Bouquet and Join Theorems for Disentanglements

Kevin Houston School of Mathematics University of Leeds Leeds, LS2 9JT, U.K. e-mail: khouston@amsta.leeds.ac.uk

July 21, 2000

Abstract The disentanglement of certain augmentations is shown to be the topological join of a disentanglement and a Milnor fibre. The kth disentanglement of a finite map is defined and for corank 1 maps from ℂ^{n} to ℂ^{n+1} it is shown that they are homotopically equivalent to a wedge of spheres. Applications to the Mond conjecture are given. AMS Mathematics Subject Classification 2000 : 32S30, 32S55, 58K65.

1 Introduction

For complex analytic map-germs f : (ℂ^{n}, 0) → (ℂ^{p}, 0) with n ≥ p there exists a well established theory of Milnor fibres and K-equivalance. The A-equivalence theory of germs f : (ℂ^{n}, 0) → (ℂ^{p}, 0) has been developing in two directions. This is because two cases present themselves, n < p and n ≥ p.

For maps with n < p the image of the map is considered. Here the image of a stabilisation of the map takes the place of the Milnor fibre in the theory and this is called the disentanglement of f. For n ≥ p then one looks at the discriminant of a stabilisation. There are a number of similarities in the theory and we wish to combine them. The definitions to achieve this are given in Section 2.

We also define and study the kth disentanglement, these are the higher multiple point spaces in the image of the disentanglement map.

In the theory of functions one can create new singularities from old ones by taking the sum of two functions. The Milnor fibre for the new singularity is homotopically equivalent to the join of the Milnor fibres of the old singularities. In the theory of disentanglements there is a method of creating new singularities from f, an unfolding and a hypersurface singularity. This process is called augmentation. In a previous paper the connection between the A_{e}-codimension of the augmented map and the Tjurina number of the hypersurface and the A_{e}-codimension of the original map has been investigated, see [7].

In Theorem 3.2 of this paper we show that the kth disentanglement of the augmentation of a finitely A-determined map by a one-parameter stable unfolding and a hypersurface singularity is homotopically equivalent to the join of the kth disentanglement of the original map and the Milnor fibre of the hypersurface singularity.

It is well known that the Milnor fibre of an isolated hypersurface singularity is homotopically equivalent to a bouquet of spheres. In Corollaries 4.7 and 4.10 we see that if the disentanglement map of a corank 1 finitely A-determined f is a hypersurface and the multiple point spaces of the disentanglement map are Milnor fibres of isolated complete intersection singularities, then the kth disentanglement is empty or is homotopically equivalent to a bouquet of spheres. For the case k = 1 this has been proved in [2] and [15] for maps from ℂ^{n} to ℂ^{n} and from ℂ^{n} to ℂ^{n+1} respectively. For k > 1 and ℂ^{n} to ℂ^{n+1} the rational cohomology was found in [4].

For corank 1 maps from ℂ^{n} to ℂ^{n+1} with A_{e}-codimension equal to 1 we calculate the homotopy type of the kth disentanglement. For k = 1 the disentanglement is one n-sphere, a fact originally proved in [1]. The essentially new result is Corollary 5.3 which shows that for k > 1 the kth disentanglement in this case is contractible. Our interest in studying these is that we can combine the results with the results on the join theorem, the kth disentanglement bouquet theorem and A_{e}-codimension 1 maps to prove interesting theorems on the topology of disentanglements. See Theorem 6.1.

Finally, we can use these results and the result on the A_{e}-codimension of augmentations in [7] to give some results supporting the Mond Conjecture. This conjecture relates the A_{e}-codimension and the number of spheres in the 1st disentanglement of f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0). In particular, augmentations of corank 1 A_{e}-codimension 1 map-germs by isolated hypersurface singularities satisfy the Mond Conjecture, see Corollary 6.8.

2 Definitions and notation

In previous papers the notion of disentanglement was stated for the multi-germ f : (ℂ^{n}, S) → (ℂ^{p}, 0) with n < p and S a finite set of points. The definition was given in [4] and in Marar’s PhD thesis [11], with the relevant parts being published in [12]. If we take the case n ≥ p then we can define the discriminant space studied in [2]. What is intended in this section is that we combine both cases and furthermore include the case of image multiple point spaces.

The notion we wish to investigate can be summed up in the statement ‘a disentanglement is the discriminant of a local stabilisation of f’. This obviously encompasses both cases n < p and n ≥ p, as the discriminant of a map with n < p is just its image.

We can broaden the scope still further to the case of image multiple point spaces by stating ‘a kth disentanglement is the kth image multiple point of the discriminant of the local stabilisation of f’.

