THOM POLYNOMIALS, SYMMETRIES AND INCIDENCES OF SINGULARITIES

RICHÁRD RIMÁNYI

Abstract. As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤ 8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N^{n} ⟶ P^{n+k} for k > 0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities.

1. Introduction

On the basis of the “generalized Pontryagin-Thom construction” of [RSz] here we present a new method to compute Thom polynomials ([T]) of singularities. Recall that the Thom polynomial T p(η) of a singularity η is a polynomial in the Chern classes of the map f between complex analytic manifolds N, P, and this polynomial equals the Poincaré dual [η(f)] of the cycle carried by the closure of

η(f) = {x ∈ N | the singularity of f at x is η }

for most maps. About the existence of such a polynomial see [AVGL] or section 6 here. By Chern classes of a map f : N ⟶ P we mean the Chern classes of the virtual bundle f^{∗}T P −T N over N. The cohomology class [η(f)] is most easily understood when η(f) is a submanifold, which is often the case if f has no more complicated singularities than η. In this case η(f) carries a fundamental homology class. We take the image of this class in the homology of N and apply Poincaré duality. The resulting class is [η(f)] ∈ H^{∗}(N; ℤ). Although the definition of [η(f)] is not much more difficult when η(f) is not a manifold (it has singularities along lower dimensional strata), the interesting thing is that we will not need this. We will only use the definition of [η(f)] in the mentioned case. Observe that this is a difference from the desingularization method (see e.g. [G]), where the behavior of η(f) near the singular part is studied.

Now we clarify what we will mean by singularities, whose Thom polynomials we are studying. Let k ≥ 0 be a fixed integer. By singularity we will mean an equivalence class of stable germs (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0) under the equivalence generated by right-left equivalence and suspension (by suspension of a germ κ we mean its trivial unfolding: (x, v) ↦ (κ(x), v)). According to the classical Mather theory singularities are in a one-to-one correspondence with (finite dimensional local) ℂ-algebras. So we might as well denote a singularity by its local algebra. Instead, we

Keywords: singularities, Thom polynomials, generalized Pontryagin-Thom construction, incidence class.

2 RICHÁRD RIMÁNYI

will use — more or less — standard notations as follows: A_{i} (= Σ^{1_{i}}) will stand for the stable germs with local algebra ℂ[[x]]/(x^{i+1}) and we will adapt Mather’s notation for the ones of Thom-Boardman type Σ^{2,0}:

I_{a,b} for those with algebra ℂ[[x, y]]/(xy, x^{a} + y^{b}) b ≥ a ≥ 2,
III_{a,b} for those with algebra ℂ[[x, y]]/(xy, x^{a}, y^{b}) b ≥ a ≥ 2 (k ≥ 1).

Recall also that among the germs in a singularity η there is a germ κ (defined up to right-left equivalence) such that all the other germs in η are right-left equivalent to a suspension of κ. We will call such a κ a prototype of the singularity η. E.g. a prototype of the singularity A_{i} (for k = 0) is

(x, u_{1} …, u_{i−1}) ↦ (x^{i+1} + ∑_{j=1}^{i−1} u_{j}x^{j}, u_{1}, …, u_{i−1}).

In [RSz] only those settings were handled where moduli do not occur. For simplicity therefore in this paper we will also avoid the settings where moduli come into the picture (recall e.g. that there is a Thom polynomial of Σ^{3} between equal dimensional manifolds, but Σ^{3} is not a “singularity” in our sense, it is rather the union of a family of singularities).

The milestones of the history of Thom polynomials are (probably among others) the works of Thom, Levine, Porteous, Ronga, Gaffney, Turnbull, Ohmoto (see the References, or the review on the history of Thom polynomials in [AVGL]). Wherever we present a Thom polynomial in this paper the origin of that result will be indicated — according to the author’s best knowledge. The author thanks A. Szűcs, I. Porteous and T. Ohmoto for some helpful conversations and/or encouragement; and the many who pointed out an error in a formal version of this paper.

2. Main theorem

Let k ≥ 0 and let η : (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0) be a stable singularity with prototype κ : (ℂ^{n}, 0) ⟶ (ℂ^{n+k}, 0).

Definition 2.1. (The total Chern class and the Euler class associated to a singularity.) The maximal compact subgroup of the right-left symmetry group

Aut κ = { (ϕ, ψ) ∈ Diff(ℂ^{n}, 0) × Diff(ℂ^{n+k}, 0) | ψ ∘ κ ∘ ϕ^{−1} = κ }

of κ will be denoted by G_{η}. Its representations (projections) on the source ℂ^{n} and the target ℂ^{n+k} spaces will be λ_{1}(η) and λ_{2}(η). The vector bundles associated to the universal principal G_{η}-bundle EG_{η} ⟶ BG_{η} using the representations λ_{1}(η) and λ_{2}(η) will be called ξ̄_{η} and ξ_{η}. The total Chern class of the singularity η is defined as

c(η) := \frac{c(ξ_{η})}{c(ξ̄_{η})} ∈ H^{Π}(BG_{η}, ℤ)

where c(ξ) = 1 + c_{1}(ξ) + c_{2}(ξ) + … is the total Chern class of the vector bundle ξ. Let the Euler class e(η) ∈ H^{2codim η}(BG_{η}, ℤ) be the Euler class of the bundle ξ̄_{η}.

Let us make a few remarks about this definition. It is clear that Aut κ is much too big to be a (finite dimensional) Lie group. Still, the concept of its maximal compact subgroup (up to conjugacy) can be defined in a sensible way, see [W], [R1] — in fact, not only the definition but the computation of G_{η} is feasible. It is clear that G_{η} can be chosen so that the images of its projection to its factors Diff(ℂ^{n}), Diff(ℂ^{n+k}) are linear, so we can indeed talk about the representations λ_{i}(η) (i = 1, 2). The ring H^{Π}(X) is almost like the usual cohomology ring H^{∗}(X) of the space X with the only difference that in its elements we allow infinitely many non-zero coefficients, see [MS] p.39. With the exception of section 7 from now on all cohomology is meant with integer coefficients without any further notice.

Let η ≠ ζ : (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0) be stable singularities.

Definition 2.2. (The hierarchy of singularities.) The singularity η will be called more difficult than ζ if in any neighbourhood of 0 in the source space of (a representative of) η there is a point where the singularity of (the representative of) η is ζ. In this case we will write ζ < η. Let us adapt the convention η ≮ η, too.

For instance in the case k = 0 we have the following relations: A_{k} > A_{l} iff k > l; I_{2,2} > A_{1}, A_{2}, A_{3}; I_{2,3} ≥ A_{1}, A_{2}, A_{3}, A_{4}, I_{2,2} etc. Some more results on the hierarchy is e.g. in [La]. A trivial but key observation is that a more difficult singularity must have larger codimension, so:

Proposition 2.3. codim ζ ≥ codim η  ⇒  ζ ≮ η.

Suppose we want to compute the Thom polynomial T p(η). Our main theorem gives its value under some substitutions.

Theorem 2.4. (Main theorem.)

T p(η)(c(ζ)) = \begin{cases}e(ζ) & if ζ = η \\ 0 & if ζ ≯ η and ζ ≠ η. \end{cases}

The proof will be given in the next chapter, and applications in chapter 5. Observe that this theorem does not state anything about the value of T p(η)(c(ζ)) for ζ > η, c.f. chapter 8.

Sometimes it will be convenient not to work with the whole maximal compact subgroup G_{η} but with a subgroup G′_{η} ≤ G_{η}. For such a G′_{η} one can define c′(η) and e′(η) in the same way as in definition 2.1. The following corollary of the main theorem will be useful in section 5.

Corollary 2.5.

T p(η)(c′(ζ)) = \begin{cases}e′(ζ) & if ζ = η \\ 0 & if ζ ≯ η and ζ ≠ η \end{cases}

for any subgroup G′_{ζ} ≤ G_{ζ}.

Proof . These equations are homomorphic images of the equations of theorem 2.4 at the natural homomorphism: H^{Π}(BG_{η}; ℤ) ⟶ H^{Π}(BG′_{η}, ℤ).

Of course, if we choose G′_{ζ} too small (e.g. the trivial subgroup) then the statement of the corollary is trivial — so to get reasonable statement we have to choose G′_{ζ} as ‘close’ to G_{ζ} as possible.

