New York Journal of Mathematics New York J. Math.

New York Journal of Mathematics

New York J. Math.25(2019) 362–373.

Degree product formula in the case of a finite group action

Piotr Bartłomiejczyk, Bartosz Kamedulski and Piotr Nowak-Przygodzki

Abstract. Let V, W be finite dimensional orthogonal representations of a finite groupG. The equivariant degree with values in the Burnside ring ofG has been studied extensively by many authors. We present a short proof of the degree product formula for local equivariant maps onV andW.

Contents

Introduction 362

1. Basic definitions 363

1.1. Local maps 363

1.2. Equivariant maps 364

1.3. Otopies 364

1.4. G-actions 364

1.5. Splitting of C_G[Ω, V] 364

1.6. Burnside ring 365

1.7. Local cross sections of a vector bundle 366

2. Degreedeg_G 366

3. Main result 366

4. Standard and polystandard maps 367

5. Proof of Main Theorem 368

Appendix A. 372

References 372

Introduction

One of the basic properties of the topological degree is the product prop- erty. Recall that a continuous map from an open subset of Rⁿ into Rⁿ is

Received August 28, 2018.

2010Mathematics Subject Classification. Primary: 55P91; Secondary: 54C35.

Key words and phrases. Equivariant degree, Burnside ring, product property.

ISSN 1076-9803/2019

362

calledlocal if its set of zeros is compact. For such maps the classical Brouwer degree degis well-defined and the product property holds. Namely,

Product property ([6, Prop. 8.7]). Let f:D_f ⊂R^m→R^m andf⁰:D_f⁰ ⊂ Rⁿ→Rⁿbe local maps. Then f×f⁰:D_f×D_f⁰ →R^m+n is also a local map and

deg(f×f⁰) = degf ·degf⁰.

Our main goal is to present a short proof of an equivariant version of the product formula for equivariant local maps in the case of a finite group action. In that case the formula has an analogous form

deg_G(f×f⁰) = deg_Gf·deg_Gf⁰,

but since the equivariant degree deg_G has its values in theBurnside ring of a finite group G, the multiplication on the right side of the formula takes place in this Burnside ring. It is worth pointing out that in [7] the authors proved the equivariant product formula in much more general setting i.e. in the case of a compact Lie group action. Unfortunately, this proof seems to be rather sketchy in some parts. We hope that our proof has the advantage of being straightforward and complete and can be seen as the first step towards proving the general case.

The paper is organized as follows. Section1contains preliminaries. In Sec- tion2we recall the concept of the equivariant degreedeg_G. Our main result is stated in Section3. In Section 4we introduce standard and polystandard maps and study their properties needed in the next section. Finally, Section 5 contains the proof of our main result.

1. Basic definitions

1.1. Local maps. The notationAbB means thatA is a compact subset ofB. For a topological spaceX, we denote byτ(X)the topology onX. For any topological spaces X and Y, let M(X, Y) be the set of all continuous maps f:Df → Y such that Df is an open subset of X. Let Rbe a family of subsets ofY. We define

Loc(X, Y,R) :={f ∈ M(X, Y)|f⁻¹(R)bDf for all R∈ R }.

We introduce a topology inLoc(X, Y,R)generated by the subbasis consisting of all sets of the form

• H(C, U) := {f ∈ Loc(X, Y,R) | C ⊂ D_f, f(C) ⊂ U} for C b X and U ∈τ(Y),

• M(V, R) := {f ∈ Loc(X, Y,R) | f⁻¹(R) ⊂ V } for V ∈ τ(X) and R∈ R.

Elements of Loc(X, Y,R) are called local maps. The natural base point of Loc(X, Y,R)is the empty map. Let tdenote the union of two disjoint local maps. Moreover, in the case when R = {{y}} we will write Loc(X, Y, y) omitting double curly brackets. For more details we refer the reader to [4].

1.2. Equivariant maps. Assume thatV is a real finite dimensional orthog- onal representation of a finite groupG. Let X be an arbitraryG-space. We say thatf:X→V isequivariant, iff(gx) =gf(x)for allx∈Xandg∈G.