This of course needs to be made precise so suppose that f : (ℂ^{n}, S) → (ℂ^{p}, 0) is a complex analytic map-germ and that F : (ℂ^{n} × ℂ, S) → (ℂ^{p} × ℂ, 0 × 0) is a 1parameter unfolding. Let F : U → V be a proper representative with F^{−1}(V) = U. Then F (U) ⊂ (ℂ^{p} × {0}) has a Milnor ball B_{ε} ⊂ ℂ^{p} × {0}.

Then we may suppose that

f_{t} : U ∩ F^{−1}(B_{ε} × {t}) → V ∩ (B_{ε} × {t})

is (topologically) stable for all t ∈ D_{δ}\{0} for some small disc D_{δ} ⊂ ℂ centred at 0 with radius δ. See [4] and [12] for more details.

Definition 2.1 The kth image multiple point space of a continuous map g is the closure of the set of points in the image that have k or more preimages. This space is denoted by M_{k}(g).

Lemma 2.2 There exists δ′ ≤ δ such that the image multiple point spaces of the maps f_{t_{1}}|f_{t_{1}}^{−1}(B_{ε}) and f_{t_{2}}|f_{t_{2}}^{−1}(B_{ε}) are homeomorphic.

Proof. The proof follows from the disentanglement material in [2], [4] and [12]. The homeomorphism arising from use of the Thom-Mather isotopy lemmas is stratum preserving and the image multiple point spaces are union of strata. □

Definition 2.3 The map f_{t} : f_{t}^{−1}(B_{ε}) → B_{ε} for t ∈ D_{δ′}\{0} is called a disentanglement map for f. The kth image multiple point space of f_{t}, denoted Dis_{k}(f) is called a kth disentanglement of f.

Remark 2.4 If f is finitely determined and in the nice dimensions or it is corank 1 then the disentanglement map is unique in the sense that any other disentanglement map is A-equivalent to it. If f is not in the nice dimensions then topological disentanglements are unique in the sense of topological A-equivalence.

The case k = 1 was defined by Mond in [15] and is just the ordinary disentanglement.

We can also define similar maps and spaces for maps f : (ℂ^{n}, 0) → (ℂ^{p}, 0) where n ≥ p. In this case we take the discriminant of f, and we can again use Dis_{k}(f) to denote the image of the discriminant.

The disentanglement is the equivalent notion of Milnor fibre in the case of fibres of isolated complete intersection singularities. Many results on disentanglements rely upon using the image computing spectral sequence (see Theorem 4.5) and K equivalence on the multiple point spaces.

3 A join theorem for augmentations

First we define the augmentation of a map.

Definition 3.1 Let F : (ℂ^{n} ×ℂ, 0×0) → (ℂ^{p} ×ℂ, 0×0) be a 1-parameter unfolding of f : (ℂ^{n}, 0) → (ℂ^{p}, 0) given by (x, t) ↦ (F̃(x, t), t) and let g : (ℂ^{q}, 0) → (ℂ, 0) be a complex analytic function. The augmentation of f with F and g is the map A_{F,g}(f) : (ℂ^{n} × ℂ^{q}, 0 × 0) → (ℂ^{p} × ℂ^{q}, 0 × 0) given by

(x, z) ↦ (F̃(x, g(z)), z).

The best situation occurs when F is stable as a map-germ. The A_{e}-codimension of the augmentation of this case was studied in [7]. One of the aims of this paper is to investigate the topology of augmentations in this situation.

Theorem 3.2 Suppose that f : (ℂ^{n}, S) → (ℂ^{p}, 0) is a finitely A-determined multigerm such that the unfolding F : (ℂ^{n} × ℂ, S × 0) → (ℂ^{p} × ℂ, 0 × 0) is stable as a map-germ. Then,

Dis_{r}(A_{F,g}(f)) ≃ Dis_{r}(f) ∗ MF (g),

where ∗ denotes the join of two spaces, MF (g) denotes the Milnor fibre of g and ≃ denotes homotopy equivalence.

Proof. The proof is similar to the case of the addition of two singular functions in [16]. Suppose F is given by (F̃(x, t), t) then the augmentation A_{F,g}(f) is given by (F̃(x, g(z)), z). By Theorem 3.8 of [7] the map A_{F,g}(f)_{t} given by (F̃(x, g(z) + t), z) is a disentanglement map. Note that F has to be stable for this to be true, since if F is not stable then A_{F,g}(f)_{t} is not stable at non-isolated points.

One can map the image of A_{F,g}(f)_{t} onto ℂ by restricting the map π : ℂ^{p} ×{t}× ℂ^{q} → ℂ given by π(x, t, z) = g(z). This gives the usual two critical values, 0 and −t.

Suppose a = g(z) ∈ (0, −t) then the fibre of π is given by points {(X, T, Z) ∈ ℂ^{p} × {t} × ℂ^{q}|(X, T) = (F̃(x, T + a), T) some x and Z ∈ g^{−1}(a)}. That is, the fibre is the product of the disentanglement of f and the Milnor Fibre of g.