4 RICHÁRD RIMÁNYI

3. Proof of the main theorem

The proof uses the generalized Pontryagin-Thom construction [RSz], whose idea goes back to earlier works of A. Szűcs (e.g. [Sz1]), see references therein. Let us now recall the main results. Meanwhile (and it the whole paper) we will use the following notation: if ξ is a vector bundle then D(ξ), S(ξ) will denote its ball and sphere bundle, respectively. If A is a submanifold in B then ν(A ⊂ B) or simply ν_{A} will denote the normal bundle of the embedding. The closure of a subspace X will be called closure X. Note that overline does not mean closure.

Let k ≥ 0 be fixed and let τ be a set of multi-singularities (ℂ, S) ⟶ (ℂ^{∗+k}, 0) (S a finite set) satisfying that ζ < η, η ∈ τ ⇒ ζ ∈ τ. For this latter notion, of course, we have to extend the definition of < to multi-singularities as in [RSz]. A map f : N ⟶ P is called a τ-map if for any y ∈ P the singularity of f at f^{−1}(y) is from τ. The result of [RSz] is a concrete description of a universal τ-map fτ : Y τ ⟶ Xτ. The topological spaces Y τ and Xτ are glued together from so called blocks, corresponding to elements of τ. The blocks in Y τ and Xτ corresponding to η are constructed as follows. Let G_{η} be the maximal compact subgroup of the right-left automorphism group of κ (the prototype of η), with the representations λ_{1}(η) and λ_{2}(η) on the source and target spaces. We can associate vector bundles to the universal principal G_{η}-bundle EG_{η} ⟶ BG_{η} using these representations, we obtain ξ̄_{η}, ξ_{η}. The block in Y τ corresponding to η is the disc bundle of ξ̄_{η} and the block in Xτ corresponding to η is the disc bundle of ξ_{η}. One can also define a natural map D(ξ̄_{η}) ⟶ D(ξ_{η}) which is η in each fibre, obtaining the block

D(ξ̄_{η}) ⟶ D(ξ_{η})
↓ ↓
K̄_{η} ⟶ K_{η} = BG_{η}

Remark 3.1. Let us remark here that in case of a multisingularity, i.e. if S has more than 1 element, some notions in the above paragraph should be meant more general. In this case λ_{1}(η) is a generalized representation, that is a representation on some disjoint (|S| copies of) vector spaces (permutations allowed). Thus the result of the association to EG_{η} ⟶ BG_{η} with λ_{1}(η) will not be a vector bundle, but a composition of a vector bundle (the left-hand arrow in the diagram above) with an |S|-sheeted covering (the bottom arrow). If |S| = 1, then K̄_{η} = BG_{η}, too, and the bottom arrow is a diffeomorphism.

The gluing of these blocks are defined recursively. Suppose we have already constructed fτ′ : Y τ′ ⟶ Xτ′ and want to attach the block D(ξ̄_{η}) ⟶ D(ξ_{η}), where both τ′ and τ′ ∪ {η} satisfy the condition on τ above. The way of gluing is detailed in [RSz], here we will only need that S(ξ̄_{η}) is glued to Y τ′ and S(ξ_{η}) is glued to Xτ.

Now we describe what we mean by fτ being a “universal τ-map”. In [RSz] — where the real case is considered — theorem 1 has two parts: (A) and (B). Although the proof of part (B) contains smooth techniques, which do not work in the complex case, part (A) holds with no change. According to this for any τ-map N ⟶ P there are maps g and h such that the diagram

\begin{matrix}P & ⟶^{g} & Xτ \\ ↑^{f} & & ↑^{fτ} \\ N & ⟶^{h} & Y τ \end{matrix}

is commutative. Also for all η ∈ τ we have that • η(f) = h^{−1}(K̄_{η}) and the normal bundle of η(f) in

N \ { ζ(f) | ζ > η }

is isomorphic to h^{∗}(ξ̄_{η}) (for simplicity here h is written instead of its restriction h|K̄_{η});

• f(η(f)) = g^{−1}(K_{η}) and the normal bundle of f(η(f)) in

P \ { f(ζ(f)) | ζ > η }

is isomorphic to g^{∗}(ξ̄_{η}) (for simplicity here g is written instead of its restriction g|_{K_{η}}).

Now we start the proof of the main (2.4) theorem. Let us fix η and let τ be the following set of (multi)singularities τ = {ζ|ζ ≯ η}. For a τ-map f : N ⟶ P the subset η(f) is a submanifold, so its cohomology class [η(f)] (the Poincaré dual of the fundamental homology class it carries) is defined. The minimum that we expect from the Thom polynomial associated to η is

T p(η)(c(f)) = [η(f)] ∈ H^{∗}(N)

for all τ-maps. Now let ζ ∈ τ different from η. If we restrict the above cohomological identity to η(f) and ζ(f) we get

\begin{aligned}T p(η)(c(ν_{f(η(f))} − ν_{η(f)})) & = & e(ν_{η(f)}) & ∈ H^{∗}(η(f)) & and \\ T p(η)(c(ν_{f(ζ(f))} − ν_{ζ(f)})) & = & 0 & ∈ H^{∗}(ζ(f)). \end{aligned} (1)

Here we implicitly identified η(f) with its f-image and used some standard facts from differential topology, such as

• c(f) = c(f^{∗}T P − T N) restricted to a submanifold M ⊂ N for which f|_{M} is a diffeomorphism onto f(M) is

\frac{c(T M ⊕ ν_{f(M)})}{c(T M ⊕ ν_{M})} = \frac{c(ν_{f(M)})}{c(ν_{M})} = c(ν_{f(M)} − ν_{M}).

• The cohomology class of a submanifold restricted to the submanifold itself is the Euler class of its normal bundle.

• The cohomology class of a submanifold M restricted to a subset which is disjoint from M is 0.

Now we use the result of the generalized Pontryagin-Thom construction: the map f : N ⟶ P induces the maps g : P ⟶ Xτ, h : N ⟶ Y τ. By abuse of notation let us denote the restriction of these maps to some subsets (e.g. f(η(f)), η(f), f(ζ(f)) and ζ(f)) by the same letter. We have that

ν_{f(η(f))} = g^{∗}ξ_{η},  ν_{η(f)} = h^{∗}ξ̄_{η},  ν_{f(ζ(f))} = g^{∗}ξ_{ζ},  ν_{η(f)} = h^{∗}ξ̄_{ζ},

and therefore formula (1) (after appropriate identifications) yields

T p(η)(h^{∗}c(ξ̄_{η} − ξ_{η})) = e(h^{∗}ξ̄_{η}) T p(η)(h^{∗}c(ξ̄_{ζ} − ξ_{ζ})) = 0.

Keeping our eyes on the definition of c(η), c(ζ), e(η), we have

T p(η)(h^{∗}c(η)) = h^{∗}e(η) T p(η)(h^{∗}c(ζ)) = 0,

that is

h^{∗}( T p(η)(c(η)) − e(η) ) = 0 h^{∗}( T p(η)(c(ζ)) ) = 0.

Since this holds for all τ-maps f (and the map h is induced from f) we have the same identities without h^{∗}. This proofs the theorem.

4. Symmetries of singularities

To effectively use theorem 2.4 we need to study the maximal compact subgroup of some singularities. In this section we will use the following notations: ρ_{i} will mean the standard idimensional representation of the unitary group U(i). The powers of representations are meant tensor powers. The following theorem is essentially from [W] (see also [R1]).

Theorem 4.1. Let η be a singularity whose prototype is κ : (ℂ^{n}, 0) ⟶ (ℂ^{n+k}, 0). The germ κ is the miniversal unfolding of another germ β : (ℂ^{m}, 0) ⟶ (ℂ^{m+k}, 0) with dβ = 0. The group G_{η} is a subgroup of the maximal compact subgroup of the algebraic automorphism group of the local algebra Q_{η} times U(k − D), where D is the defect of Q_{η}. ^{1} With β well chosen G_{η} acts as a right-left symmetry group on β with representations μ_{1} and μ_{2}. The representations λ_{1}, λ_{2} are equal to μ_{1} ⊕ μ_{V}, μ_{2} ⊕ μ_{V} respectively, where μ_{V} is the representation of G_{η} on the unfolding space V ≅ ℂ^{n−m} with the formula

\begin{aligned}(ϕ, ψ) · α := ψ ∘ α ∘ ϕ^{−1} & for & α ∈ V ≤ Germs((ℂ^{m}, 0) ⟶ (ℂ^{m+k}, 0)) \\ & and & (ϕ, ψ) ∈ G_{η} ≤ Aut β ≤ Diff(ℂ^{m}, 0) × Diff(ℂ^{m+k}, 0). \end{aligned}

Remark 4.2. Observe that the formulas given for λ_{1} and λ_{2} imply that one does not have to determine μ_{V} (so does not have to work with κ, but only with β) when wants to compute c(η). Observe also that β’s source and target dimensions are usually much smaller than those of κ.

