Thom polynomials | Thom polynomial portal

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 $Q_f=C[[x_1,\ldots,x_n]]/(f_1,\ldots,f_{n+l})$ . We often name the contact singularity 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 $({1+b_1t+b_2t^2+\ldots})/({1+a_1t+a_2t^2+\ldots})=1+c_1t+c_2t^2+\ldots$ .

Notation

For a partition $\lambda=(\lambda_1,\ldots,\lambda_r)$ we define the Schur polynomial $s_{\lambda}=\det(c_{\lambda_i+j-i})_{i,j=1,\ldots,r}$ .

Thom polynomials of contact singularities of relative dimension $l=0$ .
Thom polynomials of contact singularities of relative dimension $l=1$ .
For certain algebras we know then Thom polynomial for arbitrary l: Thom series.