If g(z) = 0 then we get the product of the disentanglement of f and the space g^{−1}(0). If g(z) = −t then we get the product of points (X, T) such that (X, T) = (F̃(x, 0), T) and the Milnor fibre of g. Note that in these special cases we have the product of a space with something contractible.

This means, by choice of balls etc, that the space is homotopically equivalent to the join of the disentanglement of f and the Milnor fibre of g. The choice of maps is as in the usual way of [16].

For the kth disentanglement with k > 1 it is easy to see that the same works for the necessary subsets: For example the homeomorphisms between the ‘non-singular’ fibres of π are stratum preserving. So the result is true for the image multiple point spaces as these are just union of strata. □

Remark 3.3 This generalises Cooper’s result, see [1], on the augmentation of corank 1 A_{e}-codimension 1 map-germs by the obvious unfolding and g(z) = z^{2}: The disentanglement of the augmentation is homotopically equivalent to the suspension of the disentanglement of f.

4 Bouquet theorems for disentanglements

4.1 Preliminaries

In this section we prove a bouquet of spheres theorem for the kth disentanglement of a finitely determined corank 1 map-germ f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0). We also prove a similar result in the f : (ℂ^{n}, 0) → (ℂ^{n}, 0) case.

The proof relies on a generalised version of the image computing spectral sequence introduced in [4] and further developed in [3] and [9]. Rather than using it to calculate the ordinary integer homology of the image of a map we shall use it to calculate the alternating homology of multiple point spaces. The notion of alternating homology is slightly more general than that in [5] and is a special case of the notion of det-twisting of [3].

Suppose H is a subgroup of S_{k}, the group of permutations on k objects and sign : H → {1, −1} is the restriction of the usual sign representation of S_{k}.

Definition 4.1 Let M be a module upon which H acts, then

M^{alt<H>} := {m ∈ M|h(m) = sign(h)m for all h ∈ H},

is the alternating module of M with respect to H.

Definition 4.2 Suppose Y is space upon which H acts cellularly and let C_{∗}(Y) denote the cellular chain complex of Y. Define the alternating homology of Y with respect to H and coefficient group G to be

H_{i}^{alt<H>}(Y; G) := H_{i}((C_{∗}(Y))^{alt<H>} ⊗ G).

We will abbreviate H_{i}^{alt<S_{k}>} to H_{i}^{alt} as this coincides with the notation of earlier papers such as [5].

Remark 4.3 The concept is a special case of what is called det-twisting in [3] and denoted with T w. For a space Y we have in the notation of [3]

H_{i}^{alt<H>}(Y; ℤ) = T w H_{i}(Y; ℤ),
and (H_{i}(Y; ℤ))^{alt<H>} = H_{i}^{T w}(Y; ℤ).

We now fix some notation for a finite and proper continuous map f : X → Y.

Definition 4.4 The kth multiple point space, D^{k}(f), of the map f : X → Y is the set of points,

D^{k}(f) := closure{(x_{1}, …, x_{k}) ∈ X^{k}|f(x_{1}) = ⋯ = f(x_{k}), x_{i} ≠ x_{j} for i ≠ j}.

For each k there exists a map ε^{k+1,k+1}(f) : D^{k+1}(f) → D^{k}(f) given by restriction of the natural projection map (x_{1}, …, x_{k}, x_{k+1}) ↦ (x_{1}, …, x_{k}). When no confusion will result we will denote ε^{k+1,k+1}(f) by ε^{k+1,k+1}. We will take ε^{1,1}(f) to be f.

According to [4], page 55, D^{j}(ε^{k+1,k+1}) is homeomorphic to D^{j+k}(f) by a homeomorphism which respects the action of S_{j} on D^{j}(ε^{k+1,k+1}) and the action of S_{j} on the last j coordinates of D^{j+k}(f). This map is given by:

((x_{1}, …, x_{k}, x_{k+1}), (x_{1}, …, x_{k}, x_{k+2}), …, (x_{1}, …, x_{k}, x_{k+j}))
↦ (x_{1}, …, x_{k}, x_{k+1}, …, x_{k+j}).

Suppose H is a group which acts on D^{k}(f) by permutation of copies of X in X^{k}, then H acts on D^{k+1}(f) by permutation of the first k coordinates. There is an action of H × S_{j} on D^{j}(ε^{k+1,k+1}) = D^{k+j}(f) given by permuting the last j coordinates in D^{k+j}(f).

To recap, the action of S_{j} on D^{j}(ε^{k+1,k+1}) is permutation of the last j coordinates of D^{k+j}(f) and the action of H on D^{j}(ε^{k+1,k+1}) is permutation of the first k coordinates of D^{k+j}(f).

Note that the image multiple point space M_{k}(f) is the image of the map from D^{k}(f) to Y that takes (x_{1}, …, x_{k}) to f(x_{1}).

