GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 2, 1997, 163-184

HYPERPLANE SINGULARITIES OF ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES

M. SHUBLADZE

Abstract. A new class of non-isolated singularities called hyperplane singularities is introduced. Special deformations with simplest critical points are constructed and an algebraic expression for the number of Morse points is given. The topology of the Milnor fibre is completely studied.

0. Introduction

This paper continues the investigation of special classes of non-isolated singularities.

In [1] and [2] germs of analytic functions having a smooth one-dimensional submanifold as a singular set were investigated, the simplest ones of such germs being obtained as limits of simple isolated singularities of series A_{k} and D_{k}. In our work germs of analytic functions f : (ℂ^{n+1}, 0) → (ℂ, 0) are considered, having singularities on the hyperplane

H = {(x, y_{1}, y_{2}, …, y_{n}) ∈ ℂ × ℂ^{n} | x = 0}.

Such singularities are called hyperplane singularities.

The paper is divided into 6 sections.

In Section 1, coordinate transformations preserving the singular hyperplane H are introduced and the equivalence of germs under such transformations is defined. Moreover, simplest germs of A_{∞} (local expression x^{2}) and D_{∞} (local expression x^{2}y_{1}) types are determined.

In Section 2 the notion of an isolated hyperplane singularity is introduced and investigated.

In Section 3, for isolated hyperplane singularities, a special deformation is constructed, having only A_{∞} and D_{∞} type singular points on the hyperplane H and only Morse points outside H.

163

1991 Mathematics Subject Classification. 57R45.

Key words and phrases. Analytic functions of several complex variables, isolated singularities, hyperplane singularities, Milnor fibre, Morse points.

1072-947X/97/0300-0163$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

In Section 4 the number of Morse points is calculated for special deformation of f.

In Section 5 the topology of the Milnor fibre is studied using the special deformation. It is shown that the Milnor fibre is homotopy equivalent to the wedge of a circle S^{1} with 2μ + σ copies of the sphere S^{n}, where μ = μ(g) is the Milnor number of the isolated singularity g(0, y_{1}, …, y_{n}), while σ is the number of Morse points of the deformation of f. To this end we investigate the problem of determining a homotopy type of the complement of a nonsingular (smooth) submanifold.

In Section 6 consideration is given to germs of analytic functions representable as f = x^{k}g(x, y_{1}, …, y_{n}), called hyperplane singularities of transversal type A_{k}. For such singularities all the results obtained in Sections 1–5 are generalized.

1. Hyperplane Singularities

Let f : (ℂ^{n+1}, 0) → (ℂ, 0) be a germ of a holomorphic complex function of n + 1 complex variables (x, y_{1}, …, y_{n}) and let the hyperplane H = {(x, y_{1}, …, y_{n}) | x = 0} consist of points z ∈ ℂ^{n+1} such that gradf(z) = 0.

In the ring O_{n+1} of all germs at zero of holomorphic functions, single out the ideals

m = {f ∈ O_{n+1} | f(0) = 0},
(x) = {f ∈ O_{n+1} | f(0, y_{1}, …, y_{n}) = 0, ∀(y_{1}, …, y_{n}) ∈ ℂ^{n+1}}.

We are going to investigate elements from the ideal (x^{2}). The following characterization of these elements is valid:

Lemma 1.1. Let f : (ℂ^{n+1}, 0) → (ℂ, 0) be a germ of a holomorphic function, having H as its singular set. Then such a germ can be represented in the form f = x^{2}g(x, y_{1}, …, y_{n}), where g is a smooth germ from the ring O_{n+1}.

In the group D_{n+1} of germs of local diffeomorphisms of ℂ^{n+1} at the origin, consider a subgroup consisting of diffeomorphisms φ ∈ D_{n+1} satisfying φ(H) = H. In the following all the coordinate transformations considered will preserve the hyperplane H, i.e., belong to D_{H}.

Let us introduce some definitions.

Definition 1.2. A singular point z ∈ H is called a point of type A_{∞}, if Hess_{x} f = \frac{∂^{2}f}{∂x^{2}} (x, y_{1}, …, y_{n}) is nonzero at this point.

Definition 1.3. A singular point is called a point D_{∞} if the gradient of the function Hess_{x} f with respect to variables y_{i}, i = 1, …, n, written as

(\frac{∂}{∂y_{1}} (\frac{∂^{2}f}{∂x^{2}} ) , \frac{∂}{∂y_{2}} (\frac{∂^{2}f}{∂x^{2}} ) , …, \frac{∂}{∂y_{n}} (\frac{∂^{2}f}{∂x^{2}} )) ,

is not equal to zero at this point.

The following simple assertions are easy to prove.

Proposition 1.4. A singular point z ∈ H is of the type A_{∞} if and only if in some neighborhood of z there exists a coordinate transform from the group D_{H} which reduces f to x^{2}.

Proposition 1.5. A singular point z ∈ H has type D_{∞} if and only if in some neighborhood of z there is a coordinate transform with respect to H which changes f to x^{2}y_{1}.

Let Orb(f) denote an orbit of the germ f under the action of D_{H}. As always the simplest orbits are of interest.

Having in mind to characterize isolated singularities, and in accord with the finite dimensional case, let us introduce a measure for germ complexity.

Definition 1.6. The number

codim(f) = dim_{ℂ} [(x^{2})/τ(f)] ,

where τ(f) is the tangent space to Orb(f) at f, is called the codimension of a hyperplane singularity f.

2. Isolated Hyperplane Singularities

Now we can give a simple criterion for finite determinacy:

Theorem 2.1. Let f ∈ (x^{2}) be a hyperplane singularity not of type A_{∞} or D_{∞} such that in the presentation f = x^{2}g(x, y_{1}, …, y_{n}) the germ g(0, y_{1}, …, y_{n}) is, as a germ of a function of y_{1}, …, y_{n}, an isolated singularity. Then the following assertions are equivalent:

(a) codim f is finite;

(b) the function g(x, y_{1}, …, y_{n}) has an isolated singularity;

(c) f has a singularity of type A_{∞} outside points with g(0, y_{1}, …, y_{n}) = 0 and a singularity of typeD_{∞} at points with g(0, y_{1}, …, y_{n}) = 0, except the origin.