We will denote by C_G(X, V) the space {f ∈Loc(X, V,0)|f is equivariant}

with the induced topology. Assume thatΩis an open invariant subset ofV. Elements of C_G(Ω, V) are calledequivariant local maps.

1.3. Otopies. LetI = [0,1]. We assume that the action ofGonIis trivial.

Any element of C_G(I×Ω, V) is called an otopy. Each otopy corresponds to a path in C_G(Ω, V) and vice versa. Given an otopy h: Λ⊂ I×Ω→ V we can define for eacht∈I:

• sets Λ_t={x∈Ω|(t, x)∈Λ},

• maps ht: Λt→V withht(x) =h(t, x).

In this situation we say thath0andh1areotopic. Otopy gives an equivalence relation onC_G(Ω, V). The set of otopy classes will be denoted byC_G[Ω, V].

Remark 1.1. Observe that iff ∈ C_G(Ω, V)andU is an open invariant subset of D_f such that f⁻¹(0) ⊂ U, then f and fU are otopic. In particular, if f⁻¹(0) =∅ thenf is otopic to the empty map.

1.4. G-actions. If H is a subgroup ofGthen

• (H)stands for the conjugacy class of H,

• N H is the normalizer ofH inG,

• W H is the Weyl group of H i.e. W H =N H/H.

Recall thatG_x ={g∈G|gx=x}. We define the following subsets ofV: V^H ={x∈V |H⊂G_x},

Ω_H ={x∈Ω|H =G_x}, Ω_(H)={x∈Ω|(H) = (Gx)}.

The set Iso(Ω) := {(H) |H is a closed subgroup ofGand Ω_H 6=∅} is par- tially ordered. Namely,(H)≤(K) if H is conjugate to a subgroup of K.

We will make use of the following well-known facts:

• V^H is a linear subspace of V and an orthogonal representation of W H,

• Ω_H is open inV^H,

• the action of W H onΩH is free,

• Ω_(H) is aG-invariant submanifold ofΩ,

• if (H)is maximal in Iso(Ω) thenΩ_(H) is closed inΩ.

1.5. Splitting of C_G[Ω, V]. Let Ω be an open invariant subset of a real finite dimensional orthogonal representation of a compact Lie groupG. As- sume that orbit types appearing in Ωare indexed (according to the partial order) by natural numbers1,2, . . . ,m. Recall two splitting results concern- ing the set C_G[Ω, V].

Theorem 1.2 ([1, Thm 5.4]). There is a natural bijection C_G[Ω, V]≈ C_{W Hm}

Ω_Hm, V^Hm

× C_G

Ω\Ω_(Hm), V

. (1.1)

Naively, it would seem that it is enough to define the above bijection by taking the otopy classes of the respective restrictions

[f]7→

fDf∩Ω_Hm ,

h

fDf\Ω_(Hm)

i .

Unfortunately, in general, it is not true thatfD_f\Ω_(Hm) ∈ C_G Ω\Ω_(Hm), V . For that reason, we first need to perturbatef within its otopy class to guar- antee that the restriction of the perturbation to the set Df \Ω_(Hm) is an element of C_G Ω\Ω_(Hm), V

. Moreover, our perturbation does not change f on Ω_(Hm). This procedure is described in detail in [1, Sec. 5].

If we apply induction to (1.1), we get immediately the following result.

Corollary 1.3 ([1, Thm 6.1]). There is a natural bijection Ψ :C_G[Ω, V]→

m

Y

i=1

C_{W Hi}

ΩHi, V^Hi

. (1.2)

Fork= 1,2, . . . , m, let πk:

m

Y

i=1

C_{W Hi}

ΩHi, V^Hi

→ C_{W Hk}

ΩH_k, V^Hk denote the natural projection.

1.6. Burnside ring. Assume again that Gis finite. LetA⁺(G)be the set of isomorphism classes of finite G-sets. While disjoint union of finiteG-sets induces addition on A⁺(G), cartesian product with diagonal action induces multiplication, i.e.

[X] + [Y] = [XtY], [X]·[Y] = [X×Y],

where[X],[Y]are isomorphism classes of finiteG-sets. The resulting struc- ture is a commutative semi-ring with identity.

