CALCULATION OF THOM POLYNOMIALS AND OTHER COHOMOLOGICAL OBSTRUCTIONS FOR GROUP ACTIONS

LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

1. Introduction

In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian in [Kaz97a]. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand we describe a way to calculate Thom polynomials: the method of restriction equations. It turned out that though the idea is quite simple the method is very powerful. We reproduced and improved earlier result in several directions: [Rim01] [FR02], [FRb], [FRa]. However a proper introduction to the basic theorems was missing. In this paper we try to pay this debt as well as we present the connections with obstruction theory and equivariant cohomology. We give some new results and outline possible generalizations and problems.

Calculating Thom polynomials has a long history. In retrospect the first definition of characteristic classes can be considered as the first appearance ([Sti36]). But it was R. Thom who initiated their study in the case of singularities of smooth maps. The major tool for calculating them was the method of resolutions (see [AVGL91] for an account of the method and results).

Works of V. Vassiliev [Vas88] and M. Kazarian [Kaz97a] clarified the connection of Thom polynomials with the underlying symmetry groups. Their works also show that the so called degeneracy loci formulas in algebraic geometry are also Thom polynomials for group actions. So in this respect the history of Thom polynomials dates back to the Giambelli formulas. Not surprisingly the tool of calculating them in algebraic geometry was also the method of resolutions.

Based on works of A. Szűcs ([Szű79]) the second author introduced a different method (the method of restriction equations) to calculate Thom polynomials for singularities of smooth maps ([Rim01]). After reading M. Kazarian’s paper [Kaz97a] we realized that the method of restriction equations can be easily generalized to the case of Thom polynomials for group actions. Similar type of methods were used first in [AB83], see in Section 10.

Supported by FKFP55/2001 as well as OTKA D29234 (first author), OTKA T029759 (second author)

Keywords: singularities, Kazarian spectral sequence, Thom polynomials, generalized Pontryagin-Thom construction, global singularity theory, degeneracy loci, equivariant cohomology Mathematical Subject Classification 2000: 14N10, 57R45, 32s20, 58K30.

2 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

The paper is organized the following way: In Section 2 we show how Thom polynomials fit into the more general problem of finding cohomological obstructions for the existence of a section. In particular we explain the connections among Thom polynomials, first obstructions and equivariant Poincaré dual. In Section 3 we prove the basic Theorems 3.2 and 3.5 on calculating Thom polynomials. We also make some comments on possible generalizations for other cohomology theories. In Section 4 we review some earlier results mainly from singularity theory and algebraic geometry. In Section 5 we discuss Thom polynomials for representations of the complex groups GL(n). Results on the adjoint representation and representations of GL(2) are new. In Section 6 we study the classical case of the Giambelli-Thom-Porteous formula in details. In Section 7 we show how the computations of [Rim01] based on the generalized Pontryagin-Thom construction fit into the framework of the present paper. In Section 8 we list some other applications of the method of restriction equations which were published elsewhere. In Section 9 we give a simple formula to calculate Thom polynomials for the projectivization of a linear group action. As a corollary we show a simple method to calculate the degree of degeneracy loci, generalizing results of Porteous, Harris and Tu, Fulton and others. In Section 10 we show how the Kazarian spectral sequence (the spectral sequence induced by the codimension filtration of the orbit stratification)—which were used to define the Thom polynomials in [Kaz97a]—can be used in finding the orbits of a representation and their stabilizer groups.

We would like to thank for A. Szűcs for several useful comments in particular for drawing our attention to the action of the stabilizer group on the normal space of an orbit.

2. Obstruction Theory for Generalized Cohomology Theories

We are looking for cohomological obstructions of sections of fiber bundles. On fiber bundle we mean a bundle F → E → M associated to a principal G-bundle G → P → M— i.e. E = P ×_{G} F. This is not really a restriction (take G to be the self homeomorphism group of F) but we are mainly interested in the case when G is a Lie group and F is a G-manifold.

Our starting point is the following elementary observation:

Observation 2.1. Sections of the fiber bundle E = P ×_{G} F are in one to one correspondence with G equivariant maps from P to F. (Here the right action on P is transformed to a left action naturally.)

In other words in the category of left G-spaces a section s of E is a lift of the collapse

map pt_{P} : P → pt: \xymatrix{ & F \ar[d]^{pt_{F}} \\ P \ar[r]_{pt_{P}} \ar[ur]^{s} & pt}

Suppose now that h is any contravariant functor from the category of (left) G-spaces into (graded) rings (the main example is equivariant cohomology). Then a section s induces a commutative diagram: \xymatrix{ & h(F) \ar@{-->}[dl]_{h(s)} \\ h(P) & h(pt) \ar[l]^{h(pt_{P})} \ar[u]_{h(pt_{F})}}

So the existence of the section s implies that

ker h(pt_{F}) ⊂ ker h(pt_{P}). (1)

Using the definitions

Definition 2.2.

h(pt) = {universal characteristic classes},

Im h(pt_{P}) = {characteristic classes of P },

O_{F} := ker h(pt_{F}) = obstruction ideal of F .

we can reformulate (1) in a more familiar form, as follows.

Theorem 2.3. If the fiber bundle E = P ×_{G} F admits a section then all characteristic classes of P from the obstruction ideal O_{F} vanish.

So the goal of a general theory would be the calculation of O_{F} for G-spaces F and G-equivariant cohomology theories.

2.1. Obstructions in Ordinary Cohomology. From any cohomology theory h we can obtain a G-equivariant cohomology theory via the Borel construction:

h_{G}(F) := h(BF) where BF = EG ×_{G} F.

The Borel construction is also a functor, so G-equivariant maps induces maps between the Borel constructions of the corresponding spaces. Notice that BP = M, Bpt = BG and B(pt_{P}) : M → BG is the classifying map of P. (Notice that G-equivariant K-theory is not obtained by the Borel construction.)

Suppose now that F is d-1 connected i.e. π_{i}(F) = 0 for i < d − 1 and suppose that π_{d−1}(F) ≅ ℤ. Then the—homotopy theoretical—first obstruction fo(E) is an element of H^{d}(M; ℤ) and for the universal F -bundle fo(BF) ∈ H^{d}(BG; ℤ). Recalling that fo(E) is the transgression of the generator of π_{d−1}(F) in the spectral sequence of the fibration E → M we can see that fo(E) can also be characterized (up to sign) by the following property:

Proposition 2.4. If fo(BF) is not zero then it is a generator of O_{F}^{d} = O_{F} ∩H^{d}(BG; ℤ) ≅ ℤ and O_{F}^{i} = O_{F} ∩ H^{i}(BG; ℤ) = 0 for i < d.

4 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

2.2. Geometric realization of the first obstruction—Poincaré duality. On geometric realization of a cohomology class α of a manifold Y we mean finding a submanifold X such that the Poincaré dual of X—denoted by [X]_{Y} or [X] for short—is equal to α. Our basic example is the Euler class:

Example 2.5. Let S → M be an S^{n−1}-bundle (or an ℝ^{n} \ {0}-bundle) associated to a principal SO(n)-bundle P over M. Then the first obstruction is the Euler class of S → M and can be realized as the Poincaré dual of the zero of a transversal section of the vector bundle E = P ×_{SO(n)} ℝ^{n}.

In general if F is a G-manifold and we want to realize α = fo(P ×_{G}F) for some principal G-bundle P → M then we try to find an open embedding of F into a contractible Gmanifold V. (In most applications F is given as an open subset of a vector space.) Then P ×_{G} V has a section s and we expect that

F^{c}(s) := {m ∈ M : s(m) ∉ F }

represents α for a generic section. The problem is that V \ F is usually not smooth (basically the only case when it is smooth is Example 2.5) so we cannot expect F^{c}(s) to be smooth either. So we have to extend the notion of Poincaré dual somewhat:

Definition 2.6. Let X be a closed subset of a topological space Y. X represents the cohomology class [X] ∈ H^{d}(Y; ℤ) if for the inclusion i : Y \ X → Y

ker H^{j}(i) ≅ \begin{cases}0 & if j < d \\ ℤ & if j = d \end{cases}

and [X] generates ker H^{d}(i).

The class [X] is defined up to sign, we can call the two choices orientations.

Examples 2.7.

(i) X → Y is a proper map of smooth oriented manifolds. Thom isomorphism for the normal bundle of X shows that X represents the Poincaré dual of X.

(ii) X can be triangulated and represents a homology class in Y and Y has a non degenerate intersection pairing (e.g. Y is a compact oriented manifold with no torsion homology).

(iii) (Y, Θ) is a stratified space and X is the closure of some strata. Then X represents a cohomology class if it represents a cocycle in the Vassiliev complex (E_{1}^{0,∗}(Θ), d_{1}) where E_{∗}^{∗,∗}(Θ) is the spectral sequence induced by the codimension filtration of the stratification Θ. (See [Vas88] and [Kaz97a] as well as Section 10 for details.) If the strata are even dimensional this condition is automatically satisfied.