4.2 An image computing spectral sequence

Now we shall state a general version of the image computing spectral sequence of [3], [4] and [9]. Theorem 4.5 Suppose f : X → Y is a proper and finite analytic map between compact subanalytic spaces and that H acts on D^{k}(f) by permutation of copies of X in X^{k}.

Then, for any coefficient group G, there exists a spectral sequence

E_{p,q}^{1,H}(ε^{k+1,k+1}(f)) ≅ H_{q}^{alt<H×S_{p+1}>}(D^{k+p+1}(f); G)

converging to

H_{∗}^{alt<H>}(ε^{k+1,k+1}(D^{k+1}(f)); G).

The direct product H × S_{p+1} ⊆ S_{k+p+1} acts on D^{k+p+1}(f) by

(h, σ)(x_{1}, …, x_{k}, x_{k+1}, … x_{k+j}) = (h(x_{1}, …, x_{k}), σ(x_{k+1}, … x_{k+j})).

Note that a disentanglement map arising from a finitely A-determined map-germ f : (ℂ^{n}, 0) → (ℂ^{p}, 0) satisfies the conditions of the theorem.

4.3 Bouquet theorems for image multiple point spaces

The bouquet theorems for disentanglements and discriminants mentioned in the introduction follow from the following more general theorem. This theorem has its antecedents in Theorem 2.8 of [4], there the assumptions of non-singularity of the multiple point spaces and reduced rational cohomology in one dimension allow one to describe the rational cohomology of the image multiple point spaces.

Theorem 4.6 Suppose f : X^{n} → Y^{n+1} is a complex analytic map such that the k^{th} multiple point space is empty or is the S_{k}-invariant Milnor fibre of an S_{k}-invariant isolated complete intersection singularity of dimension n−k+1 for all k ≥ 1. Then, M_{r}(f) is empty or homotopically equivalent to a bouquet of spheres of dimension n − r + 1 for 1 ≤ r ≤ n + 1.

The proof is delayed until after we look at a couple of consequences of the theorem. A rational homology version of the next corollary was proved in [4]. The case r = 1 was proved without the corank 1 condition in [15] and the case r = 2 is effectively proved in [6].

Corollary 4.7 Suppose f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a finitely A-determined corank 1 map-germ. Then, the r^{th} disentanglement of f, Dis_{r}(f), is empty or homotopically equivalent to a bouquet of spheres of dimension n − r + 1.

Proof. Each D^{k}(f_{t}) is the S_{k}-invariant Milnor fibre of the S_{k}-invariant isolated complete intersection singularity D^{k}(f), see Theorem 2.14 of [13]. Thus as Dis_{r}(f) = M_{r}(f_{t}) the corollary follows from the theorem. □

Definition 4.8 Define the r^{th} image Milnor number of f, denoted μ_{I_{r}} to be the number of spheres for Dis_{r}(f) in the above corollary.

This generalises the definition of image Milnor number from [15].

Remark 4.9 It is possible to prove the above corollary using the fact that the k^{th} multiple point space of the family of maps is a set-theoretic local complete intersection. (This non-trivial fact is not proved here.) The reason for not using that method is that I do not know if the same is true for the case of maps considered in the next corollary.

Corollary 4.10 Let Φ : (Σ^{n}, 0) → (ℂ^{n+1}, 0) be a projection germ of finite codimension induced from a suspension of a Whitney mapping (see Section 5 of [3]). Then, Dis_{r}(Φ), the r^{th} image multiple point space in the image of a stabilisation of Φ, is empty or homotopically equivalent to a bouquet of spheres of dimension n − r + 1.

Proof. In Section 5 of [3] it is proven that the k^{th} multiple point space of the stabilisation of Φ is the Milnor fibre of an S_{k}-invariant map-germ. Thus the corollary follows from the theorem. □

Definition 4.11 Define the r^{th} discriminant Milnor number of f, denoted μ_{Δ_{r}} to be the number of spheres in the above corollary.

This generalises the notion of discriminant Milnor number given in [2].

We shall prove Theorem 4.6 through application of a couple of lemmas. First, we fix some notation. For any finite and proper continuous map g : X → Y let D_{j}^{p}(g) denote the image of D^{p}(g) in D^{j}(g) (via restriction of the projection of X^{p} to X^{j}) and let g_{p} = g|D_{1}^{p}. Note that D^{k}(g_{r}) = D^{k}(g) for k ≥ r and D^{k}(g_{r}) = ε^{k+1,k+1}(D^{k+1}(g_{r})) for k < r. Also, observe that the image of g_{r} is M_{r}(g).