Since every finite G-set is a disjoint union of its orbits, each element of the semi-ring can be presented uniquely as P

d_(H)[^G/_H], where each d_(H) is a non-negative integer and [^G/_H] is the isomorphism class of ^G/_H, which depends only on the conjugacy class of H. The problem of decomposing

G/_H ×^G/_K into orbits makes multiplication inA⁺(G) non-trivial.

The Grothendieck ring constructed fromA⁺(G) is denoted by A(G) and called theBurnside ringofG. Additively, it is a free abelian group generated by isomorphism classes [^G/_H] of ^G/_H. A(G) is a commutative ring with the unit[^G/_G].

1.7. Local cross sections of a vector bundle. All manifolds considered are without boundary. Assume p:E → M is a smooth (i.e., C¹) vector bundle. We will identifyM with the zero section ofE. Alocal cross section of a bundle p:E → M is a continuous map s:U →E, where U is open in M,s⁻¹(M) is compact and p◦s= IdU. Let Γ(M, E) denote the set of all local cross sections of E over M.

Assume thatrankE = dimM and E is orientable as a manifold. Let us denote byI(s)the oriented intersection number of a local cross sections(see for instance [9, 10]), which is an integer. The intersection number is otopy invariant i.e., if two local cross sections are otopic then they have the same intersection number. Moreover, the following result holds. We write here Γ[M, E]for the set of otopy classes of local cross sections ofE over M. Theorem 1.4([2, Thm 5.2]). IfM is connected then the intersection number I : Γ[M, E]→Zis a bijection.

2. Degree deg_G

In papers [2, 3, 7] the authors introduce the equivariant degree deg_G : C_G(V, V) → A(G) for the action of a compact Lie group Gand prove that the degree has the following expected properties.

Additivity property. If f, f⁰ ∈ C_G(V, V) and D_f ∩D_f⁰ =∅ then deg_G(ftf⁰) = deg_Gf+ deg_Gf⁰.

Otopy invariance. Let f, f⁰ ∈ C_G(V, V). If f andf⁰ are otopic then deg_Gf = deg_Gf⁰.

Solution property. If deg_Gf 6= 0 then f(x) = 0 for somex∈D_f.

In order to formulate the next property, it is necessary to introduce some notation and make some assumptions. LetB(p, r) denote the openr-ball in V around p. Assume that G is finite, x ∈ V and f: ∪y∈GxB(y, ) → V, where < ¹₂min{|a−b| |a, b∈Gx, a6=b}.

Normalization property. If f(y+v) = v for y ∈ Gx and |v| < , then f ∈ C_G(V, V),f⁻¹(0) =Gx and deg_Gf = [G/Gx].

Remark 2.1. The equivariant degree, as an element of the Burnside ring, consists of multiple coefficients. Examination of these allows not only to find orbits of zeros, but also to analyze their orbit types.

Recall here that the main goal of this paper is to show that the degree deg_G has theproduct property as well.

3. Main result Assume that

• G is a finite group,

• V andW are real finite dimensional orthogonal representations ofG.

Recall thatC_G(V, V) denotes the space of equivariant local maps inV. Main Theorem. Iff ∈ C_G(V, V)andf⁰ ∈ C_G(W, W), thenf×f⁰ ∈ C_G(V⊕ W, V ⊕W) and

deg_G(f×f⁰) = deg_Gf·deg_Gf⁰,

where “·” denotes the multiplication in the Burnside ring A(G).

4. Standard and polystandard maps

In this section we introduce standard and polystandard maps and study their basic properties. These maps will play the crucial role in the proof of Main Theorem in the next section. Let us start with recalling the definition of an-normal neighbourhood. Assume that:

• Y is a linear subspace of Rⁿ,

• U is an open subset ofY.

Let Y^⊥ denote the orthogonal complement of Y in Rⁿ. For > 0 let us denote by U the set U ={x+v |x ∈U, v ∈Y^⊥,|v|< }. Any such set will be called an -normal neighbourhood ofU.

Now we are ready to introduce two important classes of maps: standard and polystandard. A mapf ∈ C_G(V, V)is called standard if

• f⁻¹(0) =Gx0 for somex0∈D_f,

• there is an open subset U of V_H, where H = G_x0, and > 0 such that:

– f⁻¹(0)∩U ={x₀}, – U⊂Df,

– f(x+v) =f(x) +v for allx∈U, v∈(V^H)^⊥,|v|< .