Proof. Let us show that (a) implies (b). Indeed, if codimf < +∞, then dim_{ℂ} [(x^{2})/τ(f)] < +∞, where τ(f) has type (ξxg + ξx^{2}g_{x}, η_{1}x^{2}g_{y_{1}}, …, η_{n}x^{2}g_{y_{n}}), ξ ∈ (x), η_{i} ∈ m, i = 1, …, n. To the function g associate the ideal (g_{x}, g_{y_{1}}, …, g_{y_{n}}); according to Briançon–Skoda’s theorem [3], g^{n+1} ∈ (g_{x}, g_{y_{1}}, …, g_{y_{n}}). Clearly, this implies τ^{n+1}(f) ∈ (x^{2}g_{x}, x^{2}g_{y_{1}}, …, x^{2}g_{y_{n}}) and since

dim_{ℂ} [(x^{2})/τ^{n+1}(f)] = dim_{ℂ} [(x^{2})/τ(f)] + dim_{ℂ} [τ(f)/τ^{n+1}(f)] < +∞,

one obtains

dim_{ℂ} [(x^{2})/(x^{2}g_{x}, x^{2}g_{y_{1}}, …, x^{2}g_{y_{n}})] = dim_{ℂ}^{O_{n+1}} [(g_{x}, g_{y_{1}}, …, g_{y_{n}})] < +∞;

hence g(x, y_{1}, …, y_{n}) has an isolated singularity.

(b)⇒(c). Let g have an isolated singularity and g(0, y_{1}, …, y_{n}) have an isolated singularity at zero, i.e., grad g(0, y_{1}, …, y_{n}) = 0 only at the origin. Then for an arbitrary point z of the space {g(0, y_{1}, …, y_{n}) = 0} the gradient of this function will be nonzero; assume, for definiteness, that \frac{∂g}{∂y_{1}} (0, y_{1}, …, y_{n}) is not zero at z and consider a transformation from the group D_{H}

x̃ = x, ỹ_{1} = g(x, y_{1}, …, y_{n}), ỹ_{i} = y_{i}, i = 2, …, n,

whose Jacobian is \frac{∂g}{∂y_{1}} (0, y_{1}, …, y_{n}) ≠ 0 and reduces f to the form x^{2}ỹ_{1}, i.e., f has type D_{∞} at the point z.

At the points outside the set g(0, y_{1}, …, y_{n}) = 0 on the singular hyperplane x = 0, consider the element of the group D_{H} determined by

x̃ = x\sqrt{g(x, y_{1}, …, y_{n})}, ỹ = y_{i}, i = 1, …, n,

whose Jacobian equals \sqrt{g(0, y_{1}, …, y_{n})} ≠ 0 and which transforms f to the function x^{2}, i.e., f has type A_{∞} at these points.

(c)⇒(a). Let f be some representative of the germ of a given hyperplane singularity. Define on its domain a sheaf of O_{n+1}-modules as follows:

F(u) = (x^{3})/ (τ(f) ∩ (x^{3})) ,

where (x^{3}) and τ(f) are considered as modules over the ring of holomorphic functions on u ⊂ ℂ^{n+1}, while O_{n+1} is the sheaf of holomorphic functions on ℂ^{n+1}. The sheaf F is coherent, hence we may use the fact that F has support consisting of a single point if and only if dim Γ(F) < ∞, where as usual Γ(F) denotes the space of sections of F over u.

For x ≠ 0 the function f is regular at p = (x, y_{1}, …, y_{n}), and one has dim F_{p} = 0; hence (x^{3}) ≅ (O_{n+1})_{p} and τ(f) ≅ (O_{n+1})_{p}. If x = 0, but (y_{1}, …, y_{n}) does not belong to the space {g(0, y_{1} …, y_{n}) = 0}, then the germ of f is right equivalent to x^{2} under the action of D_{H}, and (x^{2}) ≅ τ(f); hence at this points dim F_{p} = 0. Suppose now that x = 0 and g(0, y_{1}, …, y_{n}) = 0. Then f is right equivalent to the germ of x^{2}y_{1}, if (x, y_{1}, …, y_{n}) ≠ (0, …, 0); hence outside the origin one obtains τ(f) ≅ (x^{3}) and consequently dim_{ℂ} F_{p} = 0. Hence the sheaf F has support 0, whence, by the above remark about one-point supported sheaves, one concludes that dim_{ℂ} (x^{3})/ ((x^{3}) ∩ τ(f)) < ∞. This implies the finiteness of multiplicity of the hyperplane singularity f.

Definition 2.2. A hyperplane singularity f = x^{2}g(x, y_{1}, …, y_{n}) not of type A_{∞} or D_{∞} is called isolated if both g(0, y_{1}, …, y_{n}) and g(x, y_{1}, …, y_{n}) have isolated singularities.

3. Deformations of Hyperplane Singularities

For isolated hyperplane singularities one can construct special deformations having singular points of type A_{1}, A_{∞}, D_{∞}.

Theorem 3.1. Let f : (ℂ^{n+1}, 0) → (ℂ, 0) be an isolated hyperplane singularity f = x^{2}g(x, y_{1}, …, y_{n}). Then there exists a deformation f_{λ}, λ ∈ ℂ^{n+1}, within the class of isolated hyperplane singularities, which has singular points of types A_{∞} and D_{∞} on the hyperplane H and only Morse points outside H, and such a deformation can be given in the form

f_{λ} = x^{2}(g(x, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1}),

where λ_{1}, …, λ_{n+1} are sufficiently small complex numbers.

Proof. Suppose that on the hyperplane H one has

g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} ≠ 0;

then consider the transformation from the group D_{H} given by

x̃ = x\sqrt{g(x, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1}},
ỹ_{i} = y_{i}, i = 1, …, n.

Since \frac{∂_{x}}{∂x}|_{x=0} = \sqrt{g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1}} ≠ 0, the Ja-

cobian of this transformation is nonzero, and in these coordinates the singularity f̃ has type A_{∞}.

Now suppose g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} = 0 and choose λ_{n+1} from Reg(g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n}) and λ_{i}, i = 1, …, n, from Reg grad(g(0, y_{1}, …, y_{n})). This is possible, since f has an isolated hyperplane singularity and hence g(x, y_{1}, …, y_{n}) and g(0, y_{1}, …, y_{n}) have isolated singularities. Supposing, for definiteness, that \frac{∂g}{∂y_{1}} (0, y_{1}, …, y_{n}) + λ_{1} = 0, and consider a transformation of the form

x̃ = x,
ỹ_{i} = g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1},
ỹ_{i} = y_{i}, i = 2, …, n.

This is an element of D_{H}, with the Jacobian \frac{∂g}{∂y_{1}} + λ_{1}, which is nonzero.

In the new coordinates, f_{λ} has only D_{∞} type singular points on the smooth submanifold

{(y_{1}, …, y_{n}) ∈ ℂ | g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} = 0} ,

while outside the submanifold the function f_{λ} has only A_{∞} type singularities on H.

Consider the whole critical set of f_{λ}. It consists of a singular hyperplane H and a set determined by the system of equations

{x ≠ 0 ; 2g + 2λ_{1}y_{1} + ⋯ + 2λ_{n+1} + xg_{x} = 0 ; g_{y_{1}} = −λ_{1} ; ⋮ ; g_{y_{n}} = −λ_{n}  .