Let us see these notions on examples. Consider first the case k = 0, i.e. singularities between equal dimensional manifolds. Let η be the set of stable singularities of type A_{i} (i > 0). Its prototype κ maps from (ℂ^{i}, 0) to (ℂ^{i}, 0) and is the miniversal unfolding of the map β : (ℂ, 0) ⟶ (ℂ, 0), x ↦ x^{i+1} (dβ = 0). The maximal compact subgroup of the automorphism group of the algebra ℂ[[x]]/(x^{i+1}) is U(1) (defect= 0), so G_{A_{i}} = U(1) and it acts on β (as rightleft symmetry group) with the representations ρ_{1}, ρ_{1}^{i+1}. An appropriately chosen unfolding space V is generated by the germs g_{j} : x ↦ x^{j} for j = 1, 2, …, i − 1. The formula in the theorem implies that G_{η} = U(1) acts on the line spanned by g_{j} with the representation ρ_{1}^{−j+i+1}. Therefore we have proved the following theorem.

Theorem 4.3. For k = 0

G_{A_{i}} = U(1),  λ_{1}(A_{i}) = ⨁_{j=1}^{i} ρ_{1}^{j},  λ_{2}(A_{i}) = ⨁_{j=2}^{i+1} ρ_{1}^{j}.

Now we consider the case η = I_{a,b} where a, b ≥ 2, a ≠ b. Its prototype maps from (ℂ^{a+b}, 0) to (ℂ^{a+b}, 0) and is the miniversal unfolding of β : (ℂ^{2}, 0) ⟶ (ℂ^{2}, 0), (x, y) ↦ (xy, x^{a} + y^{b}) (dβ = 0). The maximal compact subgroup of Aut(ℂ[[x, y]]/(xy, x^{a} + y^{b})) is U(1) and it acts on β (as right-left symmetry group) with the representations

ρ1^{\frac{b}{d}} ⊕ ρ1^{\frac{a}{d}} , ρ1^{\frac{a+b}{d}} ⊕ ρ1^{\frac{ab}{d}} ,

1The defect of an algebra is the integer D such that if we present the algebra as a quotient of a formal power series algebra with minimal number of relations then D = #{relations} − #{generators}.

where d is the greatest common divisor of a and b. An appropriately chosen unfolding space V is generated by the germs f_{j} : (x, y) ↦ (0, x^{j}) for j = 1, 2, …, a − 1 and g_{j} : (x, y) ↦ (0, y^{j}) for j = 1, 2, …, b−1. The formula in the theorem implies that G_{η} = U(1) acts on the line spanned by f_{j} with the representation ρ_{1}^{b(a−j)/d} and on the line spanned by g_{j} with the representation ρ_{1}^{a(b−j)/d}. Therefore we have proved the following theorem.

Theorem 4.4. For η = I_{a,b} (a ≠ b) between equal dimensional manifolds G_{η} = U(1), λ_{1}(η) = μ_{1} ⊕ μ_{V}, λ_{2}(η) = μ_{2} ⊕ μ_{V}, where

μ_{1} = ρ1^{\frac{b}{d}} ⊕ ρ1^{\frac{a}{d}} ,  μ_{2} = ρ1^{\frac{a+b}{d}} ⊕ ρ1^{\frac{ab}{d}} ,  μ_{V} = ⨁_{j=1}^{a−1} ρ_{1}^{b(a−j)/d} ⊕ ⨁_{j=1}^{b−1} ρ_{1}^{a(b−j)/d}.

The following theorems use the same argument, so we give only the results.

Theorem 4.5. For η = I_{a,a}, a > 2 between equal dimensional manifolds the group G_{η} has an index 2 subgroup G′_{η} = U(1). The formulas for the representations restricted to this subgroup are the same as in theorem 4.4 (with b, d = a).

Theorem 4.6. For η = I_{2,2} between equal dimensional manifolds the group G_{η} has an index 2 subgroup G′_{η} = U(1) × U(1). The representations restricted to this subgroup are λ_{1} = μ_{1} ⊕ μ_{V}, λ_{2} = μ_{2} ⊕ μ_{V}, where

μ_{1} = ρ_{1(1)} ⊕ ρ_{1(2)},  μ_{2} = ρ_{1(1)}^{2} ⊕ ρ_{1(2)}^{2},  μ_{V} = ρ_{1(1)}^{2} ⊗ ρ_{1(2)}^{−1} ⊕ ρ_{1(2)}^{2} ⊗ ρ_{1(1)}^{−1}.

(The number i in the bracket refers to a representation of the i’th factor.)

Now we are ready to compute the total Chern classes and Euler classes associated to the singularities mentioned.

Corollary 4.7. For k = 0

c(A_{i})  = \frac{1+(i+1)x}{1+x} = 1 + ix − ix^{2} + ix^{3} − …  ∈ ℤ[[x]]
e(A_{i})  = i! · x^{i}  ∈ ℤ[x]
c′(I_{2,2})  = \frac{(1+2x)(1+2y)}{(1+x)(1+y)} =
1 + (x + y) + (−x^{2} + xy − y^{2}) + (x^{3} − x^{2}y − xy^{2} + y^{3}) + … ∈ ℤ[[x, y]]
e′(I_{2,2}) = xy(2x − y)(2y − x) ∈ ℤ[x, y]
c(I_{a,b})_{a≠b} = \frac{(1+ \frac{a+b}{d} ·x)(1+ \frac{ab}{d} ·x)}{(1+ \frac{b}{d} ·x)(1+ \frac{a}{d} ·x)}  ∈ ℤ[[x]]
e(I_{a,b})_{a≠b} = \frac{b}{d}x · \frac{a}{d}x · Π_{j=1}^{a−1}(\frac{b(a−j)}{d}x) · Π_{j=1}^{b−1}(\frac{a(b−j)}{d}x) = \frac{a!b!a^{b−1}b^{a−1}}{d^{a+b}} · x^{a+b} ∈ ℤ[x]
c′(I_{a,a})_{a>2} = \frac{(1+2x)(1+ax)}{(1+x)(1+x)}  ∈ ℤ[[x]]
e′(I_{a,a})_{a>2} = (a − 1)!^{2} · x^{2a}  ∈ ℤ[x]

The letters c′ and e′ instead of c, e mean that we used a subgroup G′_{η} (given above) instead of the whole G_{η}.

This method to compute maximal compact symmetry groups and thus to compute total Chern classes and Euler classes of singularities works for k > 0, too. As an example let us present without proof the result on Morin singularities A_{n} (n > 0) for any k ≥ 0.

Theorem 4.8. Let k ≥ 0. The prototype of A_{n} : (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0) is the miniversal unfolding of κ : (ℂ, 0) ⟶ (ℂ^{k+1}, 0), (x) ↦ (x^{n+1}, 0, …, 0). So G_{A_{n}} = U(1) × U(k) with

μ_{1} = ρ_{1},  μ_{2} = ρ_{1}^{n+1} ⊕ ρ_{k},  μ_{V} = ⊕_{j=2}^{n} ρ_{1}^{j} ⊕ ⊕_{j=1}^{n}(ρ_{k} ⊗ ρ_{1}^{−j}).

Hence

\begin{aligned}c(A_{n}) & = & \frac{1+(n+1)x}{1+x}(1 + y_{1} + y_{2} + … + y_{k}) & ∈ ℤ[[x, y_{1}, y_{2}, …, y_{k}]] \\ e(A_{n}) & = & n!x^{n}Π_{j=1}^{n}(y_{k} − jxy_{k−1} + j^{2}x^{2}y_{k−2} − … + (−1)^{k}j^{k}x^{k}) & ∈ ℤ[x, y_{1}, y_{2}, …, y_{k}]. \end{aligned}

5. Computation of Thom polynomials

In this section we will use theorem 2.4 and the computations in section 4 to compute some Thom polynomials. Let us start with the setting where the most activity took place in the past, i.e. singularities of maps between equal dimensional manifolds.