Lemma 4.12 Suppose g is a map as in Theorem 4.6 and that H acts on D^{k}(g_{r}) by permutation of copies of source. Then,

H_{i}^{alt<H>}(D^{k}(g_{r}); ℤ) = \{ \begin{matrix}free abelian & i = dim D^{k}(g_{r}), \\ ℤ & i = 0 and dim D^{k}(g_{r}) > 0 and H is trivial, \\ 0 & otherwise. \end{matrix}

Proof. If k ≥ r, then D^{k}(g_{r}) = D^{k}(g). Hence,

H_{i}^{alt<H>}(D^{k}(g_{r}); ℤ) ≅ H_{i}^{alt<H>}(D^{k}(g); ℤ)
≅ (H_{i}(D^{k}(g); ℤ))^{alt<H>} by Theorem 2.1.2 of [3],
≅ \{ \begin{matrix}free abelian & i = dim D^{k}(g_{r}), \\ (ℤ)^{alt<H>} & i = 0 and dim D^{k}(g_{r}) > 0, \\ 0 & otherwise. \end{matrix}

The last isomorphism follows from the fact that D^{k}(g) is an isolated complete intersection singularity. For the case i = 0 and dim D^{k}(g) > 0, we observe that if H is non-trivial then (ℤ)^{alt<H>} = 0. This because every point is homologous to one in the set x_{i} = x_{j} for some i and j and such points do not produce alternating chains, see for example Lemma 2.6 of [5]. If H is trivial then obviously (ℤ)^{alt<H>} = ℤ.

For the case k < r we effectively use descending induction on k. The induction argument starts with k = r − 1 and the case k = r serves as the initial case. First note that as D^{k}(g_{r}) = ε^{k+1,k+1}(D^{k+1}(g_{r})) we have,

H_{i}^{alt<H>}(D^{k}(g_{r}); ℤ) ≅ H_{i}^{alt<H>}(ε^{k+1,k+1}(D^{k+1}(g_{r})); ℤ)

for k < r. By Theorem 4.5 the latter group can be calculated using a spectral sequence, the terms of which involve the alternating homology of multiple point spaces D^{j}(g_{r}) with j > k (this is what allows us to do induction):

E_{p,q}^{1,H} = H_{q}^{alt<H×S_{p+1}>}(D^{k+p+1}(g_{r}); ℤ) ⟹ H_{∗}^{alt<H>}(ε^{k+1,k+1}(D^{k+1}(g_{r})); ℤ).

By induction

H_{q}^{alt<H×S_{p+1}>}(D^{k+p+1}(g_{r}); ℤ)
= \{ \begin{matrix}free abelian & i = dim D^{k}(g_{r}), \\ ℤ & i = 0 and dim D^{k}(g_{r}) > 0 and H is trivial, \\ 0 & otherwise. \end{matrix}

So

E_{p,q}^{1,H} = \{ \begin{matrix}free abelian & \{0 ≤ p < r and q = dim D^{r}(g)\} or \{p ≥ r and q = dim D^{p+1}(g)\}, \\ ℤ & p = q = 0 and dim D^{k}(g_{r}) > 0 and H is trivial, \\ 0 & otherwise. \end{matrix}

As the sequence converges to H_{∗}^{alt<H>}(D^{k}(g_{r}); ℤ), it is easy to see from the positions of the entries that H_{i}^{alt<H>}(D^{k}(g_{r}); ℤ) = 0 for i < dim D^{k}(g_{r}) except of course for i = 0 when H is trivial. Now D^{k}(g_{r}) is a compact complex analytic space so by Theorem 3.1 of [5] it has the S_{k}-homotopy type of a cellular complex of real dimension equal to its complex dimension. (Two complexes K and L are S_{k}-homotopically equivalent if there exist two equivariant maps α : K → L and β : L → K, such that the compositions α ∘ β and β ∘ α are homotopic to the identities in the class of S_{k}-equivariant maps.) Hence H_{i}^{alt<H>}(D^{k}(g_{r}); ℤ) = 0 for i > dim D^{k}(g_{r}) and is free abelian for i = dim D^{k}(g_{r}). □

Lemma 4.13 (i) If dim D^{k}(g_{r}) ≥ 1 then D^{k}(g_{r}) is connected.

(ii) If dim D^{k}(g_{r}) ≥ 2 then D^{k}(g_{r}) is simply connected.

Proof. Part (i) follows from taking H trivial in the previous lemma.

In part (ii) the case of k ≥ r follows from D^{k}(g_{r}) = D^{k}(g) being the Milnor fibre of an isolated complete intersection singularity of dimension greater than or equal to 2.

For k < r we use the descending induction argument seen in Lemma 4.12. Consider the group H_{0}^{alt}(D^{2}(ε^{k+1,k+1}(g_{r})); ℤ), this is equal to H_{0}^{alt}(D^{k+2}(g_{r}); ℤ). As we have dim D^{k}(g_{r}) ≥ 2 then dim D^{2}(ε^{k+1,k+1}(g_{r})) ≥ 1 and so ε^{k+1,k+1}(g_{r}) is connected by (i). This and Lemma 4.12 proves that H_{0}^{alt}(D^{k+2}(g_{r}); ℤ) = 0. By Corollary 4.19 of [5] this implies that the natural map

π_{1}(D^{k+1}(g_{r})) = π_{1}(D^{1}(ε^{k+1,k+1})) → π_{1}(ε^{k+1,k+1}(D^{k+1}(g_{r}))) = π_{1}(D^{k}(g_{r}))

is a surjection.