Now almost by definition we have the following:

Theorem 2.8. Suppose that the Lie group G acts on the vector space V and F is a Ginvariant open subspace of V. Assume moreover that π_{i}(F) = 0 for i < d−1 and π_{d−1}(F) ≅ ℤ and that for the principal G-bundle P → M the cohomology class fo(P ×_{G} F) is not zero and F^{c}(s) represents a cohomology class in H^{d}(M; ℤ). Then fo(P ×_{G} F) = ±[F^{c}(s)].

Proof. fo(P ×_{G} F) ∈ ker H^{d}(i) for i : F (s) → M by the naturality property of the first obstruction.

2.3. The theory of Thom polynomials. We use the definition of M. Kazarian [Kaz97a] for Thom polynomials of group actions: Given a representation ρ : G → GL(V) we are looking for obstruction of having a section of a V -bundle associated to this representation avoiding a certain orbit η (or more generally a G-invariant subset of V). Of course the zero section avoids any orbit different from the zero orbit but this is pathological: we want obstructions for a generic section. In effect we want to avoid the closure of η. The Thom polynomial of η is an obstruction for having a section in the complement V \ η̄: the cohomology class represented by η̄(s) for a generic section s.

We would like to show that the Thom polynomial is the first obstruction. By the previous section we only have to calculate some homotopy groups of V \ η̄. The open subspace V \ η̄ is highly connected: by a simple transversality argument

π_{i}(F) = 0 for i < d − 1 where d = codim(η).

Suppose now that G is a complex Lie group acting on the complex vector space V. Then η̄ is an algebraic subvariety of V.

Observation 2.9. Let X ⊂ ℂ^{N} be a d (real) codimensional complex algebraic variety. Then

π_{i}(ℂ^{N} \ X) ≅ \begin{cases}0 & if i < d − 1 \\ ℤ & if i = d − 1. \end{cases}

(If d = 2 then π_{1} should be replaced by H_{1}.)

Similar theorem holds for real varieties with some extra condition. However the closure of an orbit of a real representation is not necessarily a real algebraic variety, the algebraic closure can contain some other orbits with the same codimension.

Definition 2.10. Suppose that V is a G-space and η is a G-invariant subspace. Then the avoiding ideal of η is the obstruction ideal of V \ η̄:

A_{η} := O_{V \η̄}.

Suppose moreover that

A_{η}^{j} ≅ \begin{cases}0 & if j < d \\ ℤ & if j = d, \end{cases}

Then a generator Tp(η) of A_{η}^{d} is called the Thom polynomial of η.

In other words Tp(η) is the G-equivariant (generalized) Poincaré dual of η̄.

6 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

Remark 2.11. This definition of the Thom polynomial is essentially the same as Kazarian’s definition in [Kaz97a]. We hided the technical details by assuming these properties of the avoiding ideal A_{η}. The reason is that for the class of representations we are mostly interested in these assumptions can be easily verified.

Corollary 2.12. For an orbit η of the complex representation ρ : G → GL(V) we have:

Tp(η) = fo(EG ×_{G} (V \ η̄)).

3. Calculation of Thom polynomials—The method of restriction equations

If η is a homogeneous G-space—i.e. η ≅ G/G_{η} where G_{η} is the stabilizer group of a point in η—then the calculation of the obstruction ideal O_{η} can be reduced to algebra: First notice that H_{G}^{∗}(η) ≅ H_{G_{η}}^{∗}(pt) and that the map H_{G}^{∗}(pt_{η}) : H_{G}^{∗}(η) ≅ H^{∗}(BG_{η}) → H_{G}^{∗}(pt) ≅ H^{∗}(BG) is equal to H^{∗}(Bi) for i : G_{η} → G. Then choose compatible maximal tori for G and G_{η} and describe the map H^{∗}(Bi) in terms of Chern roots. The calculation for homogeneous G-spaces can be considered as the first step for calculating of the obstruction ideal for more general G-spaces (see Theorem 3.7). This is usually not an easy algebraic question (see Remark 3.1), but if we are only interested in Thom polynomials then for a certain class of representations we have an algorithm. This algorithm is based on the observation that O_{η} ⊂ O_{ξ} if ξ ⊂ η and the following property of the generalized Poincaré dual:

Lemma 3.1. For any orbit η which admits Thom polynomial H^{∗}(pt_{η})(Tp(η)) = e(η), where e(η) is the equivariant Euler class of the normal bundle of η.

Proof. Since η → V \ ∂η is a proper oriented submanifold η has a Poincaré dual [η] in V \ ∂η. Therefore [η]|_{η} = e(η). On the other hand Tp(η)|_{V \∂η} = [η] (if follows from the analogous statement for non equivariant cohomology see e.g. in [Ful96, app. B.3]).

So we get many conditions for Tp(η):

Theorem 3.2. Tp(η) ∈ O_{ξ} for orbits ξ ⊂ (V \ η̄) and H^{∗}(pt_{η})(Tp(η)) = e(η).

We call the subset of these conditions where codim ξ ≤ codim η the restriction equations. These equations are easier to handle since sometimes it is difficult to decide whether an orbit ξ belongs to the closure of η. And for a class of representations they uniquely define Tp(η):

Definition 3.3. A complex vector G-space V satisfies the Euler condition if there are finitely many orbits of V and e(ξ) is not a zero divisor for any of the orbits ξ ∈ V/G.

Remark 3.4. This condition on the Euler class appears first in [AB83] as a necessary condition for G-perfectness, see Section 10 for more details.

Let us denote the restriction maps H^{∗}(pt_{ξ}) by j_{ξ}^{∗}. Then we have Theorem 3.5. If V satisfies the Euler condition then the restriction equations

j_{ξ}^{∗} Tp(η) = \begin{cases}e(ν_{η}) & if θ = η  ‘principal equation’ \\ 0 & if ξ ≠ η, codim ξ ≤ codim η  ‘homogeneous equations’ \end{cases}

have a unique solution.

The proof is an improvement of the discussion [Rim01, Sect.6] but we translate it to the language of the obstruction ideals. The proof is based on a repeated use of the following lemma.

Lemma 3.6. Suppose that η ⊂ η ∪ ξ is a proper inclusion of (complex) manifolds and that e(η) is not a zero divisor. Then

O_{η∪ξ} = O_{η} ∩ O_{ξ}.

Proof. Look at the Mayer-Vietoris sequence of Bξ ∪ Dη = B(ξ ∪ η):

\xymatrix{H^{∗}(Bξ) ⊕ H^{∗}(Bη) & H^{∗}(B(ξ ∪ η)) \ar[l]_{a} & H^{∗−1}(Sη) \ar[l]_{δ} \\  & H^{∗}(BG) \ar@{-->}[ul] \ar[u]}

where Sη is the normal sphere bundle of Bη (equivalently you may write down here the G-equivariant Mayer-Vietoris sequence) and notice that H^{odd}(Sη) = 0 iff e(η) is not a zero divisor by looking at the spectral sequence of the fibration Sη → η.

Iterating the lemma we get the following theorem.

Theorem 3.7. If V satisfies the Euler condition and X = ⋃{ξ_{1}, …, ξ_{n}} ⊂ V is open then

O_{X} = ⋂_{i=1}^{n} O_{ξ_{i}}.

Proof of Theorem 3.5. Let d = codim η. Then using the definition of A_{η}^{d}, the definition of Tp(η), Theorem 3.7 and Theorem 2.9 we get

A_{η}^{d} = O_{V \η̄}^{d} ≅ ℤ Tp(η) = ⋂{O_{ξ}^{d} : ξ ⊄ η̄}.

On the other hand

⋂{O_{ξ}^{d} : ξ ⊄ η̄} = ⋂{O_{ξ}^{d} : codim ξ ≤ d, ξ ≠ η} ∩ ⋂{O_{ξ}^{d} : codim ξ > d}
= ⋂{O_{ξ}^{d} : codim ξ ≤ d, ξ ≠ η},

since O_{ξ}^{d} = H^{d}(BG) for codim ξ > d.

8 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

Remark 3.8. The proof shows that the principal equation is needed only to find the generator of a subgroup isomorphic to ℤ, so theoretically it is not necessary. Also it looks more complicated than the homogeneous equations. Strangely enough sometimes the homogeneous equations are more difficult to deal with (for example in the case of double or Kempf-Laksov Schur polynomials [FRa]). And in Section 6 we show some examples where the principal equation implies the homogeneous equations. This way contrary to what is expected the principal equation itself is enough to calculate the Thom polynomials.