Theorem 5.1. The polynomials in this table are Thom polynomials associated to the given singularities between equal dimensional manifolds. By c = 1 + c_{1} + c_{2} + … we mean c(T P − f^{∗}T N).

codim 1

A_{1}  c_{1}  [T]

codim 2

A_{2}  c_{1}^{2} + c_{2}  [folklore]

codim 3

A_{3}  c_{1}^{3} + 3c_{1}c_{2} + 2c_{3}  [folklore]

codim 4

\begin{matrix}A_{4} & c_{1}^{4} + 6c_{1}^{2}c_{2} + 2c_{2}^{2} + 9c_{1}c_{3} + 6c_{4} \\ I_{2,2} & c_{2}^{2} − c_{1}c_{3} \end{matrix}  {[G] \atop [P1]}

codim 5

\begin{matrix}A_{5} & c_{1}^{5} + 10c_{1}^{3}c_{2} + 25c_{1}^{2}c_{3} + 10c_{1}c_{2}^{2} + 38c_{1}c_{4} + 12c_{2}c_{3} + 24c_{5} \\ I_{2,3} & 2c_{1}c_{2}^{2} − 2c_{1}^{2}c_{3} + 2c_{2}c_{3} − 2c_{1}c_{4} \end{matrix}  {[Tu] \atop [P2]}

codim 6

\begin{aligned}A_{6} & c_{1}^{6} + 15c_{1}^{4}c_{2} + 55c_{1}^{3}c_{3} + 30c_{1}^{2}c_{2}^{2} + 141c_{1}^{2}c_{4} + 79c_{1}c_{2}c_{3} + 5c_{2}^{3}+ \\ & 202c_{1}c_{5} + 55c_{2}c_{4} + 17c_{3}^{2} + 120c_{6} \\ I_{2,4} & 2c_{1}^{2}c_{2}^{2} + 3c_{2}^{3} − 2c_{1}^{3}c_{3} + 2c_{1}c_{2}c_{3} − 3c_{3}^{2} − 5c_{1}^{2}c_{4} + 9c_{2}c_{4} − 6c_{1}c_{5} \\ I_{3,3} & c_{1}^{2}c_{2}^{2} − c_{2}^{3} − c_{1}^{3}c_{3} + 3c_{1}c_{2}c_{3} + 3c_{3}^{2} − 2c_{1}^{2}c_{4} − 3c_{2}c_{4} \end{aligned}  [P2]

codim 7

A_{7}  c_{1}^{7} + 21c_{1}^{5}c_{2} + 105c_{1}^{4}c_{3} + 70c_{1}^{3}c_{2}^{2} + 399c_{1}^{3}c_{4} + 301c_{1}^{2}c_{2}c_{3} + 35c_{1}c_{2}^{3} + 960c_{1}^{2}c_{5}+
467c_{1}c_{2}c_{4} + 139c_{1}c_{3}^{2} + 58c_{2}^{2}c_{3} + 1284c_{1}c_{6} + 326c_{2}c_{5} + 154c_{3}c_{4} + 720c_{7}
I_{2,5}  2c_{1}^{3}c_{2}^{2} + 9c_{1}c_{2}^{3} − 2c_{1}^{4}c_{3} + 14c_{2}^{2}c_{3} − 17c_{1}c_{3}^{2} − 9c_{1}^{3}c_{4} + 29c_{1}c_{2}c_{4} − 10c_{3}c_{4}−
26c_{1}^{2}c_{5} + 34c_{2}c_{5} − 24c_{1}c_{6}
I_{3,4}  2c_{1}^{3}c_{2}^{2} − c_{1}c_{2}^{3} − 2c_{1}^{4}c_{3} + 8c_{1}^{2}c_{2}c_{3} − 2c_{2}^{2}c_{3} + 13c_{1}c_{3}^{2} − 7c_{1}^{3}c_{4} − 5c_{1}c_{2}c_{4} + 10c_{3}c_{4}−
6c_{1}^{2}c_{5} − 10c_{2}c_{5}
(x^{2}, y^{3})  2c_{1}c_{2}^{3} − 2c_{1}^{2}c_{2}c_{3} + 2c_{2}^{2}c_{3} + 2c_{1}c_{3}^{2} − 4c_{1}c_{2}c_{4} + 2c_{3}c_{4} − 2c_{2}c_{5}  [P2]

codim 8

A_{8}  c_{1}^{8} + 28c_{1}^{6}c_{2} + 140c_{1}^{4}c_{2}^{2} + 140c_{1}^{2}c_{2}^{3} + 14c_{2}^{4} + 182c_{1}^{5}c_{3} + 868c_{1}^{3}c_{2}c_{3} + 501c_{1}c_{2}^{2}c_{3}+
642c_{1}^{2}c_{3}^{2} + 202c_{2}c_{3}^{2} + 952c_{1}^{4}c_{4} + 2229c_{1}^{2}c_{2}c_{4} + 364c_{2}^{2}c_{4} + 1559c_{1}c_{3}c_{4} + 332c_{4}^{2}+
3383c_{1}^{3}c_{5} + 3455c_{1}c_{2}c_{5} + 954c_{3}c_{5} + 7552c_{1}^{2}c_{6} + 2314c_{2}c_{6} + 9468c_{1}c_{7} + 5040c_{8}
I_{2,6}  2c_{1}^{4}c_{2}^{2} + 18c_{1}^{2}c_{2}^{3} + 7c_{2}^{4} − 2c_{1}^{5}c_{3} − 4c_{1}^{3}c_{2}c_{3} + 57c_{1}c_{2}^{2}c_{3} − 53c_{1}^{2}c_{3}^{2} + 4c_{2}c_{3}^{2}−
14c_{1}^{4}c_{4} + 60c_{1}^{2}c_{2}c_{4} + 74c_{2}^{2}c_{4} − 90c_{1}c_{3}c_{4} − 3c_{4}^{2} − 71c_{1}^{3}c_{5} + 166c_{1}c_{2}c_{5}−
39c_{3}c_{5} − 154c_{1}^{2}c_{6} + 162c_{2}c_{6} − 120c_{1}c_{7}
I_{3,5}  2c_{1}^{4}c_{2}^{2} + 3c_{1}^{2}c_{2}^{3} − 4c_{2}^{4} − 2c_{1}^{5}c_{3} + 8c_{1}^{3}c_{2}c_{3} + 10c_{1}c_{2}^{2}c_{3} + 13c_{1}^{2}c_{3}^{2} + 24c_{2}c_{3}^{2}−
11c_{1}^{4}c_{4} + 7c_{1}^{2}c_{2}c_{4} − 32c_{2}^{2}c_{4} + 34c_{1}c_{3}c_{4} − 20c_{4}^{2} − 26c_{1}^{3}c_{5} − 2c_{1}c_{2}c_{5}+
60c_{3}c_{5} − 24c_{1}^{2}c_{6} − 40c_{2}c_{6}
I_{4,4}  c_{1}^{4}c_{2}^{2} − c_{1}^{2}c_{2}^{3} + 2c_{2}^{4} − c_{1}^{5}c_{3} + 6c_{1}^{3}c_{2}c_{3} − 5c_{1}c_{2}^{2}c_{3} + 18c_{1}^{2}c_{3}^{2} − 11c_{2}c_{3}^{2} − 5c_{1}^{4}c_{4}−
9c_{1}^{2}c_{2}c_{4} + 13c_{2}^{2}c_{4} + 27c_{1}c_{3}c_{4} + 23c_{4}^{2} − 6c_{1}^{3}c_{5} − 29c_{1}c_{2}c_{5} − 21c_{3}c_{5} − 2c_{2}c_{6}
(x^{2} + y^{3}, xy^{2}) 2c_{1}^{2}c_{2}^{3} + c_{2}^{4} − 2c_{1}^{3}c_{2}c_{3} + 4c_{1}c_{2}^{2}c_{3} + 2c_{1}^{2}c_{3}^{2} + 2c_{2}c_{3}^{2} − 7c_{1}^{2}c_{2}c_{4} + 2c_{2}^{2}c_{4}+
5c_{1}c_{3}c_{4} + c_{4}^{2} − 9c_{1}c_{2}c_{5} + 3c_{3}c_{5} − 4c_{2}c_{6}

Proof. Consider the following table of singularities up to codimension 8 between equal dimensional manifolds, together with their Thom-Boardman class (although it it not important here, we count codimension in the source), see e.g. [WP]:

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

Let η be one of these singularities. If codim ζ ≤ codim η and ζ ≠ η then ζ ≯ η (proposition 2.3) and therefore T p(η)(c^{(′)}(ζ)) = 0 which gives a restriction on the possible values of the

coefficients in T p(η). (The sign ^{(′)} means that c is meant with or without the prime according to which is computed in section 4.) In fact if G_{ζ}^{(′)} = U(1) and hence its cohomology ring is a polynomial ring in one variable then this restriction is a linear equation. If G_{ζ}^{(′)} is U(1) × U(1) then it gives a number of linear equations. Anyway, considering all these restrictions we get a system of linear equations on the coefficients of T p(η). We can add the additional linear equation(s) coming from T p(η)(c^{(′)}(η)) = e^{(′)}(η). These all turn to be uniquely solvable in the cases given in the theorem. The actual computations were carried out by the computer algebra package called Maple. A detailed description of the calculation for T p(A_{4}) can be found in [R2].