Hence D^{k}(g_{r}) is simply connected for k = r − 1 and so on for all k < r. □

Proof (Theorem 4.6). If dim D^{r}(g) = 0 then M_{r}(g) is just a finite set of points and hence is a wedge of spheres of dimension dim D^{r}(g).

Thus assume dim D^{r}(g) ≥ 1 and consider the image computing spectral sequence for g_{r}. Then, using Lemma 4.12 we have

E_{p,q}^{1}(g_{r}) ≅ H_{q}^{alt<S_{p+1}>}(D^{p+1}(g_{r}); ℤ)

≅ \{ \begin{matrix}free abelian & \{0 ≤ p < r and q = n − r + 1\} or \{p ≥ r and q = n − p\}, \\ ℤ & p = q = 0, \\ 0 & otherwise. \end{matrix}

Just as in the proof of Lemma 4.12 we get reduced integer homology in the image of g_{r} = M_{r}(g) in middle dimension only as it has the homotopy type of a CW-complex of middle dimension and vanishing reduced homology below middle dimension.

If dim D^{r}(g) = 1 then M_{r}(g) is a connected 1-dimensional finite CW-complex. Hence it is a wedge of spheres of dimension 1.

If dim D^{r}(g) ≥ 2 then H_{0}^{alt}(D^{2}(g_{r}); ℤ) = 0 (by Lemma 4.12) and D^{1}(g_{r}) is simply connected (by Lemma 4.13). This implies by Corollary 4.19 of [5] that the image of g_{r} simply connected. The proof is completed by remarking that a simply connected (n − r + 1)-dimensional finite CW-complex with free abelian homology in dimension n − r + 1 is homotopically equivalent to a wedge of (n − r + 1)-spheres. □

Remark 4.14 For corank 1 maps ℂ^{n} to ℂ^{p} where p ≠ n+1 examples seem to show that the kth disentanglement is homotopically equivalent to a bouquet of spheres but there is no proof as yet.

5 Disentanglements for A_{e}-codimension 1 map-germs

We shall calculate the image Milnor numbers for corank 1 map-germs of A_{e}-codimension one. These maps were classified by Cooper in [1] as follows:

Example 5.1 Suppose f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a corank 1 map-germ that has A_{e}-codimension 1. Then, for some m, f is A-equivalent to

(x_{1}, …, x_{n−1}, y)

↦ (x_{1}, …, x_{n−1}, y^{m+2} + ∑_{i=1}^{m} x_{2i−1}y^{i}, y^{m+3} + y^{m+1} ∑_{i=2m+1}^{n−1} x_{i}^{2} + ∑_{i=1}^{m} x_{2i}y^{i}) .

Theorem 5.2 Suppose that f : X^{n} → Y^{n+1} is a map that satisfies the conditions of Theorem 4.6. Suppose that for some m, D^{k}(f) is non-singular and contractible for k < m, D^{m}(f) is A-equivalent by an S_{k}-equivariant map to a quadratic singularity and that D^{m+1}(f) is empty.

Then, M_{1}(f) is homotopically equivalent to an n-sphere and M_{r}(f) is homotopically trivial or empty for r ≥ 2.

Proof. Because Theorem 4.6 tells us that the spaces of interest are homotopically equivalent to bouquets of spheres, we can restrict ourselves to calculating the rational cohomology of the space. We shall use the rational cohomology image computing spectral sequence of [4] to calculate M_{r}(f), which is the image of f restricted to D_{1}^{r}. The sequence has E_{1}^{p,q} isomorphic to H^{q}(D^{p+1}(f); ℚ)^{alt} and converges to H^{∗}(f(X); ℚ).

First we note that by a paper of Wall, [17], the cocycle in the rational cohomology of D^{m}(f) is S_{m}-alternating, i.e. H^{n−m+1}(D^{m}(f); ℚ)^{alt} ≅ ℚ.

We shall apply induction on r. The case r = 1 is simple as the only non-zero entries of E_{1}(f) are E_{1}^{0,0} ≅ ℚ and E_{1}^{m−1,n−m+1} = H^{n−m+1}(D^{m}(f); ℚ)^{alt} ≅ ℚ and this implies that H^{n}(M_{1}(f), ℚ) ≅ ℚ.

Assume that the statement of the theorem is true for all M_{j}(f) with j < r, (and of course r ≤ m).

