is that of the second intrinsic derivative; if $x \in \Sigma ^ { i _ { 1 } } ( f )$, then the bilinear mapping induced by the second derivative,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050119.png" /></td> </tr></table>
+
\begin{equation*} \widetilde { d ^ { 2 } f _ { x } } : K _ { x } \times T V _ { x } \rightarrow Q _ { x }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050121.png" /> is intrinsic, as one can check easily. In the special case of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050122.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050123.png" /> is a critical point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050124.png" /> is the well-known Hessian bilinear form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050125.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050126.png" /> (cf. also [[Hessian matrix|Hessian matrix]]), whose signature determines completely the local structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050127.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050128.png" />.
+
where $K _ { x } = \operatorname { Ker } ( d f _ { x } )$, $Q _ { x } = T W _ { x } / \operatorname { Im } ( d f _ { x } )$ is intrinsic, as one can check easily. In the special case of a function $f : V ^ { n } \rightarrow \mathbf{R}$, if $x \in \Sigma ^ { n } ( f )$ is a critical point, then $\widetilde{ d ^ { 2 } f } _ { x } : \mathbf{R} ^ { n } \times \mathbf{R} ^ { n } \rightarrow \mathbf{R}$ is the well-known Hessian bilinear form of $f$ at $x$ (cf. also [[Hessian matrix|Hessian matrix]]), whose signature determines completely the local structure of $f$ near $x$.
  
The intrinsic derivative can be used to refine Thom–Boardman singularities; for example, for a generic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050129.png" />:
+
The intrinsic derivative can be used to refine Thom–Boardman singularities; for example, for a generic mapping $f : V ^ { n } \rightarrow W ^ { n }$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050130.png" /></td> </tr></table>
+
\begin{equation*} \Sigma ^ { 2 _ \text { parabolic } } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050131.png" /></td> </tr></table>
+
\begin{equation*} = \left\{ x \in \Sigma ^ { 2 } ( f ) : \begin{array}{c} { \exists \text {a line }\ \text{l} \subset K _ { x } } \\ { \text{such that} \ \widetilde{d^{2}f_x}|_{\text{l}\times \text{l}} } \end{array} \right\}. \end{equation*}
  
 
An inductive definition of the intrinsic derivatives is provided in [[#References|[a1]]]; so far, it has not been tried to refine systematically Thom–Boardman singularities using them.
 