At these points Hessf_{λ} has the form

Hessf_{λ} = |\begin{matrix}3xg_{x} + x^{2}g_{xx} & x^{2}g_{xy_{1}} & … & x^{2}g_{xy_{n}} \\ x^{2}g_{y_{1}x} & x^{2}g_{y_{1}y_{1}} & … & x^{2}g_{y_{1}y_{n}} \\ ⋮ & ⋮ & ... & ⋮ \\ x^{2}g_{y_{n}x} & x^{2}g_{y_{n}y_{1}} & … & x^{2}g_{y_{n}y_{n}} \end{matrix} | .

Consequently the set {Hessf_{λ} = 0} does not depend on λ_{1}, …, λ_{n+1} and so for almost all λ_{1}, …, λ_{n+1} the points given by the above system are the Morse ones.

Following Damon [4], one can introduce the notion of a versal deformation of hyperplane singularities. The theorem on deformation implies

Corollary 3.2. A hyperplane singularity possesses a versal deformation if and only if it is isolated, in which case the deformation can be given as F (x, y_{1}, …, y_{n}, λ) = f(x, y_{1}, …, y_{n}) + ∑_{i=1}^{σ} λ_{i}e_{i}(x, y_{1}, …, y_{n}), where σ = codim f, and e_{1}, …, e_{σ} are the representatives for the ℂ-base of the space (x^{2})/τ(f).

4. The Number of Morse Points

Let f ∈ (x^{2}) have an isolated hyperplane singularity; then according to Theorem 3.1, there exists a deformation, having, on H, A_{∞} and D_{∞} type singular points and a certain number s of Morse points outside H. It turns out that this number does not depend on the deformation choice and can be calculated in a purely algebraic way.

Theorem 4.1. The number of Morse points of a deformation of f is calculated by the formula

s = dim_{ℂ} [(x^{2})/(xf_{x}, f_{y_{1}}, …, f_{y_{n}})] .

Proof. Let F : (ℂ^{n+1} × ℂ^{σ}, 0) → (ℂ, 0) be a versal deformation of the singularity f, where f = x^{2}g(x, y_{1}, …, y_{n}). Then F = x^{2}G(x, y_{1}, …, y_{n}, λ), λ ∈ ℂ^{σ}, where G ∈ O_{x,y_{1},...,y_{n},λ} satisfies G|_{λ=0} = g.

Clearly, the number of Morse points of s is obtained as the number of solutions of the following system of equations lying outside the singular hyperplane {x = 0}, for some sufficiently small value of the parameter λ,

F_{x} = 0, F_{y_{1}} = 0, …, F_{y_{n}} = 0,

where F_{x} and F_{y_{i}} are the partial derivatives of the function F with respect to x and y_{i}, respectively.

One can trace a part consisting of values of the parameter λ_{0} which obey the transversality of the intersection of the plane λ = λ_{0} with the singular set of F_{λ_{0}} outside the singular plane {x = 0}; hence by the definition of the intersection index, the number of Morse points coincides with the intersection index of the plane {λ = 0} with the germ of the surface S ⊂ ℂ^{n+1} ×ℂ^{σ} determined as closure of the germ of the set

{F_{x} = 0, F_{y_{1}} = 0, …, F_{y_{n}} = 0, x ≠ 0}.

Since x ≠ 0, one can cancel it, which exactly corresponds to considering only the singularities outside {x = 0}; hence

S = {2G + xG_{x} = 0, G_{y_{1}} = 0, …, G_{y_{n}} = 0} ⊂ ℂ^{n+1} × ℂ^{σ}.

Since the set S is defined only by functions with isolated singularities, one gets by [5]

S = dim_{ℂ} [O_{x,y_{1},...,y_{n},λ}/(2G + xG_{x}, G_{y_{1}}, …, G_{y_{n}})] =

= dim_{ℂ} [O_{x,y_{1},...,y_{n}}/ (O_{x,y_{1},...,y_{n}}(2g + xg_{x}, g_{y_{1}}, …, g_{y_{n}}))] =

= dim_{ℂ} [(x^{2})/(2x^{2}g + x^{3}g_{x}, x^{2}g_{y_{1}}, …, x^{2}g_{y_{n}})] =

= dim_{ℂ} [(x^{2})/(xf_{x}, f_{y_{1}}, …, f_{y_{n}})] .

5. Topology of Isolated Hyperplane Singularities

Let f : (ℂ^{n+1}, 0) → (ℂ, 0) be an isolated hyperplane singularity with the singular set H = {x = 0}, and let f_{λ} be a deformation of the singularity f obtained by Theorem 3.1. Choose ε_{0} > 0 such that for any ε with 0 ≤ ε ≤ ε_{0} one has f^{−1}(0) ⋔ ∂B_{ε}, i.e., the fibre f^{−1}(0) is transversal to the boundary of a ball of radius ε in ℂ^{n+1}. (This is possible since f^{−1}(0) is an algebraic stratified set.) For such ε > 0 there is η(ε) such that f^{−1}(t) ⋔ ∂B_{ε} for any 0 < |t| < η(ε). Fix ε ≤ ε_{0} and consider 0 < η ≤ η(ε) and the restriction

f_{λ} : X_{D} = f^{−1}(D_{η}) ∩ B_{ε} → D_{η},

where D_{η} is a disc of radius η in ℂ.

Lemma 5.1. Let f_{λ} be a deformation of the isolated hyperplane singularity f. Consider the restriction

f_{λ} : X_{D,λ} = f_{λ}^{−1}(D_{η}) ∩ B_{ε} → D_{η},

for any 0 ≤ ∥λ∥ < δ and 0 < |t| < η, where δ and η are sufficiently small numbers. Then the following assertions are valid:

1. f_{λ}^{−1}(t) ⋔ ∂B_{ε}.

2. Fibrations induced over ∂D_{η} by f and f_{λ} are equivalent.

3. X_{D} and X_{D,λ} are homeomorphic.

Proof. At the points of H ∩ ∂B_{ε} one has A_{∞} and D_{∞} type singularities. If z ∈ H ∩ ∂B_{ε} is an A_{∞} type singular point, there exist coordinates (x, y_{1}, …, y_{n}) with f_{λ(s)}(x, y_{1}, …, y_{n}) ∼ x^{2}, where f_{λ(s)} is a oneparameter deformation of the singularity f and the coordinate x depends on λ smoothly. For t ≠ 0 the tangent space f_{λ(s)}^{−1}(t) is obtained from the equation

x_{0}(x − x_{0}) = 0,

i.e., x = x_{0}, which is a hyperplane parallel to H and hence transversal to ∂B_{ε}, as H is preserved under the coordinate transform involved.

Now assume that z ∈ H ∩ ∂B_{ε} is a D_{∞} type singularity; then one has

f_{λ(s)}(x, y_{1}, …, y_{n}) ∼ x^{2}y_{1}.

