View source for Thom-Boardman singularities

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:

<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>

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

<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>

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" />.

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.

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:

<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>

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" />.

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]]]).

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.

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

<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>

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>

<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>

The codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005087.png" /> 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>

<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>

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

<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>

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.

The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120050/t12005095.png" /> 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]]].

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.

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 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 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,

<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>

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" />.

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" />:

<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>

<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>

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====
<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>