Carrying out the computations mentioned, one can see that although in some cases there are some redundancies in the system of linear equations but typically the equations are “quite independent”. That is e.g. to compute T p(A_{8}) it is not enough to consider only e.g. A_{1}–A_{8} for

10 RICHÁRD RIMÁNYI

ζ, or those and I_{a,b} (a+b ≤ 8), one has to consider all singularities with codimension at most 8. Thus our method heavily depends on the classification of singularities. Since this classification is known much further than codim 8 (see [WP]), the method is probably applicable to compute further Thom polynomials. Now we are presenting a result which gives a partial information on the Thom polynomials of all Morin singularities.

Theorem 5.2. k = 0. Let π(r) denote the set of partitions of r and π(r; i) the set of partition in exactly i terms; let σ_{j} be the j’th elementary symmetric polynomial (σ_{0} = 1).

• If T p(A_{r}) = ∑_{I∈π(r)} α_{I}c_{I}, then

∑_{I∈π(r;i)} α_{I} = σ_{r−i}(1, 2, …, r − 1) for all i = 1, 2, … r.

• Let η be a codimension r singularity different from A_{r}. If T p(η) = ∑_{I∈π(r)} α_{I}c_{I}, then

∑_{I∈π(r;i)} α_{I} = 0 for all i = 1, 2, … r.

Proof . For both A_{r} and η we have A_{1}, A_{2}, …, A_{r−1} ≯ A_{r}, η. So theorem 2.4 gives r − 1 linear equations for the coefficients both of T p(A_{r}) and T p(η). We can add one more equation: T p(A_{r})(c(A_{r})) = e(A_{r}) in case of A_{r} and T p(η)(c(A_{r})) = 0 in case of η. So in both case we have r linear equations on the coefficients of the relevant Thom polynomial, whose general solution is given in the theorem.

Remark 5.3. Theorem 5.2 is sufficient to determine T p(A_{1}), T p(A_{2}), T p(A_{3}) completely. From A_{4} on, however, it is not enough, e.g. for A_{4} it only gives:

c_{1}^{4} + 6c_{1}^{2}c_{2} + Ac_{2}^{2} + Bc_{1}c_{3} + 6c_{4}, where A + B = 11,

cf. remark 6.4.

Corollary 5.4. • In the polynomial T p(A_{r}) the coefficient of c_{1}^{r} is 1, that of c_{r} is (r − 1)!,

that of c_{1}^{r−2}c_{2} is r(r − 1)/2 (r ≥ 2).

• In the polynomial T p(η) (η ≠ A_{r} is of codimension r) the coefficients of the terms c_{1}^{r}, c_{r}, c_{1}^{r−2}c_{2} are all 0.

Theorem 5.2 assures that there must be negative coefficients in T p(η) if η ≠ A_{r} (unless T p(η) = 0, is it possible?). Since the converse is true for A_{r} till it is computed (and not only for k = 0) let put it as a conjecture.

Conjecture 5.5. All the non-zero coefficients of T p(A_{r}) (for any k ≥ 0) are positive.

Since our method works for k > 0, too, let us give a few results in this setting.

Theorem 5.6. The following are Thom polynomials for k = 1

A_{1}  c_{2}  [T], [P1]
A_{2}  c_{1}c_{3} + c_{2}^{2} + 2c_{4}  [Ro1]
A_{3}  2c_{1}^{2}c_{4} + 3c_{1}c_{2}c_{3} + 10c_{1}c_{5} + c_{2}^{3} + 7c_{2}c_{4} + c_{3}^{2} + 12c_{6}
A_{4}  c_{2}^{4} + 6c_{1}c_{2}^{2}c_{3} + 2c_{1}^{2}c_{3}^{2} + 4c_{2}c_{3}^{2} + 9c_{1}^{2}c_{2}c_{4} + 16c_{2}^{2}c_{4} + 17c_{1}c_{3}c_{4}+

+11c_{4}^{2} + 6c_{1}^{3}c_{5} + 53c_{1}c_{2}c_{5} + 21c_{3}c_{5} + 54c_{1}^{2}c_{6} + 76c_{2}c_{6} + 156c_{1}c_{7} + 144c_{8}

III_{2,2} c_{3}^{2} − c_{2}c_{4}  [P1]
I_{2,2} c_{1}c_{3}^{2} − c_{1}c_{2}c_{4} + 2c_{3}c_{4} − 2c_{2}c_{5}
III_{2,3} 2c_{2}c_{3}^{2} − 2c_{2}^{2}c_{4} + 2c_{1}c_{3}c_{4} − 2c_{1}c_{2}c_{5} − 4c_{2}c_{6} + 6c_{3}c_{5} − 2c_{4}^{2}

Proof . The proof is essentially the same as that of theorem 5.1 — with the only difference that here we have to consider the hierarchy basically implied by the following table

codim_{ℂ} Σ^{0} Σ^{1} Σ^{2}
0  A_{0}

1

2  A_{1}

3

4  A_{2}

5

6  A_{3} III_{2,2}
7  I_{2,2}
8  A_{4} III_{2,3}

and of course, we have to consider the symmetries of these singularities (some of them is given in theorem 4.8).

Carrying out these kind of calculations for singularities of codim > 8 (k = 1) or for singularities with a concrete k > 1 seems to be a question of computer capacity. However, another challenge is to find Thom polynomials consisting k as a parameter. The easiest is the computation of T p(A_{1}) for any k ≥ 0, which was the first Thom polynomial computed, and is very easy with the former methods ([T], [P1]). Still let us show how it is proved with our techniques.

Theorem 5.7. T p(A_{1}) = c_{k+1}.

Proof . According to the main theorem T p(A_{1}) = e(A_{1}), hence according to theorem 4.8, we know that substituting c_{1} = −a + b_{1}, c_{2} = −a^{2} − ab_{1} + b_{2}, …, c_{k} = −a^{k} − a^{k−1}b_{1} … + b_{k}, c_{k+1} = a^{k+1} − a^{k}b_{1} − … − ab_{k} into the polynomial T p(A_{1}) we obtain −a(a^{k} + a^{k−1}b_{1} + … + b_{k}). Observe that this latter is c_{k+1} and that c_{1}, c_{2}, …, c_{k}, c_{k+1} are polynomially independent in ℤ[a, b_{1}, …, b_{k}], so T p(A_{1}) must be c_{k+1}.

A bit more delicate but still managable calculation gives the result for A_{2} for any k ≥ 0: the linear equations on the coefficients of T p(A_{2}) implied by T p(A_{2})(c(A_{1})) = 0 and T p(A_{2})(c(A_{2})) = e(A_{2}) have the unique solution

T p(A_{2}) = c_{k+1}^{2} + ∑_{j=0}^{k} 2^{k−j}c_{2k+2−j}c_{j}.

Since this result has already been known by Ronga [Ro1] and because our proof is pure linear algebra, we omit the proof.

12 RICHÁRD RIMÁNYI

The author was not able to similarly organize the solution of the linear equations implied by

T p(A_{3})(c(A_{1})) = 0, T p(A_{3})(c(A_{2})) = 0, T p(A_{3})(c(A_{3})) = e(A_{3}), T p(A_{3})(c(III_{2,2})) = 0

to obtain a general formula for T p(A_{3}) — although it can easily be done for concrete k’s.

6. Existence

In this section we are discussing the existence of Thom polynomials in terms of our techniques. This will be different from the one in the literature, see e.g. [AVGL] p. 187.

Let η : (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0) be a singularity of codimension d and let τ be the set of multisingularities τ = {ζ | ζ ≯ η}. Consider them fixed until the final remark of this section.

Let p(c_{1}, c_{2}, …, c_{d}) be a weighted homogeneous degree 2d polynomial (the weight of c_{i} is 2i). Our goal is to analyze the logical connections between the following four statements.