An inductive definition of the intrinsic derivatives is provided in [[#References|[a1]]]; so far, it has not been tried to refine systematically Thom–Boardman singularities using them.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.M. Boardman,   "Singularities of differentiable maps" ''Publ. Math. IHES'' , '''33''' (1967) pp. 383–419</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.N. Mather,   "On Thom–Boardman singularities" M.M. Peixoto (ed.) , ''Dynamical Systems, Proc. Symp. Univ. Bahia, 1971'' , Acad. Press (1973) pp. 233–248</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> B. Morin,   "Formes canoniques des singularités d'une application différentiable" ''C.R. Acad. Sci. Paris'' , '''260''' (1965) pp. 5662–5665; 6503–6506</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> B. Morin,   "Calcul jacobien" ''Thèse Univ. Paris–Sud centre d'Orsay'' (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I.R. Porteous,   "Simple singularities of maps" , ''Proc. Liverpool Singularities Symp.'' , ''Lecture Notes Math.'' , '''192''' , Springer (1971) pp. 286–312</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> I.R. Porteous,   "The second order decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t120050132.png" />"  ''Topology'' , '''11''' (1972) pp. 325–334</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> F. Ronga,   "Le calcul des classes duales aux singularités de Boardman d'ordre deux" ''Comment. Math. Helvetici'' , '''47''' (1972) pp. 15–35</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Thom,   "Un lemme sur les applications différentiables" ''Bol. Soc. Mat. Mexicana'' (1956) pp. 59–71</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Thom,   "Les singularités des applications différentiables" ''Ann. Inst. Fourier (Grenoble)'' , '''6''' (1955/6) pp. 43–87</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> H. Whitney,   "On singularities of mappings of euclidean spaces: I. Mappings of the plane into the plane" ''Ann. of Math.'' , '''62''' (1955) pp. 374–410</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top"> J.M. Boardman, "Singularities of differentiable maps" ''Publ. Math. IHES'' , '''33''' (1967) pp. 383–419 {{MR|0231390}} {{ZBL|0165.56803}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> J.N. Mather, "On Thom–Boardman singularities" M.M. Peixoto (ed.) , ''Dynamical Systems, Proc. Symp. Univ. Bahia, 1971'' , Acad. Press (1973) pp. 233–248</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> B. Morin, "Formes canoniques des singularités d'une application différentiable" ''C.R. Acad. Sci. Paris'' , '''260''' (1965) pp. 5662–5665; 6503–6506</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> B. Morin, "Calcul jacobien" ''Thèse Univ. Paris–Sud centre d'Orsay'' (1972)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> I.R. Porteous, "Simple singularities of maps" , ''Proc. Liverpool Singularities Symp.'' , ''Lecture Notes Math.'' , '''192''' , Springer (1971) pp. 286–312</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> I.R. Porteous, "The second order decomposition of $\Sigma ^ { 2 }$" ''Topology'' , '''11''' (1972) pp. 325–334</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> F. Ronga, "Le calcul des classes duales aux singularités de Boardman d'ordre deux" ''Comment. Math. Helvetici'' , '''47''' (1972) pp. 15–35</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> R. Thom, "Un lemme sur les applications différentiables" ''Bol. Soc. Mat. Mexicana'' (1956) pp. 59–71</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> R. Thom, "Les singularités des applications différentiables" ''Ann. Inst. Fourier (Grenoble)'' , '''6''' (1955/6) pp. 43–87 {{ZBL|0075.32104}}</td></tr>
 +
<tr><td valign="top">[a10]</td> <td valign="top"> H. Whitney, "On singularities of mappings of euclidean spaces: I. Mappings of the plane into the plane" ''Ann. of Math.'' , '''62''' (1955) pp. 374–410</td></tr></table>

Latest revision as of 18:08, 1 June 2023

Consider a smooth mapping $f : V ^ { n } \rightarrow W ^ { p }$, where $V ^ { n }$ and $W ^ { p }$ are smooth manifolds of dimension $n$ and $p$, respectively (by smooth one understands: class $\mathcal{C} ^ { \infty }$; cf. also Manifold). In order to understand the local structure of $f$, it is natural to distinguish among points $x \in \mathbf{V}$ according to the rank of the derivative $d f _ { x } : T V _ { x } \rightarrow T W _ { f ( x )}$, where $T ( . )$ denotes the tangent space. For $i \in \{ 0 , \dots , n \}$, set:

\begin{equation*} \Sigma ^ { i } ( f ) = \{ x \in V : \operatorname { dim } \operatorname { Ker } d f _ { x } = i \}. \end{equation*}

Using local coordinates on $V$ and $W$, this set is defined locally by the vanishing of the $( n - i + 1 ) \times ( n - i + 1 )$-minors of the $( n \times p )$-matrix of first-order partial derivatives of $f$. If one assumes that $\Sigma ^ { i } ( f )$ is a smooth submanifold of $V$, for $0 \leq i \leq i$ one can define

\begin{equation*} \Sigma ^ { i , j } ( f ) = \Sigma ^ { j } ( f | _ { \Sigma ^ { i } ( f ) } ). \end{equation*}

This can be visualized as follows: at a point $x \in \Sigma ^ { i } ( f )$ there are two vector subspaces of $T V _ { X }$, namely $\operatorname {Ker} d f_x$ and $T ( \Sigma ^ { i } ( f ) ) _ { x }$. Then $x \in \Sigma ^ { i , j } ( f )$ if and only if the intersection of these two subspaces has dimension $j$.