In such situation we also say that f is standard with respect to x₀, U and . A mapf ∈ C_G(V, V) is called m-standard if there are standard maps fi

(i= 1, . . . , m) with disjoint domains such that:

f⁻¹(0)⊂ t^m_i=1D_fi ⊂D_f.

If a map ism-standard for somem, we call itpolystandard. A finite disjoint union of standard maps is called strictly polystandard. By Remark 1.1, any polystandard map is otopic to a strictly polystandard one.

In what follows, we will need the notation that relates the classical topo- logical and equivariant degrees. Letf be standard with respect to x,U and and let α =Gx. Consider a map f_x:U ⊂V_H → V^H given by f_x =fU. Letdx= deg(fx, U).

Proposition 4.1. For eachg∈Gthe equality d_x =d_gx holds.

Proof. Let g ∈ G and K = Ggx. Then V^K = gV^H. Consider a map f_gx:gU ⊂V_K → V^K satisfying f_gx(y) =gf_x(g⁻¹y). Since g:V^H → V^K is an isomorphism of linear spaces,

dgx= deg(fgx, gU) = deg(fx, U) =dx.

Define d_α =d_y, where y is any element of α. Proposition 4.1guarantees that the integer dα is well-defined. The main advantage of standard and polystandard maps is that we can immediately compute their equivariant degree deg_G if we know the value of d_α. Namely, let f be a standard map and α=Gx=f⁻¹(0). Then

deg_Gf =dα[G/Gx] =dα[α].

More generally, letf be a m-standard map, and let {α_i}^m_i=1 denote the set of orbits of zeros of f. Then

deg_Gf =

m

X

i=1

dαi[αi].

5. Proof of Main Theorem

To prove Main Theorem we will need two lemmas.

Lemma 5.1. Let f, f⁰ be standard maps and α = f⁻¹(0), β = (f⁰)⁻¹(0).

Thenf ×f⁰ is polystandard and for each orbit γ ⊂α×β we have:

d_γ=d_α·d_β. Moreover,

deg_G(f×f⁰) = deg_Gf·deg_Gf⁰.

Proof. First we show thatf×f⁰is polystandard. Assume thatf is standard with respect tox₀,U and andf⁰ is standard with respect tox⁰₀,U⁰ and ⁰. LetH=Gx0 andK =G_x⁰

0. Note that(f×f⁰)⁻¹(0) =Gx0×Gx⁰₀ is a finite union of orbits andG_(x0,x⁰

0) =H∩K.Since U×U⁰⁰ is open inV ⊕W and (x0, x⁰₀)∈VH ×WK ⊂(V ⊕W)H∩K,

there exists an open subset U⁰⁰ ⊂(V ⊕W)H∩K and ⁰⁰ >0 such that (f × f⁰)⁻¹(0)∩U⁰⁰ ={(x₀, x⁰₀)} andU⁰⁰⁰⁰⊂U×U⁰⁰.

Now let us check that(f×f⁰)(x⁰⁰+w⁰⁰) = (f×f⁰)(x⁰⁰) +w⁰⁰for x⁰⁰∈U⁰⁰, w⁰⁰ ∈ ((V ⊕W)^H∩K)^⊥, |w⁰⁰|< ⁰⁰. Note that sinceU⁰⁰ ⊂U×U⁰⁰,x⁰⁰ has the unique representation as (x, x⁰) + (v, v⁰), where (x, x⁰) ∈ U ×U⁰ and (v, v⁰)∈(V^H)^⊥⊕(W^K)^⊥. Moreover, since

((V ⊕W)^H∩K)^⊥⊂(V^H)^⊥⊕(W^K)^⊥,

w⁰⁰ can be uniquely written as (w, w⁰) with w ∈ (V^H)^⊥ and w⁰ ∈ (W^K)^⊥. Hence

(f ×f⁰)(x⁰⁰+w⁰⁰) = (f ×f⁰)(x+v+w, x⁰+v⁰+w⁰)

= f(x+v+w), f⁰(x⁰+v⁰+w⁰)

= f(x) +v+w, f⁰(x⁰) +v⁰+w⁰

= f(x+v) +w, f⁰(x⁰+v⁰) +w⁰

= (f×f⁰)(x⁰⁰) +w⁰⁰, which proves thatf×f⁰ is polystandard.