(A) For almost all maps f : N^{n} ⟶ P^{n+k} the cohomology class [η(f)] ∈ H^{2d}(N) is
equal to p(c_{1}(f), c_{2}(f), …, c_{d}(f)), i.e. p is the Thom polynomial associated to η.
(B) For all τ-maps f : N^{n} ⟶ P^{n+k} the cohomology class [η(f)] ∈ H^{2d}(N)
is equal to p(c_{1}(f), c_{2}(f), …, c_{d}(f)).
(C) [K̄_{η}] = p(c_{1}(fτ), c_{2}(fτ), …, c_{d}(fτ)) ∈ H^{2d}(Y τ).
(D) 0 = p(c_{1}(ζ), c_{2}(ζ), …, c_{d}(ζ)) ∈ H^{2d}(K̄_{ζ}) for all ζ ∈ τ, ζ ≠ η and
e(η) = p(c_{1}(η), c_{2}(η), …, c_{d}(η)) ∈ H^{2d}(K̄_{η}).

The first thing we have to observe is that in (C) [K̄_{η}] and the c_{i}(fτ)’s are not defined (yet), since the spaces Y τ, Xτ are not manifolds (even less finite dimensional ones). So we first define these notions. Let [K̄_{η}] be the image of the Thom class of ξ̄_{η} under the composition

H^{2d}(ξ̄_{η}, ξ̄_{η} − K̄_{η}) ≅^{excision} H^{2d}(Y τ, Y τ − K̄_{η}) ⟶^{restriction}H^{2d}(Y τ).

(Here we identified vector bundles with their total spaces.) To justify this definition recall that in (B) [η(f)] could be defined similarly as the image of the Thom class of η(f)’s normal bundle under the composition

H^{2d}(ν_{η(f)}, ν_{η(f)} − η(f)) ≅^{excision} H^{2d}(N, N − η(f)) ⟶^{restristion}H^{2d}(N).

The definition of c_{i}(fτ) is a bit more delicate and uses a conjecture which holds for all singularities mentioned in the previous sections.

Conjecture 6.1. Let ζ be a singularity. The multiplication by e(ζ) in H^{∗}(BG_{ζ}) is a monomorphism.

In fact, in all the singularities occurred above the ring H^{∗}(BG_{ζ}) is a polynomial ring and the element e(ζ) is different from 0. (Let us remark that unfortunately e(ζ) = 0 quite often over the reals.) Now we define c_{i}(fτ) by recursion on τ. Let c_{i}(f{A_{0}}) be the i’th universal Chern class in H^{2i}(Y {A_{0}}) = H^{2i}(BG_{A_{0}}) = H^{2i}(BU(k)). Now suppose we have already defined c_{i}(fτ′) and want to define c_{i}(fτ) where τ = τ′ ∪ {ζ}. According to conjecture 6.1 map 4 (and so map 3) in the Gysin sequence

H^{2i−1}(Dξ̄_{ζ}) ⟶^{1} H^{2i−1}(Sξ̄_{ζ}) ⟶^{2} H^{2i}(Dξ̄_{ζ}, Sξ̄_{ζ}) ⟶^{3} H^{2i}(Dξ̄_{ζ})
||  ||
H^{2i−2codim ζ}(BG_{ζ}) ⟶^{4} H^{2i}(BG_{ζ})

is monomorphism, so map 2 is 0-map, and therefore map 1 is surjective. Then map 5 in the exact sequence

H^{2i−1}(Y τ′) ⊕ H^{2i−1}(Dξ̄_{ζ}) ⟶^{5} H^{2i−1}(Sξ̄_{ζ}) ⟶^{6} H^{2i}(Y τ) ⟶^{7} H^{2i}(Y τ′) ⊕ H^{2i}(Dξ̄_{ζ})

is also surjective and therefore map 7 is injective. So we can define c_{i}(fτ) as the unique inverse image of (c_{i}(fτ′), c_{i}(ζ)) (which maps to 0 in H^{2i}(Sξ̄_{ζ})) at map 7.

Remark 6.2. There is another way to define c_{i}(fτ) — suggested to the author by A. Szűcs, as follows. One can identify the map fτ : Y τ ⟶ Xτ with an ideal end-object of the category of all τ-maps, where the morphisms are the pull-back diagrams defined in [RSz]. Under such a pull-back morphism Chern classes of τ-maps are mapped to each other. So the Chern classes of the “ideal end-object” can be defined as their direct limit. Of course, some more work is needed to make this definition precise.

Now we study the properties (A) – (D). Clearly (A) ⇒ (B), since τ-maps must be among ‘most maps’. The definition of c_{i}(fτ) and [K̄_{η}] assures that (B) ⇔ (C) holds. What is proved in the proof of the main theorem is that (C) ⇒ (D) since (D) is in fact the restriction of (C) to the K̄_{ζ}’s (ζ ∈ τ). Now we prove that provided conjecture 6.1 holds then (D) ⇒ (C). This will be done by a Mayer-Vietoris type argument. We will use induction: first observe that (C) restricted to Y {A_{0}} holds, this is (D) for ζ = A_{0}. Now we will prove that if (C) holds restricted to Y τ′ where τ′ ⊂ τ then it holds restricted to Y τ′′ where τ′′ = τ ∪ {ζ} for a ζ ∈ τ \ τ′. That is we want to show that the element [K̄_{η}] − p(c_{1}(fτ), c_{2}(fτ), …, c_{d}(fτ)) in H^{d}(Y τ) is 0. Our hypotheses and condition (C) for ζ says that this element is mapped to (0, 0) at map 10 in the exact sequence

H^{2d−1}(Y τ′) ⊕ H^{2d−1}(Dξ̄_{ζ}) ⟶^{8} H^{2d−1}(Sξ̄_{ζ}) ⟶^{9} H^{2d}(Y τ) ⟶^{10} H^{2d}(Y τ′) ⊕ H^{2d}(Dξ̄_{ζ}).

Just like in the argument above it is clear that map 8 is surjective (provided conjecture 6.1 holds) and therefore map 10 is injective. This proves our claim.

Conclusions. On one hand we have the partial converse of the main theorem, that is the implication (D) ⇒ (B). On the other hand, if there are polynomials satisfying (D), then the implication (D) ⇒ (B) can be interpreted as a proof of the existence of Thom polynomials for τ-maps. If the polynomial satisfying (D) is unique and we know the existence of a polynomial satisfying (A) then clearly the one satisfying (D) is the Thom polynomial. Since this is the case in all cases studied by the author let us put it as a conjecture.

Conjecture 6.3. Let η be singularity. The polynomial satisfying the conditions of theorem 2.4 is unique.

Remark 6.4. In fact, one could go further and define the set of Thom polynomials T p(η; τ) of a singularity η modulo a set τ (satisfying that η ∈ τ and a closeness assumption: ζ < η ⇒ ζ ∈ τ). One can e.g. ask for the Thom polynomials of A_{n} considering only Morin-maps, i.e. τ-maps for τ_{Morin} = ∪_{m=0}^{∞} ∪_{i=0}^{m}(a_{0}A_{0} + a_{1}A_{1} + … + a_{m}A_{m}). The arguments in this section prove that the set T p(η; τ) coincides with the set of polynomials satisfying condition (D) for ζ ∈ τ. E.g. computations show that

T p(A_{4}; τ_{Morin}) = {c_{1}^{4} + 6c_{1}^{2}c_{2} + Ac_{2}^{2} + Bc_{1}c_{3} + 6c_{4} | A + B = 11},

or even more generally, the polynomials in T p(A_{n}; τ_{Morin}) are exactly the polynomials satisfying the equations of the first part of theorem 5.2.

7. Multiple points

The “universal map method” which we used to compute Thom polynomials is capable to determine multiple point formulas (see e.g. [K], [Sz3], [Sz4]), too — this will be shown in a subsequent paper. In this section we just show how to use our method to (re)prove the most basic multiple point formula (of Herbert [H] and Ronga [Ro1]) with rational coefficients. Proving it with integer coefficients needs finer cohomological analysis. In this section all cohomologies are meant with rational coefficients.

Definition 7.1. For a self-transverse immersion f : N^{n} ↬ P^{n+k} let Δ̄_{r} ⊂ N denote the (possibly open) submanifold of r-tuple points of f, and let Δ_{r} be its f-image.

Theorem 7.2. (Herbert, Ronga) For any self-transverse immersion f : N ↬ P and any r ≥ 0

[closure Δ̄_{r}] = f^{∗}[closure Δ_{r−1}] − e(f) · [closure Δ̄_{r−1}] ∈ H^{2(r−1)k}(N)

where e(f) is the top Chern class (top Stiefel-Whitney class in the real case) of the map – or what is the same, of the normal bundle of the immersion.