3.1. The role of the avoiding ideal. One of the main advantages of the point of view on Thom polynomials presented in this section is the initiation of the avoiding ideal A_{η}. First, its definition is a very straightforward topological idea, second its meaning fits into general topological studies: the ideal is the collection of “first obstructions” to a section avoiding η. Although when we concentrated on computation, we gave Theorems 3.2, 3.5 only on one element of the avoiding ideal, the Thom polynomial. However, our proof also says, that the elements of the avoiding ideal are exactly the solutions of the homogeneous equations. Thus we obtained a description of the avoiding ideal as a kernel of a homomorphism— which is in most situations the desired description. In some other cases one wants to have a generator set for this ideal, this needs additional algebraic work. This was done in the classical situation by P. Pragacz [Pra88], see also Proposition 6.5.

The greatest importance of the notion of A_{η} is, however, that this is the notion which generalizes to extraordinary cohomology theories, because of the simplicity of its definition. The notion of Thom polynomial only generalizes naturally to connected cohomology theories (h^{<0} = 0), as can be seen from the spectral sequence approach of Kazarian. The extraordinary Thom polynomial computations in the literature either depend on artificial choices of resolutions (corresponding to an artificial choice of an element in A_{η}) or use extra structure (usually in K-theory). The idea of “computing an extraordinary Thom polynomial then mapping it back to cohomology to find new obstructions”, in all the cases the authors checked produced elements in A_{η}, so could have done without leaving H^{∗}. A more detailed general study of the avoiding ideals (“Thom polynomials”) in extraordinary cohomology theories as well as their relation to different resolutions is subject to further study.

4. Classical results

4.1. Singularities of smooth maps. Consider a complex analytic map f between complex analytic manifolds N^{n} and P^{p}. Also let us fix a “singularity” η (see a discussion below). An often occurring problem is the study of the set η(f) of points x ∈ N where f has singularity η. Thom proved in [Tho56] that the cohomology class represented by the closure of η(f) in the cohomology ring of N is equal to the value of a multivariable universal polynomial—depending only on η—when we substitute the characteristic classes of N and the pull-backs of the characteristic classes of P. In this way, if we know the polynomial, and the homotopy class of f, then we can tell the (co)homology class of \overline{η(f)}, obtaining direct geometric or topological consequences.

Unfortunately—for quite long—not many of these polynomials were know explicitly. Some know examples included works of Thom [Tho56], Porteous [Por71], Ronga [Ron72] and Gaffney [Gaf83]. For a fuller list of references see [AVGL91] or [Rim01]. In [Rim01] the second author applied (basically) the method of the present paper and found an algorithm to compute these polynomials.

In section 7 we will show that these polynomials are Thom polynomials in the sense of the preceding section, and show that the method of [Rim01] is exactly the application of Theorem 3.2. In the meanwhile we will find what the good definition is for “singularities”.

Let us remark that there is a parallel theory starting with real smooth manifolds and smooth maps between them. The method of the above mentioned authors gave results for the complex and the real case simultaneously. Considering more difficult singularities the two cases become essentially different, see also Subsection 8.1.

While the techniques of the present paper seem to be powerful enough to reproduce most of the earlier results, let us mention that we never succeeded to re-calculate the results in [Ron72], i.e. the Thom polynomials associated to all second order Thom-Boardman singularities.

4.2. Lagrange and Legendre singularities. A version of singularity theory of maps is obtained when we consider maps which come from some differential geometric situations. Usually the occurring maps are Lagrange maps between symplectic manifolds. Their singularities are called Lagrange singularities and their Thom polynomial theory has been studied and solved by Vassiliev and Kazarian, see [Vas88], [AVGL91], [Kaz95], [Kaz00b].

4.3. Degeneracy loci. Thom polynomials for group actions coming from algebraic geometry are usually called “degeneracy loci formulas”. A review of the known formulas, as well as other enumerative properties of degeneracy loci, and discussions on Thom polynomials is Chow groups are found in [FP98], see also [Ful98, Ch. 14]. These investigations include the Giambelli-Porteous-Thom formula (see Section 6 in the present paper), the interpretation of Schubert calculus (see also section 8.3) and degeneracy loci formulas for other classical groups (see section 5.1, and the original references [JLP82], [HT84] as well as generalizations by Kazarian [Kaz00a]). A recent result is an algorithm finding the degeneracy loci formulas for quiver representations associated with A_{n} graphs, found by Fulton and Buch [BF99], see more in section 8.2.

4.4. Circle bundles. A spectacular topological example of Thom polynomials is given by Kazarian in [Kaz97b]. Here the Thom polynomial problem is the following. Let the (orientation preserving) diffeomorphism group of S^{1} times that of ℝ^{1} act on the space of maps from S^{1} to ℝ^{1} the natural way. Then in [Kaz97b] the Thom polynomials of a collection of orbits are computed. Translating this back to differential topology one obtains geometric interpretations of (multiples of) the powers of the Chern class of a complex line bundle.

In the next sections we show how most of these results can be obtained using the method of Section 3 together with some new results.

10 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

5. Thom polynomials for GL(n)

The naive approach would be calculating the Thom polynomials for irreducible representations and then finding the rules of calculating the Thom polynomials for the direct sums of representations if the Thom polynomials for the factors are known. The main reason that such an approach cannot work is the following elementary observation:

Observation 5.1. Let V and W be two non zero representations of a Lie group G. Then V ⊕ W has infinitely many orbits.

Proof. Dimension counting shows the product of two orbits splits into infinitely many orbits if their dimensions are different.

To get a feel of the intricacies of the geometry of the direct sum let us have a look at the “simplest” example (for details see [FRa]):

Example 5.2. GL(n) acts on V = ⨁_{i=1}^{k} ℂ^{n} ≅ Hom(ℂ^{k}, ℂ^{n}). The kernel of the map φ ∈ V is the only invariant of the GL(n)-action so the orbit space V/GL(n) will be the union of Grassmannians Gr_{i}(ℂ^{k}). Using a cell decomposition for these “moduli spaces” we can define Thom polynomials which are going to be Schur polynomials.

So to find representations with finitely many orbits we have to look at irreducible representations. An extensive list of such representations can be found in [Ric85] and [Kac80]. A simple condition is dim G ≥ dim V, otherwise we won’t have an open orbit. The classification of irreducible representations of GL(n) is well known together with a simple formula for their dimensions (the “hook length formula see e.g. [FH91, p.50]). Recalling that taking the dual or tensoring with the one dimensional representations det^{k} doesn’t change the dimension of a representation we get the following list:

Proposition 5.3. The irreducible representations V of GL(n) satisfying the condition dim GL(n) ≥ dim V are the trivial representations ℂ, the standard representations ℂ^{n}, the symmetric and antisymmetric two forms S^{2}(ℂ^{n}), Λ^{2}(ℂ^{n}), the adjoint representations of SL(n), the representations Λ^{3}(ℂ^{6}), Λ^{3}(ℂ^{7}), Λ^{3}(ℂ^{8}), and their duals and tensor products with det^{k}.

The adjoint representations have infinitely many orbits (see a detailed discussion below) but all the others have a finite orbit structure. For S^{2}(ℂ^{n}), Λ^{2}(ℂ^{n}) this is the Sylvester theorem. For the representations Λ^{3}(ℂ^{6}), Λ^{3}(ℂ^{7}), Λ^{3}(ℂ^{8}) see e.g. [FH91, ]. We calculated some Thom polynomials for the exceptional cases in [FNR].

Taking the dual or tensoring with det^{k} can change the Thom polynomials but in a controllable way so we concentrate on the list above.

The standard representation ℂ^{n} has two orbits η = ℂ^{n} \ {0} and {0}. Almost by definition the Thom polynomials are Tp(η) = 1 and Tp(0) = c_{n}.

Orbits of S^{2}(ℂ^{n}) and Λ^{2}(ℂ^{n}) are determined by the rank and their Thom polynomials were calculated in [JLP82],[HT84] (let Δ_{λ} denote the Schur polynomial associated with the partition λ, as in e.g. [FP98]):

Theorem 5.4 ([JLP82],[HT84]).

• Tp(orbit of S^{2}(ℂ^{n}) with corank r) = 2^{r}Δ_{(r,r−1,r−2,... ,1)}.

• Tp(orbit of Λ^{2}(ℂ^{n}) with corank r) = Δ_{(r−1,r−2,r−3,... ,1)}.

Since these representations satisfy the Euler condition we can apply the method of restriction equations. On the case of Λ^{2}(ℂ^{n}) let us demonstrate how simple these calculations are.

5.1. Thom polynomials for Λ^{2}(ℂ^{n})—restriction equations. The second antisymmetric (Λ^{2}) power of the standard representation of GL_{n}(ℂ) has finitely many orbits: the corank r determines the orbit Σ^{r}—where r = n, n−2, . . .. We can choose a representative of Σ^{r} as

M_{r} = (\begin{matrix}0 & I_{(n−r)/2} & 0 \\ −I_{(n−r)/2} & 0 & 0 \\ 0 & 0 & 0 \end{matrix})_{n×n} .

Now we show how to obtain Tp(Σ^{r}) = Δ_{(r−1,r−2,... ,1)}(c) using our methods. (The case of S^{2}(ℂ^{n}) is very similar.)