Hence the tangent space to f_{λ(s)}^{−1}(t) at (x_{0}, y_{1}^{0}, …, y_{n}^{0}) has the form

(x − x_{0})x_{0}y_{1}^{0} + (y_{1} − y_{1}^{0})x_{0}^{2} = 0,

i.e., x = x_{0}, y_{1} = y_{1}^{0}, which is transversal to ∂B_{ε}, since the set y_{1} = y_{1}^{0} coincides with {g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} = y_{1}^{0}}, and this set is compact and intersects ∂B_{ε} ∩ H transversally.

We have thus established that at the points z ∈ H∩∂B_{ε} the transversality condition holds, while at the points from ∂B_{ε}\H the map is a submersion. Since f^{−1}(0) ∩ ∂B_{ε} is compact and transversality is an open property, this implies f_{λ(s)}^{−1}(t) ⋔ ∂B_{ε}, 0 ≤ ∥λ∥ < δ and 0 < |t| < η, which concludes the proof of assertion (1).

Let us prove (2). Consider the mapping

F (x, y_{1}, …, y_{n}, s) = (f_{λ(s)}(x, y_{1}, …, y_{n}), s).

Define

Y_{D,s_{0}} = F^{−1}(D_{η} × [0, s_{0}]) ∩ (B_{ε} × [0, s_{0}])

and the mapping

F_{D,s} : Y_{D,s_{0}} → D_{η} × [0, s_{0}] → [0, s_{0}]

which is well defined for any s ∈ [0, s_{0}] f_{λ(s)} : X_{D,s} → D_{η}; the map F_{D,s} is submersive at the internal points of

F^{−1}(∂D_{η} × [0, s_{0}]) ∩ (intB_{ε} × [0, s_{0}]),

since df_{λ(s)} has a maximal rank over the boundary of D_{η}. The restriction of F_{D,s} to the boundary of

F^{−1}(∂D_{η} × [0, s_{0}]) ∩ (∂B_{ε} × [0, s_{0}])

is also a submersion, since f_{λ(s)}^{−1}(t) ⋔ ∂B_{ε} for any t ∈ D_{η}, λ(s), s ∈ [0, s_{0}]. Now one can apply the theorem of Ehresmann [6] to find that F_{D,s} is a trivial fibration over the contractible set [0, s_{0}] and, consequently, for any s the maps f_{λ(s)} determine equivalent fibrations over the boundary of D_{η}.

Finally, (3) follows from Thom’s lemma on isotopy [7] needed for describing a homotopy type of the Milnor fibre.

Now let us turn to the main construction.

Let b_{1}, …, b_{σ} be Morse points for the deformation f_{λ} := f̃ with critical values f̃(b_{1}), …, f̃(b_{σ}). Define B_{1}, …, B_{σ} to be disjoint 2n + 2-dimensional balls in ℂ^{n+1} centered at b_{1}, …, b_{σ} respectively, and let D_{1}, …, D_{σ} be disjoint 2-dimensional discs centered at f̃(b_{1}), …, f̃(b_{σ}). Let

f̃: B_{i} ∩ f̃^{−1}(D_{i}) → D_{i}, i = 1, …, σ,

be locally trivial Milnor fibrations satisfying the transversality condition

f̃^{−1}(t) ⋔ ∂B_{i}, t ∈ D_{i}, i = 1, 2, …, σ.

Choose furthermore a small cylinder B_{0} around H and a 2-dimensional disc D_{0} ⊂intf̃(B_{0}), satisfying

∂B_{0} ⋔ f̃^{−1}(t) for t ∈ D_{0}.

First of all, let us investigate the fibration f̃: B_{0} ∩ f̃^{−1}(D_{0}) → D_{0}. Its fibre f̃^{−1}(t) ∩ B_{0} can be in turn fibred over B_{ε} ∩ (H\U) using the projection π, where U is a tubular neighborhood of the smooth nonsingular subvariety

g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} = 0,

with π(x, y_{1}, …, y_{n}) = (0, y_{1}, …, y_{n}). This projection may have singularities. To describe them, consider the mapping

φ_{f}: f̃^{−1}(D_{0}) ∩ B_{0} → ℂ × ℂ^{n},

defined by

φ_{f}(x, y_{1}, …, y_{n}) = (f̃(x, y_{1}, …, y_{n}), y_{1}, …, y_{n}).

The Jacobi matrix of this mapping has the form

(\begin{matrix}\frac{∂_{f}}{∂x} & \frac{∂_{f}}{∂y_{1}} & ⋯ & \frac{∂_{f}}{∂y_{n}} \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \end{matrix} ) .

Hence the critical set of φ_{f} is given by the equation

\frac{∂f̃}{∂x} = 0.  (Γ)

The hypersurface Γ contains the hyperplane H, i.e., Γ = H ∪ Γ_{f}; and the projection

π := f̃^{−1}(t) ∩ B_{0} → B_{ε} ∩ (H\U)

is smooth outside Γ_{f}.

Lemma 5.2. The hypersurface Γ_{f} meets H at D_{∞} type points.

Proof. We shall prove that if Γ_{f} meets H at D_{∞} type points, then Γ_{f} coincides with H.

Let f̃ = x^{2}g̃, where g̃ = g(x, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} and g̃(0, …, 0) ≠ 0. Then

\frac{∂f̃}{∂x} = 2xg̃ + x^{2}g̃

and since g̃(0, …, 0) ≠ 0, x can be expressed by the module x^{2}. Hence

(x) ⊂ (\frac{∂_{f}}{∂x}) + (x^{2}).

By Nakayama’s lemma this implies (x) = (\frac{∂_{f}}{∂x}). Consequently, the set

defined by the equality \frac{∂_{f}}{∂x} = 0 coincides with the set x = 0, i.e., Γ_{f} = H. This concludes the proof.

The lemma implies that the projection π is a locally trivial fibration outside D_{∞} type singular points, and its fibre is given by the equation f̃ = t. And since the set π^{−1}(B_{ε} ∩ (H\U)) is compact and consists of A_{∞} type singular points, (f̃ ∼ x^{2}), the fibre locally consists of two points. By compactness of the aforementioned set one can choose a radius for B_{0} in such a way that π will define a double covering over B_{ε} ∩ (H\U).

Let us introduce the space B_{ε} ∩ (H\U) = B̃_{ε}\U, where B̃_{ε} is a 2ndimensional ball in the space ℂ^{n} and U is a small tubular neighborhood of the smooth nonsingular variety

Ṽ = {g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} = 0};

obviously, B̃_{ε}\Ṽ and B̃_{ε}\U are of the same homotopy type.