Next we show the formulad_γ=d_α·d_β. Let(x₀, x⁰₀)∈γ⊂α×β. Observe that

d_α=d_x0 = deg(f_x0, U)= deg(f, U¹ ), d_β =d_x⁰

0 = deg(f_x⁰⁰

0, U⁰)= deg(f^{1 0}, U⁰⁰).

Thus we get dγ =d_(x0,x⁰

0)= deg((f×f⁰)_(x0,x⁰

0), U⁰⁰)= deg(f¹ ×f⁰, U⁰⁰

00

)

= deg(f2 ×f⁰, U×U⁰⁰)= deg(f, U¹ )·deg(f⁰, U⁰⁰) =d_α·d_β. In the above we used two properties of the classical topological degree: the product formula (1) and the localization of zeros (2).

Finally, we prove the product formula for standard maps. As we have shown, f×f⁰ ism-standard for somem. Decomposeα×β into the disjoint union of orbitsFm

i=1γi. We havedγi =dα·dβ for eachi= 1,2, . . . , m. Thus we get

deg_G(f ×f⁰) =

m

X

i=1

dγi[γi] =

m

X

i=1

dαdβ[γi] =dαdβ m

X

i=1

[γi]

=dαdβ[α×β] =dα[α]·dβ[β] = deg_Gf·deg_Gf⁰,

which establishes the desired formula.

We precede the next lemma by recalling the following notation. Orbit types inV are indexed (according to the partial order) by natural numbers i = 1,2, . . . , m. In particular, H₁ = G. Write M_i = V_Hi/W H_i and E_i =

V_Hi ×V^Hi

/W H_i. Recall that p_i: E_i → M_i is a vector bundle such that rankEi = dimMi and Ei is orientable as a manifold. Moreover, the bundle E_i →M_i is naturally isomorphic to the tangent bundleT M_i →M_i.

Recall that Γ(M_i, E_i) (Γ[M_i, E_i]) stands for the set of (otopy classes of) local cross sections of the bundlepi. Moreover, let us denote by

Θ_i:C_{W Hi} V_Hi, V^Hi

→Γ(M_i, E_i)

the function defined by the formula Θ_i(f)([x]) = [(x, f(x))], wherex ∈V_Hi and by

Ξ_i:C_{W Hi}

V_Hi, V^Hi

→Γ[M_i, E_i]

the function given byΞi([f]) = [Θi(f)]. SinceW Hi acts freely onVHi, both Θ_i and Ξ_i are bijections (see [2, Thm 4.1]).

Let {M_ij}_j denote the set of connected components of the manifold Mi

andn(i)denote the number of these components, which is finite or countable.

Observe that, by Theorem 1.4, the function I_i: Γ[M_i, E_i]→

n(i)

X

j=1

Z

defined by Ii([s]) ={I sMij

}ⁿ⁽ⁱ⁾_j=1 is a bijection.

Now we are ready to define the function Φ : C_G(V, V)→

m

Y

i=1

ⁿ⁽ⁱ⁾ X

j=1

Z

required in the formulation of the next lemma. Namely, let Φ(f) :={(I_i◦Ξ_i◦π_i◦Ψ)([f])}^m_i=1

with the notation Ψ and π_i introduced in Subsection 1.5. By [2, Thm 5.2]

the function Φhas the following properties

• Φ(C_G(V, V)) =





 Qm

i=1

Pn(i) j=1Z

if dimV^G>0, {0,1} ×Qm

i=2

Pn(i) j=1Z

if dimV^G= 0,

• the function induced byΦonC_G[V, V], which will be denoted by the same letter, is an injection.

Lemma 5.2. For any system {c_ij} ∈ Φ(C_G(V, V)) there is a strictly poly- standard map f such that Φ(f) ={c_ij}.

Proof. We need to consider two cases.