Proof. Easy computation shows that the maximal compact symmetry group of M_{r} is G_{r} = U(r) × Sp(\frac{n−r}{2}), with cohomology ring H^{∗}(BG_{r}) = ℤ[c_{1}, …, c_{r}, p_{1}, …, p_{\frac{n−r}{2}} ] (c_{i} are the Chern classes of rank 2i and p_{i} are the Pontryagin classes of rank 4i). In other words this ring is ℤ[c_{i}, p_{i}], i ∈ ℤ with the “substitutions”

cnegative = 0,  pnegative = 0,  c_{>r} = 0,  p_{> \frac{n−r}{2}} = 0.

Let us call these substitutions (∗r). With this notation the map between cohomology groups induced by the inclusion j_{r} : G_{r} → GL_{n}(ℂ) is

j_{r}^{∗} : c_{i} ↦ ∑_{l=0}^{∞} c_{i−2l}p_{l}|_{(∗r)},

and the Euler class of Σ^{r} is

Δ_{(r−1,r−2,... ,1,0,0,0,...)}(c)|_{(∗r)}.

To prove the theorem let us compute j_{s}^{∗}(Δ_{(r−1,r−2,... ,1)}(c)). It is

Δ_{(r−1,r−2,... ,1,0,0,...)}( ∑_{l=0}^{∞} c_{i−2l}p_{l}|_{(∗s)}) =

= ∑_{f:ℕ→ℕ}p_{1}^{|f^{−1}(1)|}p_{2}^{|f^{−1}(2)|} . . . · Δ_{(r−1−2f(1),r−2−2f(2),...)}(c)|_{(∗s)}.

The Δ’s in the last sum corresponding to f ≢ 0 are all zero, since the i’th and the i+f(i)’th rows are identical for the greatest i such that f(i) ≠ 0. The term corresponding to f ≡ 0 is e_{r} if r = s, and has only 0’s in the first row if s < r. These prove the principal and the homogeneous equations, so the theorem.

We have calculated the whole avoiding ideals for these orbits in [FNR].

5.2. The adjoint representations. We show that the adjoint representation of GL(n) has no interesting Thom polynomials. This is true for the adjoint representation of a large class of Lie groups including the complex semisimple Lie groups but the proof is essentially the same so we only indicate how the general case works.

The adjoint representation V = Ad(GL(n)) is not irreducible: V = Ad(SL(n)) ⊕ ℂ. We state the following proposition for V but it is true for Ad(SL(n)), too and the proof works the same way.

Proposition 5.5. If the invariant subset ξ ⊂ V has a Thom polynomial then Tp(ξ) = 0 or 1.

Proof. The orbits of V are described by the Jordan normal forms. A generic v ∈ V has n different eigenvalues and the stabilizer group G_{η} of the orbit η = GL(n)v is isomorphic to GL(1)^{n}. The inclusion G_{η} → GL(n) induces a map j_{η}^{∗} : H_{GL(n)}^{∗}(pt) → H_{GL(1)^{n}}^{∗}(pt). This map is injective. (This fact is usually called the splitting lemma. For semisimple Lie groups the corresponding map is still injective rationally by a theorem of Borel.) In other words O_{η} = 0. It implies that for any invariant subset ξ ⊂ V for which η ⊄ ξ̄ the avoiding ideal A_{ξ} ⊂ O_{η} = 0 so Tp(ξ) = 0. And if ξ̄ contains every generic orbit then ξ̄ = V and Tp(ξ) = 1.

Remark 5.6. In other words the first obstruction for finding a section of an Ad(GL(n))- bundle with different eigenvalues is zero. But there are other obstructions: Suppose that E is a complex vector bundle over a simply connected base and Ad(E) is its adjoint bundle. If Ad(E) admits a section with different eigenvalues at every point then E = ⨁ L_{i} where the L_{i}’s are the one dimensional eigenspace bundles (if the base is not simply connected we get only an n-line distribution), but not all bundles split.

5.3. Representations of GL(2). The relevant irreducible representations of GL(2) are of the form

S^{n}ℂ^{2} = {homogeneous polynomials of order n in two variables}.

As we mentioned at the beginning of the section Thom polynomials for the other irreducible representations can be easily obtained from these. For n ≥ 4 these representations have families of orbits. A homogeneous polynomial p = p(x, y) defines a 0-dimensional variety V (p) in the projective line ℙ^{1}. Let us call the points of V (p)—counted with multiplicity— the roots of p. Since the cross ratio of 4 points is an invariant of the GL(2)-action we will get families of orbits for n ≥ 4. However if p has at most 3 different roots then the orbit of p doesn’t have a family. It turns out that the existence of these orbits allows us to calculate the Thom polynomial for the orbits with at most two different roots:

η_{i} := orbit of x^{i}y^{n−i} for i = 0, …, [n/2].

Notice that x^{i}y^{n−i} is in the same orbit as x^{n−i}y^{i}. This is the reason why we get a slightly different formula for n even and n odd.

First we calculate the geometric input: The stabilizers (more precisely the maximal compact subgroups of the stabilizers) G_{η_{i}} of these orbits and the the action of G_{η_{0}} on the normal space N_{η_{0}} of η_{0}:

Proposition 5.7.

(i) G_{η_{i}} is the image of the homomorphism h_{i} : U(1) → GL(2) where h_{i}(α) = ( \begin{matrix}α^{n−i} \\ & α^{i} \end{matrix} ). (ii) N_{η_{0}} = ⟨x^{n}, x^{n−1}y, …, x^{2}y^{n−2}⟩.

The calculation are elementary so we omit the proof.

Proposition 5.7 implies that

h_{i}^{∗}(c_{1}) = (n − 2i)c_{1} h_{i}^{∗}(c_{2}) = −i(n − i)c_{1}^{2} for i = 1, …, [n/2] and
h_{0}^{∗}(c_{1}) = c_{1} h_{0}^{∗}(c_{2}) = 0 and e(η_{0}) = n!c_{1}^{n−1}.

Using Theorem 3.2 we see that if η_{i} is not in the closure of an orbit η then Tp(η) is divisible by i(n − i)c_{1}^{2} + (n − 2i)^{2}c_{2}. Taking the principal equation into account we get that for example:

Theorem 5.8.

Tp(η_{0}) = \begin{cases}n ∏_{i=1}^{t} (i(n − i)c_{1}^{2} + (n − 2i)^{2}c_{2}) & if n is odd \\ n\frac{n}{2}c_{1} ∏_{i=1}^{t} (i(n − i)c_{1}^{2} + (n − 2i)^{2}c_{2}) & if n is even. \end{cases}

Remark 5.9. Since Tp(0) = e(S^{n}ℂ^{2}) we have the factorization Tp(0) = nc_{2} Tp(η_{0}) (use e.g. the formula in [Ful98, ex.3.2.6 p.57]). This is not an accident. The orbit η_{0} is the smallest (therefore it is a cone over a smooth projective variety) so a version of the Duistermaat-Heckman formula can be applied which gives the Thom polynomial in this factorized form. (Thanks for A. Knutson for drawing our attention to the Duistermaat-Heckman formula.)

We can associate an invariant subset (n_{1}, …, n_{k}) to any partition of n where the partition encodes the required multiplicities of the roots of the polynomials in the subset. We calculated the Thom polynomials Tp(n_{1}, …, n_{k}) with different methods [FNR]. Recently B. Kőműves found a closed formula using incidences in the sense of [Rim01].

Remark 5.10. The Thom polynomials Tp(1^{k}, n − k) were calculated in [Kir84] by Kirwan for the SL(2)-action (notice that for the SL(2)-action c_{1} = 0 so the SL(2)-Thom polynomial contains less information).

6. The classical case: Giambelli-Thom-Porteous formula

In this section we show how to recover the classical Thom polynomial formula (the so called Giambelli-Thom-Porteous formula, see [Tho56], [Por71]) in our theory. We choose our field to be ℂ and our cohomology theory to be H^{∗}( , ℤ). Suppose that f : N → P is a smooth map of manifolds. The Giambelli-Thom-Porteous formula describes the cohomology class defined by Σ_{s}(f), the subset of N where df has corank s.

14 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

In terms of the theory described above we calculate the Thom polynomials of the representation ρ = Hom(ρ(n), ρ(n + k)) of the group G := GL(n) × GL(n + k) on the linear space ℂ^{(n+k)×n} where ρ(n) is the standard representation of GL(n). So (R, L) ∈ G acts on an (n + k) × n matrix X by: (R, L) · X := LXR^{−1}. We can assume that k ≥ 0.

As it is well known the orbits Σ_{s} of this action are characterized by corank. A repre-

sentative from Σ_{s} is X_{s} := (\begin{matrix}0 & 0 \\ 0 & I_{n−s} \end{matrix})_{(n+k)×n}. The maximal compact stabilizer subgroup

of X_{s} is

G_{s} := G_{X_{s}} = \{ (( \begin{matrix}A & 0 \\ 0 & C \end{matrix} ), ( \begin{matrix}B & 0 \\ 0 & C \end{matrix} )) | (A, B, C) ∈ U(s) × U(s + k) × U(n − s)\}.