Our aim is to investigate a homotopy type of the complement to Ṽ. The homology of that space is easily computed from Leray’s exact ho-

mological sequence

⟶^{δ_{∗}} H_{q}(B̃_{ε}\Ṽ) ⟶^{j_{∗}} H_{q}(B̃_{ε}) ⟶^{i_{∗}} −H_{q−2}(Ṽ) ⟶^{δ_{∗}} H_{q−1}(B̃_{ε}\Ṽ) ⟶^{j_{∗}}

obtained from Leray’s exact cohomological sequence [8] by the Poincaré duality

⟶^{δ_{∗}} H^{p}(B̃_{ε}\Ṽ) ⟶^{j_{∗}} H^{p}(B̃_{ε}) ⟶^{i_{∗}} H^{p}(Ṽ) ⟶^{δ_{∗}} H^{p+1}(B̃_{ε}\Ṽ) ⟶^{j_{∗}},

where j_{∗} are induced by the embedding j : B̃_{ε}\Ṽ ⊂ B̃_{ε}, i_{∗} is the intersection of cycles from H^{∗}(B̃) and H^{∗}(Ṽ), and δ_{∗} is the Leray coboundary.

Since Ṽ is a smooth nonsingular submanifold of real codimension two which is homotopy equivalent to the wedge of μ(g) copies of the n − 1spheres, where μ(g) is the Milnor number of the isolated singularity g, one obtains H_{0}(Ṽ) = ℤ, H_{n−1}(Ṽ) = ℤ^{μ(g)}, and H_{i}(Ṽ) = 0 for i ≠ 0, n − 1. Taking this in account, one obtains from Leray’s exact homological sequence that

H_{1}(B̃_{ε}\Ṽ) = ℤ, H_{n}(B̃_{ε}\Ṽ) = ℤ^{μ(g)} and H_{i}(B̃_{ε}\Ṽ) = 0, if i ≠ 0, 1, n.

[figure: U; D_{0}; t; t_{0}; v_{1}; v_{2}; v_{μ}; t_{1}; t_{2}; t_{μ}; Figure 1]

To find the homotopy type of the Milnor fibre, let us prove

Lemma 5.3. For sufficiently small t the complement B̃_{ε}\Ṽ of the nonsingular hypersurface Ṽ inside the ball B̃_{ε} is homotopy equivalent to the space obtained from the direct product S^{1} × Ṽ by filling all the vanishing spheres S_{i}^{n−1}, i = 1, 2, …, μ, with n-dimensional balls, in one of the fibres {t_{0}} × V for some t_{0} ∈ S^{1}, where μ is the Milnor number of the isolated singularity g(0, y_{1}, …, y_{n}).

Proof. Take a small neighborhood u of the point t_{0} ∈ D_{0} and let ū be its closure. Let t_{0} ∈ ∂u. Connect the critical values t_{i} of the mapping g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} with t by disjoint paths v_{i}(τ), where v_{i}(0) = t_{i} and v_{i}(1) = t_{0} (see Figure 1).

The disc D_{0}\t is a deformation retract of the set ⋃_{i=1}^{μ} v_{i}(τ)∪(ū\t). Since g̃(0, y_{1}, …, y_{n}) = g(0, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1} is a locally trivial Milnor fibration in the ball B̃_{ε}, by the homotopy lifting property one obtains that g^{−1}(D_{0}\t) is homotopy equivalent to g^{−1} ⋃_{i=1}^{μ} v_{i}(τ) ∪ (ū\t). Restrictions of the locally trivial fibration on the contractible set are trivial; consequently g^{−1}(ū\t) is a total space of the trivial fibration over ū\t, i.e., over a circle, with fibre {g̃(0, y_{1}, …, y_{n}) = t}∩B̃_{ε}, diffeomorphic to Ṽ, hence g^{−1}(ū\t) is homotopy equivalent to the direct product S^{1} × Ṽ.

Following [9], we shall show that the space Y = g^{−1}(⋃_{i=1}^{μ} v(τ)) is obtained, up to homotopy type, from the fibre V by filling all the spheres Δ_{i}^{n−1}, i = 1, 2, …, μ with n-dimensional balls T_{i}. Let

S_{i}(t) : S_{i}^{n−1} ⟶ S_{i}(t) ⊂ F_{v_{i}(t)} (0 ≤ t ≤ 1)

be the family of maps of the standard n − 1-dimensional sphere S_{i}^{n−1} (the index i counts copies of the sphere), determining the vanishing cycle Δ_{i} = S_{i}(1)(S_{i}(0) : S_{i}^{n−1} → P_{i}). Let T_{i} be the n-dimensional ball constructed as the cone over the sphere S_{i}^{n−1},

T = [0, 1] × S_{i}^{n−1}/{0} × S_{i}^{n−1}.

The space Ṽ ∪_{Δ_{i}} {T_{i}} obtained from the fibre Ṽ by filling in the vanishing cycles Δ_{i} with n-balls T_{i} is the quotient of Ṽ ∪⋃_{i=1}^{μ} T_{i} under the equivalence relation

S_{i}(1)(a) ∼ (1, a), a ∈ S_{i}^{n−1}, (1, a) ∈ T_{i}, i = 1, …, μ,

and its mapping to the space Y can be written as

φ(x) = x for x ∈ Ṽ ⊂ Y; φ(t, a) = S_{i}(t)(a) for (t, a) ∈ T_{i}, 0 ≤ t ≤ 1,

a ∈ S_{i}^{n−1}. Let us construct the inverse mapping ψ : Y → Ṽ ∪_{Δ_{i}} {T_{i}} by putting ψ(y) = y for y ∈ Ṽ, ψ(y) = (t, a) for y ∈ Ṽ_{v_{i}(t)}, if under the homotopy equivalence between Ṽ_{v_{i}(t)} and the wedge ⋁_{i=1}^{μ} Δ_{i} the point y passes S_{i}(t)(a) for a ∈ S_{i}^{n−1}. Consider the composition

ψ ∘ φ : Ṽ ∪_{Δ_{i}} {T_{i}} → Ṽ ∪_{Δ_{i}} {T_{i}}.

Then ψ(φ(x)) = x for x ∈ Ṽ and ψ ∘φ is homotopic to the identity mapping of Ṽ ∪_{Δ_{i}} {T_{i}}, since Ṽ_{v_{i}(t)} for 0 < t ≤ 1 is homotopy equivalent to the wedge of the spheres Δ_{i}, while Ṽ_{v_{i}(0)} – to that without one of them (the one vanishing along v_{i}). Similarly, φ ∘ ψ : Y → Y is homotopic to Id_{y}, which proves the homotopy equivalence.