Again, if one assumes that $\Sigma ^ { i , j } ( f )$ is a smooth submanifold, then one can define a subset $\Sigma ^ { i , j , k } ( f ) \subset \Sigma ^ { i , j } ( f )$, etc. At the end, one has partitioned the manifold $V$ into a collection of locally closed submanifolds, such that the restriction of $f$ to each submanifold is of maximal rank; in fact, if the local equations defining the various submanifolds of the collection are of maximal rank, it turns out that for $n \leq p$ the restriction of $f$ to each submanifold is an immersion, while for $n > p$ the same holds except at the points of rank $p - n$, where it is a submersion.

This program has been initiated by R. Thom in his seminal paper [a9], inspired by earlier work of H. Whitney [a10]. Thom handles completely the first-order case, by showing that for a generic mapping (i.e. for mappings in a dense subset of all smooth mappings from $V$ to $W$), $\Sigma ^ { i } ( f )$ is a locally closed submanifold of codimension $i ( p - n + i )$ of $V$, and that for the closure one has:

\begin{equation*} \overline { \Sigma } \square ^ { i } ( f ) = \bigcup _ { h \geq i } \Sigma ^ { i } ( f ). \end{equation*}

This is done by writing the mapping locally as $f : U \rightarrow {\bf R} ^ { n }$, $U \subset \mathbf{R} ^ { n }$ open, then associating to each $x \in U$ the graph $\Gamma _ { x } \subset \mathbf{R} ^ { n } \times \mathbf{R} ^ { p }$ of the derivative $d f _ { x } : \mathbf{R} ^ { n } \rightarrow \mathbf{R} ^ { p },$. Clearly, $x \in \Sigma ^ { i } ( f )$ if and only if $\operatorname { dim } ( \Gamma _ { X } \cap ( \mathbf{R} ^ { n } \times \{ 0 \} ) ) = i$, a condition defining a Schubert variety $F_{i}$ in the Grassmann manifold $G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$ of $n$-planes in $\mathbf{R} ^ { n } \times \mathbf{R} ^ { p }$. Thus, $\Sigma ^ { i } ( f )$ is seen locally as the pull-back of $F _ { i } \subset G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$ by the mapping $x \mapsto \Gamma _ { x }$.

This approach is exemplary, because it presents the singular locus $\Sigma ^ { i } ( f )$ as the pull-back of a universal situation, namely $F _ { i } \subset G _ { n } ( \mathbf{R} ^ { n } \times \mathbf{R} ^ { p } )$; it is then straightforward to show that for most mappings $f$, the induced local mappings $U \rightarrow G _ { n } ( {\bf R} ^ { n } \times {\bf R} ^ { p } )$ are transversal to $F_{i}$, and hence that $\Sigma ^ { i } ( f )$ is a locally closed smooth submanifold of codimension $i ( p - n + i )$ of $V$. Moreover, this approach can serve as basis for the computation of the cohomology class that is Poincaré dual to $\overline { \Sigma } \square ^ { i } ( f )$, which can be interpreted as the first obstruction to having a homotopy from $f$ to a mapping $g : V \rightarrow W$ for which $\Sigma ^ { i } ( g ) = \emptyset$ (see [a9], p. 80, or [a5], Prop. 1.3; the dual classes for second-order singularities have been computed in [a7] and [a6]).

The complete proof that the process of decomposition of the source of a generic smooth mapping can be carried out successfully has been given by J.M. Boardman [a1]. See Singularities of differentiable mappings for the notions of jet space $J ^ { r } ( V , W )$ and $r$-jet extension $j ^ { r } ( f )$ of a mapping $f : V \rightarrow W$, used below.