An invariant normal slice Σ_{s} at X_{s} is N_{s} = \{( \begin{matrix}M^{(s+k)×s} & 0 \\ 0 & 0 \end{matrix} )\}. It implies that the— complex—codimension of Σ_{s} is s(s + k). To determine the principal equation for the Thom polynomial associated to Σ_{s} we need two data: the equivariant Euler class of the normal slice N_{s} of Σ_{s} and the map H^{∗}(Bi_{s}) : H^{∗}(BG) → H^{∗}(BG_{s}) where i_{s} is the inclusion of G_{s} into G. We will use the notation

H^{∗}(BG) = ℤ[R_{1}, …, R_{n}, L_{1}, …, L_{n+k}]
= ℤ^{S_{n}×S_{n+k}}[r_{1}, …, r_{n}, l_{1}, …, l_{n+k}]
H^{∗}(BG_{s}) = ℤ[A_{1}, …, A_{s}, B_{1}, …, B_{s+k}, C_{1}, …, C_{n−s}]
= ℤ^{S_{s}×S_{s+k}×S_{n−s}}[a_{n−s+1}, …, a_{n}, b_{n−s+1}, …, b_{n+k}, c_{1}, …, c_{n−s}],

where the capitals mean universal Chern classes while the lower letters mean Chern roots, and e.g. ℤ^{S_{n}×S_{n+k}}[ ] means the part of the polynomial ring ℤ[ ] invariant under the action of S_{n} × S_{n+k}, i. e. the permutations of the a_{i}’s and b_{j}’s.

The action of (A, B, C) ∈ G_{s} on the invariant normal slice N_{s} is given by changing M to BMA^{−1}. So the equivariant Euler class of the normal slice N_{s} is (written in terms of Chern roots):

e(N_{s}) = ∏_{i=n−s+1}^{n} ∏_{j=n−s+1}^{n+k}(b_{j} − a_{i}) ∈ H^{s(s+k)}(BG_{s}).

Since e(ν_{s}) is not zero for any s the representation satisfies the Euler condition and by Theorem 3.5 the restriction equations have a unique solution. Even more is true:

Proposition 6.1. For every orbit Σ_{s} the principal equation H^{∗}(Bi_{s}) Tp(Σ_{s}) = e(N_{s}) has a unique solution.

Proof. Since G_{s} ⊂ G_{s+1} the restriction map H^{∗}(Bi_{s}) factors through H^{∗}(Bi_{s+1}) so the homogeneous equations contain no extra information.

Remark 6.2. The same condition applies to the representations Λ^{2}ℂ^{n} and S^{2}ℂ^{n} studied in Section 5. However in that cases it was easy to prove the homogeneous equations.

Remark 6.3. The fact G_{s} ⊂ G_{s+1} has even stronger consequences. It implies that O_{Σ_{s+1}} ⊂ O_{Σ_{s}} therefore the avoiding ideal

A_{Σ_{s+1}} = ⋂_{i≤s} O_{Σ_{i}} = O_{Σ_{s}} = ker H^{∗}(Bi_{s}). (2)

Using the explicit description of H^{∗}(Bi_{s}) below it makes it easy to check whether a characteristic class in H^{∗}(BG) belongs to A_{Σ_{s+1}}. Also, since A_{Σ_{s+1}} ∩ H^{d}(BG) = 0 for d < codim Σ_{s+1} formula (2) implies the injectivity of H^{codim Σ_{s}}(Bi_{s}) which in turn implies Proposition 6.1 directly without using the unicity theorem 3.5.

The map H^{∗}(Bi_{s}) is given by (again in terms of Chern roots):

r_{i} ↦ \{ \begin{matrix}c_{i} & if & i ≤ n − s \\ a_{i} & if & i > n − s \end{matrix}  l_{i} ↦ \{ \begin{matrix}c_{i} & if & i ≤ n − s \\ b_{i} & if & i > n − s \end{matrix}

Lemma 6.4. Using the notation

\frac{1 + L_{1}t + L_{2}t^{2} + … + L_{n+k}t^{n+k}}{1 + R_{1}t + R_{2}t^{2} + … + R_{n}t^{n}} = 1 + H_{1}t + H_{2}t^{2} + …

det(H_{s+i−j})_{(s+k)×(s+k)} maps to ∏_{i=n−s+1}^{n} ∏_{j=n−s+1}^{n+k}(b_{j} − a_{i}) under the map H^{s(s+k)}(Bi_{s}).

Proof. The image H′_{i} of H_{i} are the coefficients of the Taylor series

\frac{∏(1 + a_{i}t) ∏(1 + c_{k}t)}{∏(1 + b_{j}t) ∏(1 + c_{k}t)} = \frac{∏(1 + a_{i}t)}{∏(1 + b_{j}t)} = \frac{1 + B_{1}t + B_{2}t^{2} + … + B_{s+k}t^{s+k}}{1 + A_{1}t + A_{2}t^{2} + … + A_{s}t^{s}} .

But det(H′_{s+i−j})_{(s+k)×(s+k)} equals to the resultant of the two polynomials 1+B_{1}t+B_{2}t^{2}+ … + B_{s+k}t^{s+k} and 1 + A_{1}t + A_{2}t^{2} + … + A_{s}t^{s}. (This is a less known form of the resultant R(p, q) which can be obtained by multiplying the Sylvester-matrix by a matrix obtained from the coefficients of the Taylor series of 1/p see [ACGH85, p.87].) On the other hand the resultant is equal to the product of differences of the roots of the two polynomials.

This lemma and Proposition 6.1 together proves that there is only one polynomial that satisfies the principal equation:

Tp(Σ_{s}) = det(H_{s+i−j})_{(s+k)×(s+k)}.

Notice that a ρ-bundle in this case is a pair of vector bundles E, F of rank n and n + k. In the classical situation of a map f : N → P these bundles are T N and f^{∗}T P, and H_{i} can be interpreted as the i^{th} Chern class of the virtual bundle F ⊖ E. Also notice that the formula doesn’t depend on n.

Using Section 3.1 one could easily calculate the whole avoiding ideal A_{Σ_{s}}, but here we only give the result, since it has been computed by P. Pragacz.

Proposition 6.5 ([Pra88], [FP98, Section 4.2]). A_{Σ_{s}} = {Δ_{λ}(c, c′) : λ ⊃ (s + k)^{s}}.

16 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

7. Singularities

In this section we show how our theory applies to the case of singularities of maps between manifolds—the case where Thom polynomials were originally defined by Thom in [Tho56]. We will work over the complex field, so manifolds and maps are assumed to be complex analytic. What we really show is that the equations we get by Theorem 3.2 for the Thom polynomials of simple singularities are the same as were studied and solved in [Rim01], so we will not repeat their solution here.

Now we recall some standard definitions of singularity theory (see e.g. [AVGL91]): E^{0}(n, n+k) will be the vector space of smooth germs (ℂ^{n}, 0) → (ℂ^{n+k}, 0). We will think of E^{0}(n, n+k) as a subset of E^{0}(n+a, n+k+a) by trivial unfolding. Fixing k let E^{0}(∞, ∞+k) be the union (or formally the direct limit): ∪_{n=0}^{∞}E^{0}(n, n + k). This space will play the role of V of the general theory. Also we have maps u_{∞} : E^{0}(n, n + k) → E^{0}(∞, ∞ + k) (u_{∞} stands for infinite unfolding). Let Hol(ℂ^{n}, 0) denote the group of biholomorphism germs of (ℂ^{n}, 0). The group

A(n, n + k) := Hol(ℂ^{n}, 0) × Hol(ℂ^{n+k}, 0)

acts on E^{0}(n, n + k) by (φ, ψ) · f := ψ ∘ f ∘ φ^{−1}. Similarly the limit group

A(∞, ∞ + k) := ∪_{n=0}^{∞}Hol(ℂ^{n}, 0) × Hol(ℂ^{n+k}, 0)

acts on E^{0}(∞, ∞ + k) by the same formula.

We will mainly be concerned with the bigger—contact—groups

K(n, n + k) := {(φ, M) : φ ∈ Hol(ℂ^{n}, 0), M is a germ (ℂ^{n}, 0) → Hol(ℂ^{n+k}, 0)},

acting on E^{0}(n, n + k) by ρ^{K(n,n+k)}(f) = M(x) ∘ f ∘ φ^{−1}(x). The limit group K(∞, ∞ + k) acts on E^{0}(∞, ∞ + k) by the same formula. This group will play the role of G in the general theory. We use the notations E^{0}, A and K if the value of n (n = ∞ allowed) is clear from the context.