The space (ū\t) ∪ ⋃_{i=1}^{μ} v_{i}(τ) is the amalgam [10] of the diagram

⋃_{i=1}^{μ} v_{i}(τ) ⟵ {t_{0}} ⟶ ū\t,

whence the inverse image of this space under the mapping g̃ will be the amalgam of the inverse image of the diagram [10], i.e., of the diagram

g^{−1} (⋃_{i=1}^{μ} v(τ)) ⟵ Ṽ_{t_{0}} ⟶ g^{−1}(ū\t),

where Ṽ_{t_{0}} is diffeomorphic to Ṽ. We arrived at the amalgam of

Ṽ ∪_{Δ_{i}} ⋃_{i=1}^{μ}{T_{i}} ⟵ Ṽ ⟶ Ṽ × S^{1},

which is the space Ṽ × S^{1} with all the vanishing spheres in the fibre over t_{0} filled with n-balls.

An even more general fact can be proved.

Proposition 5.4. Let g : (ℂ^{n}, 0) → (ℂ, 0) be a germ of an isolated singularity; then V = {g = t} ∩ B_{ε}, where B̃_{ε} is a small ball in ℂ^{n}, is, for small t, homotopy equivalent inside B̃_{ε} to S^{1} × V with n-dimensional balls filling in all the vanishing spheres of one of its fibres V.

Proof. Let g̃ be the morsification of the isolated singularity g in the ball B̃_{ε}, having nondegenerate critical points p_{i} with different critical values t_{i} = g̃(p_{i}); then V = {g = t} ∩ B̃_{ε} is diffeomorphic to V.

We shall show that B̃_{ε}\V is homotopy equivalent to B̃_{ε}\Ṽ, which by Lemma 5.3 will imply our proposition.

Let ε > 0 and δ > 0 be chosen in such a way that g^{−1}(t) = g_{λ}^{−1}(t) is transversal to ∂B̃_{ε} for any 0 ≤ ∥λ∥ ≤ δ. Consider the mapping given by F (x, λ) = (g_{λ}(x), λ) and its restriction

F_{t,δ} = F^{−1}({t} × [0, δ]) ∩ (B̃_{ε} × [0, δ]) → {t} × [0, δ] → [0, δ].

The mapping F_{t,δ} is submersive at the interior points of

F^{−1}({t} × [0, δ]) ∩ (intB̃_{ε} × [0, δ]),

since dg_{λ} has a maximal rank over {t}. Moreover, the restriction of F_{t,δ} on the boundary of F^{−1}({t}×[0, δ])∩(δB̃_{ε}×[0, δ]) is submersive as g^{−1}(t) ⋔ ∂B̃_{ε} for any λ ∈ [0, δ]. By Ehresmann’s theorem [6] one obtains a locally trivial fibration over the contractible space [0, δ], which is trivial. Consequently, V is diffeomorphic to the fibre Ṽ.

Let T be a tubular neighborhood of the submanifold V. Choose 0 < δ_{1} < δ sufficiently small for the fibre F^{−1}(δ_{1}) to lie inside T. Making δ_{1} still smaller, one can make T into a tubular neighborhood for F^{−1}(δ_{1}) too. This will imply that B̃_{ε}\T is homotopy equivalent to B̃_{ε}\V and B̃_{ε}\F^{−1}(δ_{1}) simultaneously; hence B̃_{ε}\V is homotopy equivalent to B̃_{ε}\F^{−1}(δ_{1}). By the compactness of [0, δ] in a finite number of steps one obtains the homotopy equivalence of B̃_{ε}\V to B̃_{ε}\Ṽ.

Corollary 5.5. The complement B̃_{ε}\V is homotopy equivalent to a wedge of S^{1} and μ copies of the n-sphere S^{n}, where μ is the Milnor number of the isolated singularity g(y_{1}, …, y_{n}).

Proof. By Lemma 5.4, B̃_{ε}\V is homotopy equivalent to the direct product S^{1} × V, where over a point t_{0} ∈ S^{1} the fibre V is contracted to a point, hence such a space is homotopy equivalent to the suspension of V with the identified vertices, i.e., suspension of the wedge of (n − 1)-spheres S_{i}^{n−1}, i = 1, …, μ(g), with identified vertices, which is obviously a wedge of μ(g(y_{1}, …, y_{n})) copies of the n-sphere and a circle (see Figure 2).

[figure: ≃; ≃; ≃; Figure 2]

We obtain that B_{ε} ∩ (H\U), where U is a tubular neighborhood of the smooth nonsingular subvariety g(0, y_{1}, …, y_{n})+λ_{1}y_{1} +⋯+λ_{n}y_{n} +λ_{n+1}, is homotopy equivalent to the wedge of a circle S^{1} and μ = μ(g(0, y_{1}, …, y_{n})) copies of the n-sphere and one has a double covering

π : f̃^{−1}(t) ∩ B_{0} → B_{ε} ∩ (H\U).

Represent B_{ε} ∩ (H\U) as a union of V_{1} and V_{2}, where V_{1} has homotopy type of a circle, while V_{2} has homotopy type of a wedge of n-spheres, and where V_{1}∩V_{2} is contractible. Since π is a double cover, π^{−1}(V_{1}) is homotopy equivalent to the circle S^{1}, while over the simply connected space V_{2} the covering π is trivial, hence π^{−1}(V_{2}) consists of a disjoint union of two wedges of μ = μ(g(0, y_{1}, …, y_{n})) copies of the n-sphere S^{n}, and, since π^{−1}(V_{1}) is a doubly winded circle, one obtains that f̃^{−1}(t)∩B_{0} is homotopy equivalent to the wedge of S^{1} and 2μ(g(0, y_{1}, …, y_{n})) copies of the n-spheres S^{n}. Hence we arrive at

Lemma 5.6. Let an isolated hyperplane singularity be not of type A_{∞}; then the fibre of the Milnor fibration in a small cylinder B_{0}

X̃_{t} = f̃^{−1}(t) ∩ B_{0}

is homotopy equivalent to the wedge of a circle S^{1} and 2μ copies of the nsphere, where μ is the Milnor number of the isolated singularity g(0, y_{1},...,y_{n}).

We have already established that the critical set f̃ consists of

(a) the hyperplane H;

(b) Morse points b_{1}, …, b_{σ}.

We have defined the small discs D_{i} around the points f̃(b_{i}), the disjoint disc balls B_{i} over the points b_{i} and the cylinder B_{0} over the singular set H. Now choose t_{i} ∈ ∂D_{i}, t ∈ D_{η} and a system of separate paths γ_{0}, γ_{1}, …, γ_{σ} from the point t to t_{i} (see Figure 3)

[figure: B_{0}; H; B_{σ}; B_{2}; B_{1}; f̃; D_{1}; t_{1}; D_{2}; t_{2}; D_{0}; D_{σ}; t_{σ}; t; γ_{1}; γ_{2}; γ_{σ}; t_{0}; γ_{0}; B_{ϵ}; D_{η}; Figure 3]

Let us introduce the notations

D = ⋃_{i=0}^{σ} D_{i}, X_{A} = f^{˜ −1}(A) ∩ B_{ε}, A ⊂ D_{η}, X̃_{s} = f^{−1}(s) ∩ B_{ε}, s ∈ D_{η}.