For smooth manifolds $V ^ { n }$ and $W ^ { p }$, and integers $i _1 , \ldots , i _ { r }$, with $n \geq i _ { 1 } \geq \ldots \geq i _ { r } \geq 0$, one defines the subsets $\sum ^ { i _ { 1 } , \dots , i _ { r }}$ of the space of $r$-jets $J ^ { r } ( V , W )$; it can be proved that these are locally closed smooth submanifolds, and that if $f : V \rightarrow W$ is a mapping whose jet extensions $j ^ { s } ( f ) : V \rightarrow J ^ { s } ( V , W )$ are transversal to $\sum ^ { i _ { 1 } , \dots , i _ { s }}$, $s = 1 , \dots , r$, then, setting

\begin{equation*} \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f ) = j ^ { r } ( f ) ^ { - 1 } ( \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( V , W ) ), \end{equation*}

one obtains:

\begin{equation*} \Sigma ^ { i _ { 1 } } ( f ) = \{ x \in V : \operatorname { dim } \operatorname { Ker } ( d f _ { x } ) = i _ { 1 } \}, \end{equation*}

\begin{equation*} \dots \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f ) = \Sigma ^ { i _ { r } } ( f | _ { \Sigma ^ { i _ { 1 } } , \ldots , i _ { r - 1 } ( f ) } ). \end{equation*}

The codimension of $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$ equals

\begin{equation*} ( p - n + i _ { 1 } ) \cdot \mu _ { i _ { 1 } , \dots , i _ { r } } - ( i _ { 1 } - i _ { 2 } ) \cdot \mu _ { i _ { 2 } , \dots , i _ { r } } \dots \end{equation*}

\begin{equation*} \ldots - ( i _ { r - 1} - i _ { r } ) \cdot \mu _ { i _ { r } }, \end{equation*}

where $\mu _ { i _ { 1 } , \ldots , i _ { s } }$ is the number of sequences $( j _ { 1 } , \dots , j _ { s } )$ satisfying

\begin{equation*} \left\{ \begin{array} { l } { j _ { 1 } \geq \ldots \geq j _ { s }; } \\ { i _ { s } \geq j _ { s } \geq 0 \quad \forall s , 1 \leq s \leq r, } \\ { j _ { 1 } > 0 .} \end{array} \right. \end{equation*}

Moreover, local equations for $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$ can be given explicitly, in terms of the ideal generated by the components of $f$, in some local coordinates, and its Jacobian extensions, an operation which adds to an ideal of functions certain minors of the matrix of their first-order derivatives.

The $\Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( V , W )$ are called Thom–Boardman singularities.

An alternative, more concise approach to Thom–Boardman singularities has been given later by J.N. Mather [a2], and an algebraic approach can be found in [a4].

Thom's transversality theorem [a8] implies that the set of mappings that are transversal to all possible Thom–Boardman singularities, that one may call generic mappings, is dense in the space of all mappings from $V$ to $W$. So now one may ask how useful are Thom–Boardman singularities in the understanding of generic mappings.

In some cases, they allow a full classification. This is so, for example, if $n \leq p$ and $i _ { 1 } = \ldots = i _ { r } = 1$, or $n > p$ and $i_ 1 = n - p$, $i _ { 2 } = \ldots = i _ { r } = 1$, by a result of B. Morin [a3]; for $n = p$ and $r = 1,2,3,4$ one finds the catastrophes of the fold, the cusp, the swallowtail, and the butterfly, respectively (see Thom catastrophes).

In general, Thom–Boardman singularities allow a very useful first approach to the understanding of the structure of a mapping; however, they are not fine enough to provide an, even coarse, classification. Indeed, as pointed out by I.R. Porteous [a6], a generic mapping $f : \mathbf{R} ^ { 5 } \rightarrow \mathbf{R} ^ { 5 }$ can present the singularities $\Sigma ^ { 2 }$ and $\Sigma ^ { 1,1,1,1 }$, both of dimension $1$, and some isolated points of $\Sigma ^ { 2 }$, called parabolic $\Sigma ^ { 2 }$-points by Porteous, can be in the closure of $\Sigma ^ { 1,1,1,1 }$; the structure of such a mapping is definitely different at $\Sigma ^ { 2 }$-parabolic and $\Sigma ^ { 2 }$-non-parabolic points. Similar phenomena occur in other dimensions.

In fact, Thom–Boardman singularities provide a partition of the source of a generic mapping into locally closed submanifolds, but the closure of a submanifold is not necessarily a union of similar submanifolds.