So, consider the action of K(∞, ∞ + k) on E^{0}(∞, ∞ + k). The nicest orbits are the so called simple ones: an orbit (or a representative) is simple, if a neighborhood intersects only finitely many different orbits. Simple orbits will be strata in an appropriate Vassiliev stratification. Let η be a simple orbit, and let us choose a representative f ∈ E^{0}(n, n + k) with minimal n. In other words we choose a minimal dimensional representative with η = orbit of u_{∞}(f). Such an f (defined up to K(n, n + k)-equivalence) is called a genotype for η in [AVGL91, p. 157]. The contact automorphism group Stab^{K}(f) = {(φ, M) ∈ K(n, n+k)|(φ, M)·f = f} and the analogously defined Stab^{A}(f) are not finite dimensional (moreover they do not possess convenient topologies) so we need the following definition— inspired by the classical Bochner theorem—using that GL(n)×GL(n+k) ⊂ A(n, n+k) ⊂ K(n, n + k).

Definition 7.1 ([Jän78]). If M is a subgroup of A(n, n + k) or K(n, n + k) then M is compact if M is conjugate to a compact subgroup N ⊂ GL(n) × GL(n + k).

Luckily enough the groups Stab^{K}(f) and Stab^{A}(f) share many properties with finite dimensional groups, as follows.

Theorem 7.2.

1. Stab^{K}(f) ( Stab^{A}(f)) has a maximal compact subgroup G_{f} = G_{f}^{K} (G_{f}^{A}).

2. Any two maximal compact subgroups are conjugate.

3. B Stab^{K}(f) ≃ BG_{f}^{K} and B Stab^{A}(f) ≃ BG_{f}^{A}.

The proof of 1 and 2 can be found in [Jän78], [Wal80]. As we mentioned Stab(f) does not possess convenient topology, so strictly speaking B Stab(f) is not defined. However, it is possible to define the notion of Stab(f)-principal bundle over a smooth manifold and BG_{f} classifies those bundles ([Rim96, Thm 1.3.6] or [Rim]). So from our point of view we can replace B Stab(f) by BG_{f}. In particular we have BK(n, n + k) ≃ BA(n, n + k) ≃ BGL(n) × BGL(n + k).

Remark 7.3. Theorem 7.2 allows us to use the same algorithm to calculate the Thom polynomials as in the finite dimensional case.

Definition 7.1 and Theorem 7.2 shows that by choosing f carefully from its K-orbit, we can assume that G_{f}^{K} ⊂ GL(n)×GL(n+k), so we have representations μ_{0}, μ_{1} of G_{f}^{K} on the source and target spaces respectively. By part 2 of Theorem 7.2 the isomorphism classes of these representations are uniquely defined. The groups G_{f}^{K} and representations μ_{0}, μ_{1} were calculated for low codimensional singularities in [Rim].

For the identification of G_{η} for G = K(∞, ∞ + k) we cannot directly use Theorem 7.2, but it is not difficult to get around:

Definition 7.4.

G_{η} := G_{f} × U(∞) ⊂ Stab(η)

where the inclusion of U(∞) into Stab(η) corresponds to the diagonal action on the unfolding dimensions.

Though G_{η} is not compact in any reasonable sense, it is still true that B Stab(η) ≃ BG_{η} (in the sense of our remark after Theorem 7.2) and as we will see the U(∞) summand acts trivially on N_{η} anyway.

Below we explain how to calculate the two inputs of the algorithm for computing the Thom polynomials for an orbit η, i. e. the map H^{∗}(Bi) : H^{∗}(BG) → H^{∗}(BG_{η}) and the representation ρ_{η} : G_{η} → GL(N_{η}).

Proposition 7.5. The homomorphism H^{∗}(BK) = H^{∗}(BU(∞) × U(∞)) → H^{∗}(BG_{η}) induced by the inclusion G_{η} → A ⊂ K is given by

\begin{matrix}ℤ[\underline{a}, \underline{b}] & ⟶ & H^{∗}(BG_{f}) ⊗ ℤ[\underline{d}] \\ a & ↦ & c(μ_{0}) · d \\ b & ↦ & c(μ_{1}) · d, \end{matrix}

where \underline{a} = a_{1}, a_{2}, … and \underline{b} = b_{1}, b_{2}, … are the universal Chern classes of the two factors of U(∞) × U(∞); a = 1 + a_{1} + a_{2} + …, b = 1 + b_{1} + b_{2} + … are the total Chern classes,

18 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

and the definitions for \underline{d}, d are similar. The class c(μ_{i}) is the total Chern class of the vector bundle E_{μ_{i}} over BG_{f}.

Let us turn to our second goal. It is enough to calculate the G_{f}^{K}-action ρ_{f} on the normal space N_{f}^{K} to the K(n, n + k)-orbit in E^{0}(n, n + k) since u_{∞}(N_{f}^{K}) is normal to η in E^{0}(∞, ∞ + k) too, and once the action is G_{f}-invariant, it is also G_{η}-invariant with the trivial action of the U(∞) factor.

The representation ρ_{f} is also explicitly computable for low codimensional singularities: The miniversal unfolding F of f is a stable germ (ℂ^{n} ⊕ U, 0) → (ℂ^{n+k} ⊕ U, 0) where U is the unfolding space. The group G_{f}^{K} acts linearly on U: let us denote this representation by μ_{U}. Then we have ρ_{f} ≅ μ_{0} ⊕ μ_{U}. We can also see that the source space of F is naturally isomorphic to N_{f}^{K}. (For more details see [Wal80] or [Rim].)

Remark 7.6. Calculation of the Thom polynomials for the stable orbits of the A(∞, ∞+ k)-action doesn’t give anything new: every such orbit is a dense open subset of a K(∞, ∞+ k)-orbit —we usually use the same notation for the two orbits—so their Thom polynomials are the same.

Now, we might as well write down the equations of Theorem 3.2 for the Thom polynomials for K, but it is possible to simplify these equations as we will see in Theorem 7.13.

We can also calculate Thom polynomials for K(n, n+k). These results are not independent: If η is an orbit of K(n, n + k) then its d-dimensional unfolding u_{d}(η) is an orbit of K(n + d, n + k + d) with the same codimension. This is a consequence of the fact that the codimension of the K-orbit can be read off from its local algebra, which doesn’t change by trivial unfolding. To understand the connection between the Thom polynomial of η and of u_{d}(η) we look at the unfolding map u_{d} : K(n, n + k) → K(n + d, n + k + d). It induces a map

H^{∗}(Bu_{d}) : H^{∗}(BK(n + d, n + k + d) ≅ ℤ[a_{1}, …, a_{n+d}, b_{1}, …, b_{n+k+d}] → H^{∗}(BK(n, n + k)) ≅ ℤ[a_{1}, …, a_{n}, b_{1}, …, b_{n+k}]

(d = ∞ is allowed) such that

H^{∗}(Bu_{d})(a_{i}) = \{ \begin{matrix}a_{i} & if & i ≤ n \\ 0 & if & i > n \end{matrix} H^{∗}(Bu_{d})(b_{i}) = \{ \begin{matrix}b_{i} & if & i ≤ n + k \\ 0 & if & i > n + k \end{matrix}

Proposition 7.7. Tp(η) = H^{∗}(Bu_{d})( Tp(u_{d}(η))) where u_{d} denotes the d-fold trivial unfolding (d = ∞ is allowed).

This is a special case of a more general fact used frequently in calculating Thom polynomials:

Proposition 7.8. Let V_{1} be a G_{1}-vector space and V_{2} be a G_{2}-vector space Let φ : G_{1} → G_{2} be a homomorphism and J : V_{1} → V_{2} a φ-equivariant map transversal to the G_{2}-action. Suppose that η ⊂ V_{2} has a Thom polynomial Tp(η). Then Tp(φ^{−1}(η)) = φ^{∗} Tp(η).

Proof. This is a straightforward generalization of the pullback property of the ordinary Poincaré dual.

Proof of Proposition 7.7. The unfolding map u_{d} is transversal so we can apply Proposition 7.8.

So using the generators a_{i}, b_{i} for all of these groups we can say that for d large enough (n+ d ≥ codim η) unfoldings don’t change the Thom polynomial. Pursuing these arguments further, we find another important property of these Thom polynomials (first appeared probably in [Dam72]):

Proposition 7.9 (folklore, see [AVGL91]). Tp(η) ∈ Q where Q is the subring of H^{∗}(BK) generated by 1, h_{1}, h_{2}, …, where 1 + h_{1} + h_{2} + … = \frac{1+b_{1}+b_{2}+...}{1+a_{1}+a_{2}+...}.

Before getting into the proof we need some definitions:

Definition 7.10. If A^{n} and B^{n+k} are vector bundles over a manifold M, then let E^{0}(A, B) denote the E^{0}(n, n + k)-bundle over M such that

E_{m}^{0}(A, B) = {germs of smooth maps (A_{m}, 0) → (B_{m}, 0)}.