Proof . Let k be fixed and let τ be the set of the multisingularities of self-transverse immersions, i.e. τ = {A_{0}, 2A_{0}, 3A_{0}, … }. Associated to this τ we can construct the universal τ-map fτ : Y τ ⟶ Xτ, i.e. the universal self-transverse immersion, which we now describe.

Both Y τ and Xτ are glued together from blocks corresponding to the elements of τ. The block corresponding to sA_{0} in Y τ is the disc bundle of a vector bundle ξ̄_{sA_{0}} over the bases space K̄_{sA_{0}}, where

K̄_{sA_{0}} = B[U(k) × (U(k)^{s−1} ⋊ S_{s−1})]  ξ̄_{sA_{0}} = (γ_{2}^{k} × … × γ_{s}^{k}) ⋊ S_{s−1}.

Here S_{i} means the symmetric group on i elements and its action on the other factor is permutation. The bundle γ_{i}^{k} is the universal k-bundle over the i’th U(k) factor.

The block corresponding to sA_{0} in Xτ is the disc bundle of a vector bundle ξ_{sA_{0}} over the base space K_{sA_{0}}, where

K_{sA_{0}} = B[U(k)^{s} ⋊ S_{s}] ξ_{sA_{0}} = (γ_{1}^{k} × … γ_{s}^{k}) ⋊ S_{s}.

The pull back of ξ_{sA_{0}} by the s-sheeted covering (fτ)|K̄_{sA_{0}} is

(fτ)|∗ ; K̄_{sA_{0}}(ξ_{sA_{0}}) = γ_{1}^{k} × [(γ_{2}^{k} × … × γ_{s}^{k}) ⋊ S_{s−1}].

To prove the formula of the theorem we only have to prove it for fτ, that is

[closure K̄_{rA_{0}}] = (fτ)^{∗}[closure K_{(r−1)A_{0}}] − e(fτ) · [closure K̄_{(r−1)A_{0}}] ∈ H^{2(r−1)k}(Y τ). (2)

Our strategy is to prove this first restricted to K̄_{sA_{0}} for all s = 1, 2, … (I) and then use a Mayer-Vietoris argument to prove it in Y τ (II).

(I) We need to determine the restrictions of [closure K̄_{rA_{0}}], (fτ)^{∗}[closure K_{rA_{0}}] and e(fτ) to H^{∗}(K̄_{sA_{0}}). Since this latter cohomology ring is not easy to determine, we go further and pull back the mentioned classes by the (s − 1)!-sheeted covering ι : B[U(k)^{s}] ⟶ K̄_{sA_{0}} and compute them in the polynomial ring

H^{∗}(B[U(k)^{s}]) = ℤ[v_{1}^{(1)}, …, v_{k}^{(1)}, v_{1}^{(2)}, …, v_{k}^{(2)}, …, v_{1}^{(s)}, …, v_{k}^{(s)}].

Here v_{i}^{(j)} is a generator in dimension i, the upper index is just a distinguishing sign.

Lemma 7.3.

(i)  ι^{∗} is injective
(ii) ι^{∗} [closure K̄_{rA_{0}}]|K̄_{sA_{0}}  = σ_{r−1}(v_{k}^{(2)}, …, v_{k}^{(s)})
(iii) ι^{∗} (fτ)^{∗}[closure K_{rA_{0}}]|K̄_{sA_{0}} = σ_{r}(v_{k}^{(1)}, …, v_{k}^{(s)})
(iv) ι^{∗} e(fτ)|K̄_{sA_{0}}  = v_{k}^{(1)}

Here, again, σ_{i} is the i’th elementary symmetric polynomial, with the convention that σ_{0}( anything ) = 1 and σ_{i}( less than i elements ) = 0.

Proof of the lemma. Statement (i) is clear since ι is a covering, and in the coefficient group (rationals) one can divide by any integer. Statement (iv) comes from the above description of fτ as follows:

ι^{∗}c(fτ)|K̄_{sA_{0}} = \frac{(1 + v_{1}^{(1)} + … + v_{k}^{(1)}) · . . . · (1 + v_{1}^{(s)} + … + v_{k}^{(s)})}{(1 + v_{1}^{(2)} + … + v_{k}^{(2)}) · . . . · (1 + v_{1}^{(s)} + … + v_{k}^{(s)})} = 1 + v_{1}^{(1)} + … + v_{k}^{(1)},

so ι^{∗}e(fτ)|K̄_{sA_{0}} = v_{k}^{(1)}.

Now let us turn to the computation of ι^{∗}[closure K̄_{rA_{0}}]|K̄_{sA_{0}} for r ≤ s, since the r > s case is trivial (indeed, for r > s the subspaces closure K̄_{rA_{0}} and K̄_{sA_{0}} are disjoint). The reader here is advised to picturize first a special case, e.g. k = 1, r = 3, s = 4. Recall that the neighbourhood of K̄_{sA_{0}} in Y τ is the total space of the bundle (γ_{2}^{k} × … × γ_{s}^{k}) ⋊ S_{s−1}. Pulling back this bundle by the map ι we get the bundle γ_{2}^{k} × … × γ_{s}^{k}. In this bundle we can identify the inverse images L_{i} := ι^{−1}(K̄_{iA_{0}}) as follows.

L_{s} = 0-section
L_{s−1} = γ_{2}^{k} ∪ γ_{3}^{k} ∪ … ∪ γ_{s}^{k}  \ L_{s}
L_{s−2} = span(γ_{2}^{k}, γ_{3}^{k}) ∪ span(γ_{2}^{k}, γ_{4}^{k}) ∪ … ∪ span(γ_{s−1}^{k}, γ_{s}^{k}) \ closure L_{s−1}
…
L_{1} = span(γ_{2}^{k}, γ_{3}^{k}, …, γ_{s}^{k})  \ closure L_{2}

So the closure of L_{r} has \binom{s−1}{s−r} branches through L_{s}. These all considered we have that

ι^{∗}[closureK̄_{rA_{0}}]|K̄_{sA_{0}} = [closure L_{r}]|_{L_{s}} = e(ν(closure L_{r} ⊂ closure L_{1}))|_{L_{s}} =
= e(ν(closure L_{r} ⊂ closure L_{1})|_{L_{s}}) = e( ⨁(γ_{i_{1}}^{k} × … × γ_{i_{r−1}}^{k})) =
= ∑ v_{k}^{(i_{1})} · . . . · v_{k}^{(i_{r−1})} = σ_{r−1}(v_{k}^{(2)}, …, v_{k}^{(s)}).

(The ⨁ and the ∑ is taken for all the r − 1-element subsets {i_{1}, …, i_{r−1}} of {2, 3, …, s}. The proof of (iii) is similar.

To complete the proof of part (I) we only have to substitute the cohomology classes computed in the lemma (keeping in mind that ι^{∗} is injective), i.e. we have to check the equation

σ_{r−1}(v_{k}^{(2)}, …, v_{k}^{(s)}) = σ_{r−1}(v_{k}^{(1)}, …, v_{k}^{(s)}) − v_{k}^{(1)} · σ_{r−2}(v_{k}^{(2)}, …, v_{k}^{(s)})

in ℤ[v_{1}^{(1)}, …, v_{k}^{(1)}, …, v_{1}^{(s)}, …, v_{k}^{(s)}]. This is an identity, so part (I) is proved.

(II) Now we are going to use a Mayer-Vietoris type argument to prove formula (2) in H^{2(r−1)k}(Y τ). This will be done by induction. The formula clearly holds restricted to the first block, i.e. restricted to Y τ_{1}, where τ_{1} = {A_{0}}, since this is the result of part (I) for s = 1. Now we are going to prove that if the formula holds for Y τ_{s}, τ_{s} = {A_{0}, 2A_{0}, …, sA_{0}} then it must hold for Y τ_{s+1}, too. Consider the following portion of the Mayer-Vietoris sequence:

H^{2(r−1)k−1}(Y τ_{s}) ⊕ H^{2(r−1)k−1}(Dξ̄_{(s+1)A_{0}})⟶^{1}H^{2(r−1)k−1}(Y τ_{s} ∩ Dξ̄_{(s+1)A_{0}})⟶^{2}

H^{2(r−1)k}(Y τ_{s} ∪ Dξ̄_{(s+1)A_{0}})⟶^{3}H^{2(r−1)k}(Y τ_{s}) ⊕ H^{2(r−1)k}(Dξ̄_{(s+1)A_{0}}).

