Thom-Boardman singularities

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