Using Theorem 7.2 it is not difficult to see that

a_{i}(E^{0}(A, B)) = c_{i}(A) and b_{i}(E^{0}(A, B)) = c_{i}(B)

through the obvious identifications. (In fact it also follows from Theorem 7.2 that every E^{0}(n, n + k)-bundle is isomorphic to a bundle of the form E^{0}(A, B) but we don’t use this in this paper.)

Definition 7.11. Let C^{d} be a d-dimensional vector bundle. Then

u_{C} : E^{0}(A, B) → E^{0}(A ⊕ C, B ⊕ C)

denotes the twisted unfolding map:

u_{C}(φ) := φ ⊕ Id_{C} .

Proof of Proposition 7.9. We have a commutative diagram

\xymatrix{E_{ρ^{K(n,n+k)}} \ar[r]^{ũ_{d}} & E_{ρ^{K(n+d,n+k+d)}} \\ E^{0}(A, B) \ar[r]^{u_{c}} \ar[u]^{k̃} & E^{0}(A ⊕ C, B ⊕ C) \ar[u]^{k̃_{C}}}

where k̃ and k̃_{C} are the bundle maps induced by the corresponding classifying maps. It shows that

k^{∗} Tp(η) = k_{C}^{∗} Tp(u_{d}(η))

Choose C to be the ‘inverse’ of A. Then using that we can think of Tp as a polynomial of \underline{a} and \underline{b} we get (with some abuse of notation):

Tp(η)(c(A), c(B)) = Tp(u_{d}(η))(1, c(B)/c(A))

Definition 7.12. Let tp(η) be the unique polynomial with the property

tp(η)(1, h_{1}, …) = Tp(η)(\underline{a}, \underline{b}),

where 1 + h_{1} + h_{2} + … = \frac{1+b_{1}+b_{2}+...}{1+a_{1}+a_{2}+...}.

Proposition 7.5 is enough to write down the equations for the Thom polynomials of simple singularities in the cohomology ring of BA or BK, namely in ℤ[\underline{a}, \underline{b}]. In the light of Proposition 7.9 we write it in terms of the ‘quotient’ variables h_{i}:

Theorem 7.13.

tp(η)(c(θ)) = \begin{cases}Euler class of E_{ρ_{η}} & if θ = η  ‘principal equation’ \\ 0 & if θ ≯ η  ‘homogeneous equations’, \end{cases}

where c(θ) = c(E_{μ_{1}(θ)})/c(E_{μ_{0}(θ)}).

These are exactly the equations that were solved for many cases in [Rim01].

Remark 7.14. This representation of K doesn’t satisfy the Euler condition but a closer look at the proof of the unicity theorem 3.5 shows that the restriction equations have a unique solution in the codimension range where there are only finitely many orbits and the euler classes of the normal spaces are non zero.

Remark 7.15. At this point we would like to comment on the history of these ideas. The systematic study of classifying spaces of the symmetry groups of singularities and a powerful construction out of these spaces was pioneered by A. Szűcs (see e.g. [Szű79]). In the language of the present paper he calculated G_{η} for several singularities, and described a way how to glue these spaces together to get a space whose algebraic topological properties can be translated to differential topological theorems. He applied his construction to various differential topological questions, such as e.g. the cobordism groups of maps with given singularities (e.g. [Szű80], [Szű91], [Szű94]). A general method of calculating more of the symmetry groups was given in [Rim96], [RS98], [Rim]. In the present paper we explored the fact that roughly speaking (the ‘source space’ in) Szűcs’ construction is a union of strata in the stratification of BA defined by the A-action, and that the ideas fruitful there (e.g. Thom polynomial calculations [Rim01]) turn out to be fruitful viewing any G-action.

Example 7.16. Let us start with V = E^{0}(1, k + 1). The group K(1, k + 1) acts on it as usual. Here the finite codimensional strata will be the contact orbits A_{n}^{1}(k) represented by the germ x ↦ (x^{n+1}, 0, …, 0). Just like above one can write up the equations for the Thom polynomials of these strata. Carrying out the computation (or using Proposition 7.8) one finds that here the ‘principal’ and ‘homogeneous’ equations are enough (see Remark 7.14) to determine the Thom polynomials:

Tp(A_{n}^{1}(k)) = ∏_{j=0}^{k} ∏_{i=1}^{n}(b_{j} − ia) ∈ H^{∗}(BU(1) × BU(k + 1)) = ℤ^{S_{k+1}}[a, b_{0}, …, b_{k}]. (3)

Since the unfolding map u_{∞} : E^{0}(1, k + 1) → E^{0}(∞, k + ∞) is transversal to the orbits by Proposition 7.8 we have u_{∞}^{∗}( Tp(A_{n}(k))) = Tp(A_{n}^{1}(k)). The homomorphism u_{∞}^{d} is injective for d ≤ 2k +2 so this way we get a simple way to calculate the Thom polynomials of A_{2}(k) (the so called Ronga formulas). u_{∞}^{3k+3} has a kernel so formula (3) is not enough to calculate the Thom polynomials of A_{3}(k). (It makes it easier however. We published a closed formula for Tp(A_{3}(k)) in [BFR02].)

8. Other applications of the method of RE

8.1. Thom polynomials for singularities with integer coefficients. There is a theory of real singularities parallel to the complex case discussed in Section 7. References for the rich geometry and topology of this local theory are e.g. [AVGL91] and [PW95]. The globalization of the theory, i.e. in the study of the Thom polynomials of real singularities has two levels: one can ask for the Thom polynomials with mod 2 coefficients or with integer coefficients. The mod 2 case can be basically solved by the method of [BH61]. Already in this case, and very essentially in the ℤ-coefficient case one has to answer the question: what (formal linear connections of singularities) defines a Thom polynomial? The answer uses a detailed analysis of the Kazarian spectral sequence (or at least its 0’th row, the so called Vassiliev universal complex). Both the answer to this question and the actual Thom polynomials are presented in [FRb].

If a singularity η is not cooriented then there is no Thom polynomial with integer coefficients. However the avoiding ideal may not be empty. A simple way to find element in A_{η} is to look at the ”realization” of an element in A_{η_{ℂ}}. For example p_{1} is an obstruction for finding an immersion of an n-dimensional manifold into ℝ^{n−1} since Tp(Σ_{ℂ}^{1}(−1)) = c_{2}. p_{1} as an obstruction was used in [OSS].

8.2. Quivers. A surprisingly rich class of representations are the quiver representations. Let E_{0} be the set of vertices and E_{1} be the set of edges of a connected oriented graph (double edges, loops allowed). The tail and the head of an edge e is denoted be t(e) and h(e). If there is a vector space V_{v} assigned to any vertex v then we can consider the group G = ∏_{v∈E_{0}} GL(V_{v}) and its action on the vector space

⨁_{e∈E_{1}} Hom(V_{t(e)}, V_{h(e)})

given by

(M_{v})_{v∈E_{0}} · (φ_{e})_{e∈E_{1}} = (M_{h(e)} ∘ φ_{e} ∘ M_{t(e)}^{−1})_{e∈E_{1}}.

This representation is called the quiver representation associated to the graph and the dimension vector (dim(V_{v}))_{v∈E_{0}}.

22 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

This representation, including its orbit structure was thoroughly studied in representation theory (see e.g. [ARS95]). It turns out that this action has finitely many orbits if and only if the graph is of Dynkin type (with some orientations on the edges).

The Thom polynomials of the action associated to the A_{n} series were studied by Buch and Fulton [BF99]. They showed on one hand that these Thom polynomials generalize many objects in algebraic combinatorics (see also [FP98]), including different versions of Schubert polynomials (for which there has been no explicit determinantal formula known, they are usually computed by recursion). On the other hand in [BF99] an algorithm is given to compute the Thom polynomials of any orbit of A_{n}. Buch and Fulton conjecture a formula for these Thom polynomials, and prove it for special cases.

In [FR02] the authors applied the method of the present paper to quiver representations associated with arbitrary Dynkin graphs, and thus obtained a straightforward procedure (but not a formula) to get any Thom polynomials.

8.3. Schur and Schubert polynomials. The cohomology ring structure of Grassmann and flag manifolds are governed by Schur and Schubert polynomials, see e.g. [FP98]. In a recent application ([FRa]) the authors found a way to obtain these Schur and Schubert polynomials as Thom polynomials.

The starting point is that we act on the vector space Hom(ℂ^{n}, ℂ^{p}) by triangular matrices from both sides (flag case) or by triangular matrices from one side and GL from the other side (Grassmann case). Then the orbits will correspond to Schubert varieties, whose equivariant Poincaré dual are the double Schubert (flag case) and double Schur (Grassmann case) polynomials. If we disregard one set of indeterminates we obtain the ordinary Schubert and ordinary Schur polynomials.

So it is enough to apply the method of the present paper to the above triangular and half-triangular actions, and we recover (or give a new definition as well as a new way to compute) double and simple Schur and Schubert polynomials. We proved a Jacobi-Trudi type determinantal formula for the double Schur polynomials and gave an illuminating proof for the Lascoux-Schützenberger recursion ([LS82]) for double Schubert polynomials.