According to the induction hypothesis the cohomology class

( [closure K̄_{rA_{0}}] − (fτ)^{∗}[closure K_{(r−1)A_{0}}] + e(fτ) · [closure K̄_{(r−1)A_{0}}] )|_{Y τ_{s+1}}

in H^{2(r−1)k}(Y τ_{s} ∪ Dξ̄_{(s+1)A_{0}}) = H^{2(r−1)k}(Y τ_{s+1}) maps to (0, 0) at map 3. Now we prove that map 2 is the 0-map, which then shows our claim. To show this we will prove that map 1 is surjective, in fact already the map H^{2(r−1)k−1}(Dξ̄_{(s+1)A_{0}}) ⟶ H^{2(r−1)k−1}(Y τ_{s} ∩ Dξ̄_{(s+1)A_{0}}) is surjective. Observe that this latter cohomology group is equal to H^{2(r−1)k−1}(Sξ̄_{(s+1)A_{0}}) (where S means “sphere bundle”), and the map is contained in the Gysin sequence (as map 4)

H^{2(r−1)k−1}(Dξ̄_{(s+1)A_{0}})⟶^{4}H^{2(r−1)k−1}(Sξ̄_{(s+1)A_{0}})⟶^{5}

H^{2(r−1)k}(Dξ̄_{(s+1)A_{0}}, Sξ̄_{(s+1)A_{0}})⟶^{6}H^{2(r−1)k}(Dξ̄_{(s+1)A_{0}}).

Map 6 is — after appropriate identifications — a multiplication by the Euler class e of ξ̄_{(s+1)A_{0}} in the cohomology ring of K̄_{(s+1)A_{0}}. Considering the commutative diagram

H^{∗}(K̄_{(s+1)A_{0}}) ⟶^{·e} H^{∗+2sk}(K̄_{(s+1)A_{0}})
↓^{ι^{∗}} ↓^{ι^{∗}}
H^{∗}(BU(k)^{r}) ⟶^{·ι^{∗}(e)} H^{∗+2sk}(BU(k)^{r})

as well as the facts that in the bottom line we have a polynomial ring, and ι^{∗}(e) = v_{k}^{(1)} ≠ 0, we see that map 6 is injective. Therefore map 5 is a 0-map, so map 4 is surjective. The proof is complete.

Remark 7.4. Let us remark that the history of the universal immersion used in this section (or at least its target space) goes back to works of Szűcs much before [RSz], see e.g. [Sz2].

8. Incidences

Let η and ζ be singularities (non-multi ones for now), i.e. sets of germs (ℂ^{∗}, 0) ⟶ (ℂ^{∗+k}, 0), with the same k. Let their codimensions be d and b, respectively.

Definition 8.1. The incidence class of η, ζ is defined as

I(η, ζ) := [K̄_{η}]|K̄_{ζ} ∈ H^{2d}(BG_{ζ}).

Although this definition assumes familiarity with the generalized Pontryagin-Thom construction, according to our theory an alternative definition can be given without any mentioning of [RSz]:

Proposition 8.2. I(η, ζ) = T p(η)(c(ζ)).

Our main theorem in these terms can be composed as

Proposition 8.3. • If ζ ≯ η, ζ ≠ η then I(η, ζ) = 0; • I(η, η) = e(η).

Let us remark, that at least one of ζ ≯ η and η ≯ ζ necessarily holds, so at least one of I(η, ζ) = 0, I(ζ, η) = 0 necessarily holds.

While definition 8.1 gives us a geometric idea of I(η, ζ), proposition 8.2 is easily computable. This duality can be used in both ways. The geometric ⇒ computable direction was, in fact, our method to compute Thom polynomials. One can, however, consider the other direction, too. We are going to show it on an example.

Let ζ be the singularity between equal dimensional manifolds corresponding to the local algebra ℂ[[x, y]]/(x^{2}, y^{3}). Its codimension is 7, so proposition 2.3 allows it to be > than A_{6} whose codimension is 6. Still, if we compute

I(A_{6}, ζ) = T p(A_{6})(c(ζ)) = T p(A_{6})(\frac{(1 + 2a)(1 + 3b)}{(1 + a)(1 + b)})

we find that it is 0, so we might suspect that η ≯ A_{6}. In fact it it true, and can be proved by long algebraic computations (the author used the computer algebra package Maple/grobner). The same method can be used to calculate I(A_{8}, (x^{2}, y^{4})) = 0 and so conjecture that (x^{2}, y^{4}) ≯ A_{8} although codim(x^{2}, y^{4}) = 10. Just like above, long calculation can prove that indeed (x^{2}, y^{4}) ≯ A_{8}.

There are two directions which the author believes may turn to be interesting in the study of incidences of singularities this way. The first challenge is a deeper geometric understanding of the incidence class. The other is the possible proof of the converse of proposition 8.3:

Conjecture 8.4. If I(η, ζ) = 0 then ζ ≯ η.

References

[AVGL] V. Arnold, V. Vasil’ev, V. Goryunov, O. Lyashko: Singularities. Local and global theory, Enc. Math. Sci. Vol. 6. (Dynamical Systems VI) Springer-Verlag 1993

[G] T. Gaffney: The Thom polynomial of \overline{Σ^{1111}}, in Proc. Symp. in Pure Math., Vol. 40 (1983), Part 1 [H] R. J. Herbert: Multiple points of immersed manifolds, Mem. Amer. Math. Soc. 34 (1981), no. 250 [K] S. L. Kleiman: Multiple point formulas II. The Hilbert scheme, in Enumerative Geometry (S. Xambó-

Descamps ed.), LNM 1436 (1990) Springer, 101–138 [La] L. Lander: The structure of the Thom-Boardman singularities of stable germs with type Σ^{2,0}, Proc.

London Math. Soc. (3) 33 (1976) 113–137

[Le] H. Levine: The singularities S_{1}^{q}, Ill. J. Math. 8, 152–168 (1964)

[MS] J. Milnor, J. Stasheff: Characteristic Classes, Princeton 1974

[O] T. Ohmoto: A geometric approach to Thom polynomials for C^{∞} stable mappings, J. London Math. Soc. (2) 47 (1993) 157–166

[P1] I. Porteous: Simple singularities of maps, in Proc. Liverpool Singularities I, LNM 192 (1971) 286–307 [P2] I. Porteous: The second-order decomposition of Σ^{2}, Topology, Vol. 11, pp. 325–334, 1972

[RSz] R. Rimányi, A. Szűcs: Generalized Pontrjagin-Thom construction for maps with singularities, Topology, Vol 37, No 6, pp. 1177-1191, 1998

[R1] R. Rimányi: On right-left symmetires of stable singularities, preprint

[R2] R. Rimányi: Computation of the Thom polynomial of Σ^{1111} via symmetries of singularities, to appear in the Proc. of the 5th Sao Carlos meeting on Real and Complex Singularities

[Ro1] F. Ronga: La calcul des classes duales aux singularités de Boardman d’ordre 2, Comm. Math. Helv. 47 (1972), 15–35

[Ro2] F. Ronga: On multiple points of smooth immersions, Comment. Math. Helv. 55, 521–527 (1980)

[SS] O. Saeki, K. Sakuma: Elimination of singularities: Thom polynomial and beyond, preprint

[Sz1] A. Szűcs: Analogue of the Thom space for mapping with singularity of type Σ^{1}, Math. Sbornik (N. S.) 1979, 108 (150) (3), 438–456 (in Russian), English translation: Math. of the USSR-Sbornik, 1979 (36)(3), 405–426

[Sz2] A. Szűcs: Cobordism groups of immersions with restricted self-intersection, Osaka J. Math. 21 (1984), 71–80

[Sz3] A. Szűcs: Multiple points of singular maps, Math. Proc. Camb. Phyl. Soc. 100 (1986) 331–346

[Sz4] A. Szűcs: The linking number of singular maps, Comment. Math. Helv. 61 (1986) 360–369

[T] R. Thom: Les singularités des applications différentiables, Ann. Inst. Fourier 6, (1955-56) 43–87 [Tu] R. Turnbull: The Thom-Boardman singularities Σ^{1},Σ^{1,1}, Σ^{1,1,1},Σ^{1,1,1,1},Σ^{1,1,1,1,1} and their closures,

Ph. D. Thesis, Liverpool University, 1989

[W] C. T. C. Wall: A second note on symmetry of singularities, Bull. London Math. Soc., 12 (1980), 347–354

[WP] C. T. C. Wall, A. A. du Plessis: The Geometry of Topological Stability, OUP 1995

Department of Analysis, ELTE TTK, Rákóczi út 5., Budapest 1088, Hungary

  E-mail address: rimanyi@cs.elte.hu