The multiple point spaces of f|D_{1}^{r} are given on page 56 of [4]:

D^{k}(f|D_{1}^{r}) = \{ \begin{matrix}M_{r−k}(ε^{k+1,k+1}(f)) & for k < r, \\ D^{k}(f) & for k ≥ r. \end{matrix}

The map ε^{k+1,k+1} : D^{k+1} → D^{k+1} satisfies the requirements of the statement of the theorem and so by the induction hypothesis D^{k}(f|D_{1}^{r}) is contractible for k < r − 1 and D^{r−1}(f|D_{1}^{r}) has reduced rational alternating cohomology possible only in dimension dim_{ℂ} D^{k}(f|D_{1}^{r}) = n − r + 1. This vector space can have at most dimension 1, (it will turn out to have exactly dimension 1 but the induction hypothesis above cannot guarantee that the alternating cohomology is not zero). For k ≥ r the rational alternating cohomology of D^{k}(f|D_{1}^{r}) is the same as that of D^{k}(f).

Thus E_{1}^{∗,∗} = E_{1}^{∗,∗}(f|D_{1}^{r}) has only three possible non-zero groups: E_{1}^{0,0} ≅ ℚ, E_{1}^{m−1,n−m+1} = H^{n−m+1}(D^{m}(f); ℚ)^{alt} ≅ ℚ and dim_{ℚ} E_{1}^{r−2,n−r+1} ≤ 1. This implies that the only possible non-zero differential is

∂_{m−r+1} : E_{m−r+1}^{r−2,n−r+1} → E_{m−r+1}^{m−1,n−m+1}.

This is not zero because if it were then the cocycle from E_{m−r+1}^{m−1,n−m+1} would survive in E^{∞} and would appear in dimension n of the cohomology of M_{r}(f). However, the complex dimension of M_{r}(f) is n − r + 1 which is less than n by assumption and hence this situation cannot occur.

As the differential is non-zero we deduce that E_{1}^{r−2,n−r+1} ≅ ℚ and that the map is an isomorphism. So only E_{1}^{0,0} ≅ ℚ survives in E^{∞} in the sequence and hence the rational cohomology of M_{r}(f) is trivial. □

Corollary 5.3 Suppose that f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a corank 1 map germ with A_{e}-codimension equal to 1. Then, Dis_{1}(f) is homotopically equivalent to an nsphere, and Dis_{k}(f) is contractible or empty for k ≥ 2.

Proof. By [1] such maps (see Example 5.1) have disentanglement maps that satisfy the conditions of the theorem. □

Remark 5.4 The case in which f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is corank 1, of A_{e}- codimension 1 and k = 1 was proved first in Theorem 8.14 of [1].

Corollary 5.5 Suppose that f : (ℂ^{n}, 0) → (ℂ^{n}, 0) has A_{e}-codimension 1 and is induced from the projection of a Whitney mapping, see [3]. Then, Dis_{1}(f) is homotopically equivalent to a n-sphere, and Dis_{k}(f) is contractible or empty for k ≥ 2.

Proof. By [3] such maps have disentanglement maps that satisfy the conditions of the theorem. □

Remark 5.6 The case f : (ℂ^{n}, 0) → (ℂ^{n}, 0) with f induced from a map as in Corollary 4.10 with k = 1 was proved in [3].

6 Consequences of the main results

In this section we apply the join result for disentanglements of augmentations, (Theorem 3.2), and the bouquet of spheres theorem for kth disentanglements, (Theorem 4.6) to the topology of disentanglements and the Mond Conjecture.

6.1 Topology of disentanglements

Theorem 6.1 Suppose f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a finitely A_{e}-determined corank 1 map germ, then Dis_{k}(A_{F,g}(f)) (if non-empty) is homotopically equivalent to the join of Dis_{k}(f) and the Milnor fibre of g:

Dis_{k}(A_{F,g}(f)) ≃ (⋁_{μ_{I_{k}}} S^{n−k+1}) ∗ MF (g).

Proof. Follows from Theorem 3.2 and Corollary 4.7. □

Remark 6.2 This theorem provides us with examples of disentanglements of nonfinitely determined map-germs as the following example shows.

Example 6.3 Let f(x, y, z) = (x, y, yz + z^{5}, xz + z^{3}). This is the map P_{1,2} of [10]. We have μ_{I_{1}}(f) = 2, μ_{I_{2}}(f) = 1, see [8], and the other image multiple point spaces are empty.

We can unfold with

F (x, y, z, u) ↦ (x, y, u, yz + z^{5} + uz^{2}, xz + z^{3}).

The 3-jet of this is a stable map which is 3-determined and hence F is stable.

Let g(u_{1}, u_{2}, u_{3}) = u_{1}u_{2}u_{3}; the Milnor fibre for this non-isolated singularity is S_{1} × S_{1}. Then, a disentanglement exists for the map-germ

(x, y, z, u_{1}, u_{2}, u_{3}) ↦ (x, y, u_{1}, u_{2}, u_{3}, yz + z^{5} + u_{1}u_{2}u_{3}z^{2}, xz + z^{3}).