9. Projective Thom polynomials and degree calculations

If a group G acts linearly on a vector space V then this action ρ induces an action ℙρ of G on the projective space ℙV. If the image of ρ contains the scalars then there is a bijection between the orbits of ℙρ and the non zero orbits of ρ. Strangely enough the projective Thom polynomials formally contain more informations than the affine ones: Suppose that η is a d complex codimensional invariant subset of ρ and ℙη is the corresponding invariant subset of ℙρ. Then Tp(ℙη) is an element in H_{G}^{2d}(ℙV) ≅ H^{2d}(BG)[ξ]/ ∏ ξ − β_{i} where β_{i} are the weights of the representation ρ. So Tp(ℙη) = ∑ p_{i}ξ^{i} where p_{i} ∈ H^{2(d−i)}(BG). It is easy to see that p_{0} is the affine Thom polynomial Tp(η) and p_{d} is the degree of the closure of ℙη:

p_{0} = Tp(η) p_{d} = deg(η) (4)

In fact the main application of the projective Thom polynomial is that we can calculate the degree of certain varieties.

The main result of this section is that a simple substitution into the affine Thom polynomial Tp(η) provides the projective Thom polynomial. To state the result we need to give names to the generators of H^{∗}(BG). Let m : U(1)^{n} → G be a (coordinatized) maximal torus of G and let α_{i} are the corresponding roots. So by the Borel theorem (or splitting principle) Tp(η) is a polynomial in the roots α_{i} and Tp(ℙη) is a polynomial in the roots α_{i} and ξ. We assume that the image of ρ contains the scalars i.e. there is a homomorphism φ : GL(1) → G and a non zero integer q such that ρ∘φ(λ) = λ^{q}v for all v ∈ V, λ ∈ GL(1). We assume that Im m ⊃ Im φ|_{U(1)} so we have a homomorphism φ̃ : U(1) → U(1)^{n} such that φ|_{U(1)} = m ∘ φ̃. The homomorphism φ̃ is necessarily of the form φ̃(t) = (t^{w_{1}}, …, t^{w_{n}}) where t ∈ U(1) and w_{i} are integers. Notice that the choice of φ is not unique.

Theorem 9.1 ([FNR]). Let ρ : G → GL(V) be a representation of the Lie group G such that the image of ρ contains the scalars. Let α_{i}, q, w_{i} be as above and let η be an invariant subset of ρ. Then

Tp(ℙη)(α_{1}, …, α_{n}, ξ) = Tp(η)(α_{1} + \frac{w_{1}}{q} ξ, …, α_{n} + \frac{w_{n}}{q} ξ).

Corollary 9.2. Using the notation of Theorem 9.1

deg(ℙη) = q^{−d} Tp(η)(w_{1}, …, w_{n}).

This a generalization of results on degrees of certain degeneracy loci of Porteous [Por71], Harris-Tu [HT84], Fulton [Ful98]. In [FNR] this formula is used to calculate the degree of the dual of some of the Grassmannians Gr_{k}(ℂ^{n}).

10. The Kazarian spectral sequence

The Kazarian spectral sequence gives interesting relations (e.g. linear equations or bounds) for the number of different strata—i.e. in some sense, the number of different singularity types—in a fixed codimension. In this section we are giving two examples of these.

Let us briefly recall the Kazarian spectral sequence [Kaz97a]. Associated with the group representation ρ : G → GL(V) we consider the universal ρ-bundle: BV := EG ×_{ρ} V ⟶ BG (the letter B in BV stands for Borel construction). A key observation is that if η is an invariant subset of ρ then η can be identified in every fiber. Let their collection be Bη (= the Borel construction with η). So a stratification of V by invariant submanifolds induces a stratification of BV. If F_{i} is the union of Bη’s with codim η at most i, then the spectral sequence associated with the filtration ∅ ⊂ F_{0} ⊂ F_{1} ⊂ … ⊂ BV is called the Kazarian spectral sequence. Now suppose that the stratification we started with is special: it satisfies Vassiliev’s condition [Vas88, 8.6.5]—i.e. it is locally finite, the stabilizer subgroups in one stratum are constant and the moduli spaces stratum/G are contractible. Then using excision and Thom isomorphism one can easily see that the E_{1} term of this spectral sequence is the following: the q’th column contains the cohomology groups of the

24 LÁSZLÓ M. FEHÉR AND RICHÁRD RIMÁNYI

classifying spaces of the stabilizer subgroups of q-codimensional strata. In this general situation the Thom polynomial is defined as the edge homomorphism of this spectral sequence: E_{2}^{p,0} → E_{∞}^{p,0} ⊂ H^{∗}(BV) = H^{∗}(BG).

So this spectral sequence clearly organizes the points to consider when there are neighboring dimensional strata and one has to glue some together to obtain one which satisfies the conditions we put in Section 2.3. Considering complex representations this spectral sequence often (but not always) degenerates at E_{1}. Since this was the situation in almost all the examples we considered, we restricted ourselves to the simple version of Section 2.

The study of this spectral sequence was initiated in [AB83] to study the betti numbers of certain moduli spaces. Let us now, however, give an other and simpler application of the Kazarian spectral sequence.

Consider the Kazarian spectral sequence associated to the representation of GL(n) × GL(n+k) (over ℂ) discussed in Section 6. It collapses at E_{1} since the odd rows and columns are zero. Particularly interesting is the limiting case n = ∞. We assume for simplicity that k = 0. One finds that in the s^{th} column one has to write the cohomology groups of U(s)^{2} ×U(∞), see Section 6. Summing up the ranks in the skew diagonals we must obtain the ranks of the cohomologies of U(∞)^{2}. Some combinatorics shows that “one can drop a ×U(∞) term” from everywhere, i.e. one can write the cohomology groups of U(s)^{2} in the s^{th} column and get the ranks of the cohomologies of U(∞) (i.e. 1, 1, 2, 3, 5, 7, 11, 15, …) in the skew diagonals, as follows:

8. . 5 . . 14 …. 16
6. . 4 . . 8 …. 10
4. . 3 . . 5 …. 5
2. . 2 . . 2 …. 2
0. 1 1 . . 1 …. 1
0. 2. 4. 6. 8. 10. 12. 14. 16. 18.

In this table only the ranks of the free Abelian groups are written (i.e. a is written instead of ℤ^{a}) and only the terms with two even coordinates are indicated, since everything else is 0. This leads to the combinatorial identity

π(n,[1, 1, …])=π(n−1,[2, 0, 0, …])+π(n−4,[2, 2, 0, 0, …])+π(n−9,[2, 2, 2, 0, 0, …])+⋯,

where π(n, [a_{1}, a_{2}, …]) denotes the number of degree n monomials in terms of a_{i} copies of variables of degree i. This identity—already known to Euler as a very effective way to compute the number of partitions—can be directly proved by using Young diagrams (thanks to A. Blokhuis for these informations).

Let us now turn to the Kazarian spectral sequence of E^{0}(∞, k + ∞) of Section 7. For simplicity let k = 0. The list of simple singularities (for codimension ≤ 8) is given in the following table.

codim_{ℂ} 0 1 2 3 4 5 6  7  8
A_{0} A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7}  A_{8}
I_{2,2} I_{2,3} I_{2,4} I_{2,5}  I_{2,6}
I_{3,3} I_{3,4}  I_{3,5}
I_{4,4}
(x^{2}, y^{3}) (x^{2} + y^{3}, xy^{2})

The maximal compact symmetry groups of these singularities can be computed as in [Rim], which gives us the E_{1} term for the Kazarian spectral sequence. Let us suppose however, that we only know the easy part of the classification, i.e. only up to codimension 3. The symmetry group of the A_{i} singularity is easily computed as U(1) × U(∞) for i > 0. So hereby we show the (degenerated) spectral sequence with the ‘U(∞) terms dropped’ (as above), and writing only the ranks of the occurring groups:

10. 0 1 1 1
8. 0 1 1 1
6. 0 1 1 1
4. 0 1 1 1
2. 0 1 1 1
0. 1 1 1 1 n
0. 2. 4. 6. 8.

Here thus n is the number of singularities of complex codimension 4 (which we assumed not to know). The point here is that the value of n can be found from the spectral sequence above, by observing that the the sums of skew diagonals should be equal to the ranks of the cohomologies of BU(∞): 1, 1, 2, 3, 5, … (the number of partitions). This gives us that n must be 2. So we could predict the number of different strata of codimension d knowing only information about strata of codimension < d. The interested reader can extend the above spectral sequence using the further classification of singularities, and try to speculate about the number of different strata in codimensions 16, 17, 18, … (but be careful about moduli spaces!).

Applications of this method in algebraic geometry and singularity theory are subject to further study.

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Department of Analysis, ELTE TTK, Budapest Hungary

  E-mail address: lfeher@math-inst.hu

Department of Mathematics, Ohio State University

  E-mail address: rimanyi@math.ohio-state.edu