Define suitable neighborhoods of the critical sets as follows:

(1) for the hyperplane H : r_{0}(z) = |x|^{2} and let

B_{0}(ε̃) = {z ∈ B_{ε} | r_{0}(z) ≤ ε̃, ε̃ ≪ ε};

(2) for the Morse points b_{i} : r_{i}(z) = |z − b_{i}|^{2} and

B_{i}(ε̃) = {z ∈ B_{ε} | r_{i}(z) ≤ ε̃, ε̃ ≪ ε}.

As shown by Lemma 5.1, for f̃ there exists ε_{0} such that for any 0 < ε̃ ≤ ε̃_{0} the set X_{0} is transversal to ∂B_{0}(ε̃) and there is ε̃_{i} such that for any ε̃ with 0 < ε̃ ≤ ε̃_{i} the set X_{f(b_{i})} is transversal to ∂B_{i}(ε̃), i = 1, 2, …, σ, as the points b_{i} are the Morse ones [11].

Since the transversality condition is open, for any 0 < ε̃ ≤ ε̃_{i}, i = 0, 1, …, σ, there exists τ_{i} = τ_{i}(ε̃), such that X_{i} ⋔ ∂B_{i}(ε) for any

0 < |t − f̃(b_{i})| ≤ τ_{i}, i = 0, 1, …, σ, where f(b_{0}) = 0.

Now fix ε̃ > 0 and τ > 0 and require B_{i}(ε̃) and D_{i}(τ) to be disjoint balls and discs, respectively.

Denote

B_{i} = B_{i}(ε), D_{i} = D_{i}(τ), E^{i} = B_{i} ∩ X_{D_{i}},
E = B_{ε} ∩ X_{D_{η}}, F^{i} = B_{i} ∩ X_{t_{i}}, F = B_{ε} ∩ X_{t}.

We shall need the isomorphism (see [2])

H_{∗}(E, F) ≃ ⨁_{i=0}^{σ} H_{∗}(E^{i}, F^{i}).

This implies that the homology groups H_{∗}(E, F) are direct sums of homology groups over all critical sets. The situation for the Morse points b_{1}, …, b_{σ} is very well known [11]

H_{k+1}(E^{i}, F^{i}) = H_{k}(F^{i}) = \begin{cases}ℤ, & k = 0, n, \\ 0, & k ≠ n. \end{cases}

Hence we finally obtain

\begin{cases}H_{n+1}(E, F) = H_{n}(E^{0}, F^{0}) ⊕ ℤ^{σ}, \\ H_{k}(E, F) = H_{k}(E^{0}, F^{0}), & k ≠ n + 1. \end{cases}

To calculate the homology groups H_{k}(E^{0}, F^{0}), write down an exact sequence of the pair (E^{0}, F^{0}) [12]

⋯ → H_{k}(E^{0}) → H_{k}(E^{0}, F^{0}) → H_{k−1}(F^{0}) → H_{k−1}(E^{0}) → ….

The spaces E^{0} = E ∩ B_{0} and E are homotopy equivalent, while E is contractible [11]. Therefore

0 → H_{k}(E^{0}, F^{0}) → H_{k−1}(F^{0}) → 0.

This implies H_{k}(E^{0}, F^{0}) ≅ H_{k−1}(F^{0}).

Similarly, one obtains H_{k}(E, F) = H_{k−1}(F) so that we have

\begin{cases}H_{n}(F) = H_{n}(F_{0}) ⊕ ℤ^{σ} & , \\ H_{k−1}(F) = H_{k−1}(F_{0}) & , k ≠ n. \end{cases}

By Lemma 5.6 we obtain

Proposition 5.7. Homology groups of the Milnor fibre are calculated as follows:

\begin{cases}H_{0}(F) = ℤ, \\ H_{1}(F) = ℤ, \\ H_{n}(F) = ℤ^{2μ+σ}, \\ H_{i}(F) = 0, & i ≠ 0, 1, n, \end{cases}

where μ = μ(g(0, y_{1}, …, y_{n})) is the Milnor number of an isolated singularity and σ is the number of the Morse critical points for f̃, which by Theorem 4.1 equals

σ = dim_{ℂ} [(x^{2})/(xf_{x}, f_{y_{1}}, …, f_{y_{n}})] .

[figure: D_{η}; D_{0}; D_{σ}; D_{1}; t_{σ}; t_{1}; t_{0}; ψ_{σ}; ψ_{1}; Figure 4]

Now let us determine a homotopy type of the fibre F. We have

Theorem 5.8. Let f be an isolated hyperplane singularity (not of A_{∞} type). Then the Milnor fibre of f is homotopy equivalent to the wedge of a circle S^{1} and 2μ + σ copies of the n-dimensional sphere, where μ = μ(g) is the Milnor number of the isolated singularity g(0, y_{1}, …, y_{n}), while σ is the number of Morse points of the deformation f̃.

Proof. Let D_{η}, D_{0}, …, D_{σ} and B_{ε}, B_{0}, …, B_{σ} be as before. Let t be a point in ∂D_{0}, and choose a system of separate paths ψ_{1}, …, ψ_{σ} from t to D_{1}, …, D_{σ} (see Figure 4).

Applying the Morse lemma [13] to |f| which has b_{1}, …, b_{σ} as Morse points of the index n + 1, one obtains the homotopy equivalences

(X_{D_{η}}, X_{t}) ≅ (X_{D_{0}} ∪_{ψ_{1}} e_{1}^{n+1} ∪ ⋯ ∪ ψ_{σ}e_{σ}^{n+1}, X_{t}),
(X_{D_{0}}, X_{t}) ≃ (X_{D_{0}} ∩ B_{0} ∪ X_{t}, X_{t}).

Let φ_{1}, φ_{2}, …, φ_{2μ} : S^{n} → F^{0} and φ_{0} : S^{1} → F^{0} represent the generators of π_{n}(F^{0}) and π_{1}(F^{0}), respectively. Use φ_{0}, …, φ_{2μ} to attach a 2-cell and n + 1-cells e_{0}^{2}, e_{1}^{n+1}, …, e_{2μ}^{n+1} to F_{0} = X_{t} ∩ B_{0}.

The inclusion X_{t} ∩ B_{0} ⊂ X_{D} ∩ B_{0} extends to the homotopy equivalence

X_{t} ∪_{φ_{0}} e_{1}^{n+1} ∪ ⋯ ∪_{φ_{2μ}} e_{2μ}^{n+1} → X_{D_{0}} ∩ B_{0}

as both spaces are contractible. This gives the homotopy equivalence

(X_{D_{0}}, X_{t}) ≃ (X_{t} ∪_{φ_{0}} e_{1}^{n+1} ∪ ⋯ ∪_{φ_{2μ}} e_{2μ}^{n+1}, X_{t}).