We have

Dis_{1}(f) ≃ S^{3} ∗ ((S^{1} × S^{1}) ∨ (S^{1} × S^{1})) ,
and Dis_{2}(f) ≃ S^{2} ∗ (S^{1} × S^{1}) .

Corollary 6.4 Suppose f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a finitely A_{e}-determined corank 1 map germ and g : (ℂ^{q}, 0) → (ℂ, 0) defines an isolated singularity then Dis_{k}(A_{F,g}(f)) is homotopically equivalent to

⋁_{μ_{k}} S^{n−k+q−1},

where μ_{k} = μ_{I_{k}} (A_{F,g}(f)) = μ_{I_{k}}(f)μ(g).

Proof. This corollary follows from Theorem 6.1 and the fact the Milnor fibre of g is a bouquet of spheres of dimension q − 1. Of course, the homotopy description could have been found from Theorem 4.6 as f is finitely A-determined. □

Corollary 6.5 Suppose f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a corank 1 map-germ of A_{e}- codimension 1 then Dis_{k}(A_{F,g}(f)) is homotopically equivalent to the n-fold suspension of the Milnor fibre of g for k = 1 and is contractible for k ≥ 2.

Proof. This follows from Theorem 6.1, Corollary 5.3 and the facts that the join with an n-sphere is equivalent to n-fold suspension and the join with a contractible space is contractible. □

Similar results are true for f : (ℂ^{n}, 0) → (ℂ^{n}, 0) as the next example shows.

Example 6.6 Consider the A_{e}-codimension 1 map-germ

(x_{1}, …, x_{n−1}, y) ↦ (x_{1}, …, x_{n−1}, y^{n+2} + ∑_{i=1}^{n−1} x_{i}y^{i}).

If we augment this with the stable unfolding

(x_{1}, …, x_{n−1}, y, t) ↦ (x_{1}, …, x_{n−1}, y^{n+2} + ty^{n} + ∑_{i=1}^{n−1} x_{i}y^{i})

and the singularity g : ℂ^{q} → ℂ then we deduce that the disentanglement of the augmentation is homotopically equivalent to μ(g) n-spheres. This has already been shown in section 6 of [3] by another method.

Note that the series ‘∗’ and 4_{1}^{k}’ of [14] are of the form in the example above. They are augmentations of y ↦ y^{3} and (x, y) ↦ (x, y^{4} + xy) respectively. This helps explain the presence of the A, D, and E series in their classification.

6.2 Application to the Mond Conjecture

A finitely A-determined map-germ f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) satisfies the Mond Conjecture if A_{e}-cod(f) ≤ μ_{I}(f) with equality if f is quasihomogeneous. Maps from ℂ^{2} to ℂ^{3} satisfy the Mond conjecture, see [15], and so do corank 1 folding maps, see [7]. We can now show how augmentations and the join theorem can lead to germs satisfying the Mond Conjecture.

In the following let cod(f) denote the A_{e}-codimension of the map f.

Theorem 6.7 Suppose that f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0) is a finitely A_{e}-determined map-germ that satisifies the Mond Conjecture and that g defines an isolated hypersurface singularity. If f or g is quasihomogeneous then

τ(g)cod(f) = cod(A_{F,g}(f)) ≤ μ_{I}(A_{F,g}(f)) = μ_{I}(f)μ(g)

with equality if both f and g are quasihomogeneous. That is, A_{F,g}(f) satisfies the Mond Conjecture.

Proof. If f is quasihomogeneous then the one-parameter unfolding F is quasihomogeneous as well. So if F or g is quasihomogeneous then,

cod(A_{F,g}(f)) = τ(g)cod(f), by Theorem 3.3 of [7],
≤ τ(g)μ_{I}(f), by the Mond Conjecture assumption,
≤ μ(g)μ_{I}(f), as τ(g) ≤ μ(g),
= μ_{I}(A_{F,g}(f)), by Corollary 6.4.

If f and g are quasihomogeneous then the inequality parts above are equalities. □

Corollary 6.8 Suppose that A_{F,g}(f) is an augmentation with g of a corank 1 A_{e}- codimension 1 map-germ f : (ℂ^{n}, 0) → (ℂ^{n+1}, 0). Then,

τ(g) = cod(A_{F,g}(f)) ≤ μ_{I}(A_{F,g}(f)) = μ(g),

with equality if g is quasihomogeneous. That is, A_{F,g}(f) satisfies the Mond Conjecture.

Proof. The map f is quasihomogeneous by Cooper’s classification and μ_{I}(f) = cod(f) = 1, see [1]. So the result follows from the theorem. □

Remark 6.9 This last result is not too surprising if one views augmentations of A_{e}-codimension 1 maps as the embedding of the theory of hypersurface singularities into the theory of maps from ℂ^{n} to ℂ^{p}.

References

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