When studying the equations of Thom–Boardman singularities, an interesting device shows up: the intrinsic derivative, first studied by Porteous (see [a5]). In general, derivatives of order higher than $1$ are not intrinsic, in the sense that are affected by higher derivatives of coordinate changes, not only the linear part of them. However, it turns out that if $x \in \Sigma ^ { i _ { 1 } , \ldots , i _ { r } } ( f )$, then a suitable combination of the first $r + 1$ derivatives, restricted to appropriate subspaces, is intrinsic. The simplest case is that of the second intrinsic derivative; if $x \in \Sigma ^ { i _ { 1 } } ( f )$, then the bilinear mapping induced by the second derivative,

\begin{equation*} \widetilde { d ^ { 2 } f _ { x } } : K _ { x } \times T V _ { x } \rightarrow Q _ { x }, \end{equation*}

where $K _ { x } = \operatorname { Ker } ( d f _ { x } )$, $Q _ { x } = T W _ { x } / \operatorname { Im } ( d f _ { x } )$ is intrinsic, as one can check easily. In the special case of a function $f : V ^ { n } \rightarrow \mathbf{R}$, if $x \in \Sigma ^ { n } ( f )$ is a critical point, then $\widetilde{ d ^ { 2 } f } _ { x } : \mathbf{R} ^ { n } \times \mathbf{R} ^ { n } \rightarrow \mathbf{R}$ is the well-known Hessian bilinear form of $f$ at $x$ (cf. also Hessian matrix), whose signature determines completely the local structure of $f$ near $x$.

The intrinsic derivative can be used to refine Thom–Boardman singularities; for example, for a generic mapping $f : V ^ { n } \rightarrow W ^ { n }$:

\begin{equation*} \Sigma ^ { 2 _ \text { parabolic } } = \end{equation*}

\begin{equation*} = \left\{ x \in \Sigma ^ { 2 } ( f ) : \begin{array}{c} { \exists \text {a line }\ \text{l} \subset K _ { x } } \\ { \text{such that} \ \widetilde{d^{2}f_x}|_{\text{l}\times \text{l}} } \end{array} \right\}. \end{equation*}

An inductive definition of the intrinsic derivatives is provided in [a1]; so far, it has not been tried to refine systematically Thom–Boardman singularities using them.

References

[a1] J.M. Boardman, "Singularities of differentiable maps" Publ. Math. IHES , 33 (1967) pp. 383–419 MR0231390 Zbl 0165.56803
[a2] J.N. Mather, "On Thom–Boardman singularities" M.M. Peixoto (ed.) , Dynamical Systems, Proc. Symp. Univ. Bahia, 1971 , Acad. Press (1973) pp. 233–248
[a3] B. Morin, "Formes canoniques des singularités d'une application différentiable" C.R. Acad. Sci. Paris , 260 (1965) pp. 5662–5665; 6503–6506
[a4] B. Morin, "Calcul jacobien" Thèse Univ. Paris–Sud centre d'Orsay (1972)
[a5] I.R. Porteous, "Simple singularities of maps" , Proc. Liverpool Singularities Symp. , Lecture Notes Math. , 192 , Springer (1971) pp. 286–312
[a6] I.R. Porteous, "The second order decomposition of $\Sigma ^ { 2 }$" Topology , 11 (1972) pp. 325–334
[a7] F. Ronga, "Le calcul des classes duales aux singularités de Boardman d'ordre deux" Comment. Math. Helvetici , 47 (1972) pp. 15–35
[a8] R. Thom, "Un lemme sur les applications différentiables" Bol. Soc. Mat. Mexicana (1956) pp. 59–71
[a9] R. Thom, "Les singularités des applications différentiables" Ann. Inst. Fourier (Grenoble) , 6 (1955/6) pp. 43–87 Zbl 0075.32104
[a10] H. Whitney, "On singularities of mappings of euclidean spaces: I. Mappings of the plane into the plane" Ann. of Math. , 62 (1955) pp. 374–410
How to Cite This Entry:
Thom-Boardman singularities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom-Boardman_singularities&oldid=23075
This article was adapted from an original article by F. Ronga (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article