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The Thom polynomial of a singularity $\Sigma$ measures how the appearance of $\Sigma$ is forced by global topology. It is named after French mathematician René Thom, honoring his seminal paper R. Thom: Les singularités des applications différentiables, Ann. Inst. Fourier 6, (1955-56) 43–87.

Definition

Let $G_n$ be the group of analytic diffeomorphism germs of $(C^n,0)$ . Consider the vector space $V_{n,n+l}=\{ f:(C^n,0)\to (C^{n+l},0)$ analytic $\}$ acted upon by the group $G_n \times G_{n+l}$ , and let $\Sigma$ be an invariant subset, the singularity.

The Thom polynomial of $\Sigma$ is the $G_n\times G_{n+l}$ -equivariant fundamental cohomology class of $\overline{\Sigma}$ , that is

Tp( $\Sigma$ )= $[\overline{\Sigma}\subset V_{n,n+l}] \in H^*_{G_n\times G_{n+l}}(V_{n,n+l})$ .

Since $G_n$ is homotopy equivalent to $GL_n(C)$ , and $V_{n,n+l}$ is contractible, the Thom polynomial of $\Sigma$ is a $GL_n(C)\times GL_{n+l}(C)$ -characteristic class. Therefore, the Thom polynomial is a polynomial in the variables $a_1,a_2,\ldots,a_n$ , $b_1,b_2,\ldots,b_{n+l}$ , where $a_i$ ( $b_i$ ) is the i’th Chern class of the tautological bundle over $BGL_n(C)$ ( $BGL_{n+l}(C)$ ).

Degeneracy locus interpretation

The Thom polynomial has the following degeneracy locus characterization. For a map $f:N^n\to M^{n+l}$ (satisfying appropriate transversality properties) the cohomological fundamental class of the closure of $\Sigma(f)=\{x\in N :$ the germ of $f$ at $x$ is from $\Sigma \}$ is equal to the value of the Thom polynomial after substituting the Chern classes of N for $a_i$ and the Chern classes of M (pulled back by $f^*$ ) for $b_i$ .

Thom polynomials of contact singularities

An important class of singularities is the so-called contact singularities. One of their equivalent characterizations is the following. Let Q be a finite dimensional, local, commutative C-algebra. A contact singularity is $\Sigma_Q=\{ f:(C^n,0)\to (C^{n+l},0) : Q_f$ is isomorphic with $Q\}$ , where the local algebra $Q_f$ of the germ $f=(f_1(x_1,\ldots,x_n),\ldots,f_{n+l}(x_1,\ldots,x_n))$ is defined by $Q_f=C[[x_1,\ldots,x_n]]/(f_1,\ldots,f_{n+l})$ . The contact singularity is often denoted by its local algebra or the ideal $(f_1,\ldots,f_{n+l})$ .

It is a fact that the Thom polynomial of a contact singularity depends only on the “quotient” classes $c_1,c_2,\ldots$ , defined by the Taylor expansion $({1+b_1t+b_2t^2+\ldots})/({1+a_1t+a_2t^2+\ldots})=1+c_1t+c_2t^2+\ldots$ .

Examples

For $Q=C[x]/(x)$ and $\Sigma_Q\subset V_{n,n+l}$ the Thom polynomial is $1$ .

For $Q=C[x]/(x^2)$ and $\Sigma_Q\subset V_{n,n+l}$ the Thom polynomial is $c_{l+1}$ .

For $Q=C[x]/(x^3)$ , and $\Sigma_Q\subset V_{n,n+l}$ the Thom polynomial is $c_{l+1}^2+\sum_{i=1}^{l+1} 2^{i-1} c_{l+1-i}c_{l+1+i}$ .

For $Q=C[x,y]/(x^2,xy,y^2)$ , and $\Sigma_Q\subset V_{n,n+l}$ the Thom polynomial is $c_{l+2}^2-c_{l+1}c_{l+3}$ .

Variations

The following variations of Thom polynomials are also studied:

Thom polynomials of singularities of real maps, in cohomology with $Z_2$ coefficients, using Stiefel-Whitney characteristic classes.
Thom polynomials of singularities of real maps, in cohomology with $Z$ coefficients, using Pontryagin characteristic classes.
Thom polynomials of multi-singularities.
Thom polynomials in generalized cohomology theories.
Avoiding ideals of singularities.

References

Rene Thom wikipedia page https://en.wikipedia.org/wiki/Ren%C3%A9_Thom

Thom Polynomial Portal http://tpp.web.unc.edu