Finally, we obtain the contractible space X_{D_{η}} from the fibre X_{t} by attaching σ + 2μ copies of the n + 1-cell and one 2-cell, and since attaching n + 1cells does not change homotopy groups in dimension n − 1, it follows that X_{t} ∪_{φ_{0}} e_{0}^{2} is n − 1-connected.

The homology group H_{n}(X_{t} ∪_{φ_{0}} e_{1}^{n+1}) must be free abelian, since any torsion elements would give rise to nonzero elements in the (n + 1)th cohomology group, which would contradict the fact that X_{t} ∪_{φ_{0}} e_{0}^{2} is an ndimensional CW-complex. According to Hurewich’s theorem [12] there is an isomorphism π_{n}(X_{t} ∪_{φ_{0}} e_{0}^{2}) ≃ H_{n}(X_{t} ∪_{φ_{0}} e_{0}^{2}). Hence π_{n}(X_{t} ∪_{φ_{0}} e_{0}^{2}) is a free abelian group, and one can choose a finite number of maps

(S_{i}^{n−1}, basepoint) → (X_{t} ∪_{φ_{0}} e_{0}^{2}, basepoint)

representing the basis in the group π_{n}(X_{t}∪_{φ_{0}} e_{0}^{2}). Wedging these maps gives the map

S^{n} ⋁ ⋯ ⋁ S^{n} → X_{t} ∪_{φ_{0}} e_{0}^{2}

inducing an isomorphism in homology, which, consequently, by Whitehead’s theorem [12], is a homotopy equivalence. Therefore X_{t} ∪_{φ_{0}} e_{0}^{2} is homotopy equivalent to the wedge S^{n} ⋁ ⋯ ⋁ S^{n} of n-spheres.

This implies that π_{1}(X_{t}) is generated by one element, and since H_{1}(X_{t}) = ℤ, we obtain π_{1}(X_{t}) = ℤ.

Consider the map

(S^{1} ∨ S^{n} ∨ ⋯ S^{n}, basepoint) → (X_{t}, basepoint)

defined as follows: the sphere S^{n} maps to X_{t} as the representative of a generator in the homology group H_{n}(X_{t}), whereas S^{1} maps as the representative of the generator of π_{1}(X_{t}). The constructed map induces an isomorphism of homology groups and fundamental groups, hence, by Whitehead’s theorem, the constructed map is a homotopy equivalence. This concludes the proof of the theorem.

6. Hypersurface Singularities of Transversal Type A_{k}

In this section the germs of analytic functions f : (ℂ^{n+1}, 0) → (ℂ, 0) of n + 1 complex variables shall be considered, having the hyperplane H = {(x, y_{1}, …, y_{n}) | x = 0} as their singular set and representable in the form f = x^{k}g(x, y_{1}, …, y_{n}), where k > 2.

Let us introduce some definitions.

Definition 6.1. A singular point z in H is called an A_{k∞} type singular point if in some neighborhood U of the point z there exists a local coordinate system (x, y_{1}, …, y_{n}) such that

H = {x = 0}, x(z) = 0, y_{i}(z) = 0, i = 1, 2, …, n,

and in U the identity f = x^{k} holds.

Definition 6.2. A singular point z in H is called a D_{k∞} type singular point if in some neighborhood U of the point z there exists a local coordinate system (x, y_{1}, …, y_{n}) such that

H = {x = 0}, x(z) = 0, y_{i}(z) = 0, i = 1, 2, …, n,

and in U the identity f = x^{k}y_{1} holds.

Definition 6.3. The codimension of the singularity of a germ is called

codim f = dim_{ℂ} [(x^{k})/τ(f)] .

Definition 6.4. A singularity f ∈ (x^{k}) is called an isolated hyperplane singularity of transversal type A_{k} if codimf < +∞.

Similarly to the isolated hyperplane singularity case one can prove

Theorem 6.5. Let f ∈ (x^{k}) be not of type A_{k∞} or D_{k∞}, and, moreover, let the germ g(0, y_{1}, …, y_{n}) from the representation f = x^{k}g(x, y_{1}, …, y_{n}) be not identically zero; then the following assertions are equivalent:

(a) codim f is finite;

(b) the function g(x, y_{1}, …, y_{n}) has an isolated singularity at zero;

(c) outside the points with g(0, y_{1}, …, y_{n}) = 0 the germ f has type A_{k∞}, while at the points with g(0, y_{1}, …, y_{n}) = 0, except for the origin, it has D_{k∞} type singular points.

Theorem 6.6. Let f ∈ (x^{k}) have an isolated hyperplane singularity of transversal type A_{k} on the hyperplane H; then there exists a deformation f̃ of the form

f̃ = x^{k}(g(x, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n} + λ_{n+1}),

where

λ_{n+1} ∈ Reg(g(x, y_{1}, …, y_{n}) + λ_{1}y_{1} + ⋯ + λ_{n}y_{n})

and

λ_{i} ∈ Reg grad g(0, y_{1}, …, y_{n})),

satisfying the condition: f̃ has only A_{k∞} and D_{k∞} type singular points on H and only Morse type singular points outside H. Moreover, the number σ of Morse points is calculated by the formula

σ = dim_{ℂ} [(x^{k})/(xf_{x}, f_{y_{1}}, …, f_{y_{n}})] .  (1)

Lemma 6.7. The Milnor fibre f^{−1}(t)∩B_{0}, where B_{0} is a cylinder around the singular set H, admits a k-fold covering of a wedge of the circle S^{1} and μ copies of the n-sphere S^{n}, where μ = μ(g(0, y_{1}, …, y_{n})) is the Milnor number of the isolated singularity g(0, y_{1}, …, y_{n}).

[figure: ≅; Figure 5]

This enables the proof of

Lemma 6.8. The Milnor fibre f^{−1}(t) ∩ B_{0} has the homotopy type of a wedge of the circle S^{1} and k · μ copies of the n-sphere S^{n} (see Figure 5).

This implies

Theorem 6.9. The Milnor fibre of the isolated hyperplane singularity of transversal type A_{k} in the ball B_{ε} ⊂ ℂ^{n+1} is homotopy equivalent to the wedge of a circle and μk + σ copies of the n-sphere, where μ is the Milnor number of the isolated singularity g(0, y_{1}, …, y_{n}), while σ is the number of Morse points calculated by formula (1).

Acknowledgment

The author acknowledges the financial aid of INTAS in the framework of the project “Investigations in Singularity Theory” #4373.

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(Received 11.04.1995)

Author’s address: Department of Mathematics No. 99 Georgian Technical University 77, M. Kostava St., Tbilisi 380075 Georgia