Case 1: dimV^G >0. Under that assumption we havedimMi >0 for each i= 1,2, . . . , m. Fix {c_ij} ∈Φ(C_G(V, V)), wherec_ij ∈Z. On the component Mij choose|c_ij|points together with their disjoint disc neighbourhoods. Let us denote byP_ij the set of these points and by F_ij the union of their neigh- bourhoods. By CorollaryA.2from AppendixA, there is a local cross section s_ij:F_ij ⊂M_ij →E_i such that

s⁻¹_ij (Mij) =Pij and I(sij) =cij.

Next we define a local cross sectionsi:Fi ⊂Mi →Eias a disjoint unionsi= tⁿ⁽ⁱ⁾_j=1sij. Note that the set∪_i,jPij is finite, because only a finite number ofcij

are nonzero. Setf_i = Θ⁻¹_i (s_i). By the definition ofΘ_i,f_i ∈ C_{W Hi} V_Hi, V^Hi and by the construction of si, the domain Dfi is the union of disjoint discs around all points in f_i⁻¹(0). Consequently, the setD :=∪^m_i=1Dfi is a finite disjoint union of discs R_k such that every disc contains one zero of a given f_i. Thus D = t_kR_k. Observe that there is > 0 such that the sets R_k are pairwise disjoint. Since any point of t_kR_k can be uniquely represented in the form gx+gv, where x ∈ Dfi, v ∈ (V^Hi)^⊥, |v| < , let us define f:D_f :=t_kR_k→V by

f(gx+gv) =gf_i(x) +gv

for all g ∈ G, x ∈ D_fi, v ∈ (V^Hi)^⊥, |v|< . Our construction guarantees that

• f ∈ C_G(V, V),

• f is strictly polystandard,

• Φ(f) ={c_ij}.

Case 2: dimV^G = 0. In that situation, M₁ = {0} and dimM_i > 0 for i > 1. Analogously as in the previous case, we define f:Df → V, but now in the construction off we take into accountM_i only fori >1. By choosing the disc neighbourhoods small enough, we can guarantee that 0 6∈ cl(D_f).

Hence there is δ >0 such that B(0, δ)∩D_f =∅, whereB(0, δ) denotes the openδ-ball inV around the origin. Set

fe= (

f if c11= 0, ftIdB(0,δ) if c₁₁= 1.

It is easy to see thatΦ fe

={c_ij}.

Corollary 5.3. In each otopy class inC_G(V, V) there is a strictly polystan- dard map.

Proof. Recall that Φ : C_G[V, V] → Q P Z

is an injection. Let [f] ∈ C_G[V, V].By Lemma5.2, there is a strictly polystandard mapf⁰ ∈ C_G(V, V) such thatΦ([f⁰]) = Φ([f]).From the injectivity ofΦ,f⁰ ∈[f].

It occurs that Main Theorem is now a consequence of Lemma 5.1 and Corollary 5.3.

Proof of Main Theorem. The fact that f ×f⁰ ∈ C_G(V ⊕W, V ⊕W) is obvious. By Corollary5.3,f andf⁰ are otopic to strictly polystandard maps t_kf_k and t_lf_l⁰ respectively, where f_k and f_l⁰ are standard. In consequence, f×f⁰ is otopic tot_kf_k× t_lf_l⁰ =t_k,l(f_k×f_l⁰). Hence

deg_G(f ×f⁰)= deg¹ _Gt_k,l(fk×f_l⁰)=² X

k,l

deg_G(fk×f_l⁰)

=3 X

k,l

deg_Gfk·deg_Gf_l⁰ = X

k

deg_Gfk

· X

l

deg_Gf_l⁰

=2

deg_Gt_kf_k

·

deg_Gt_lf_l⁰ ₁

= deg_Gf ·deg_Gf⁰ from the otopy invariance property (1), the additivity property (2) and

Lemma5.1(3). This completes the proof.

Remark 5.4. Apart from the equivariant degree deg_G, the equivariant gra- dient degree deg^∇_G with values in the Euler-tom Dieck ring U(G) is also considered, studied and applied (see for example [3,5,8,11]). In many sit- uationsdeg^∇_G gives more information thandeg_G. However, for a finite group the Burnside ring A(G) and the Euler-tom Dieck ring U(G) are identical.

Moreover, deg_G = deg^∇_G if we restrict ourselves to equivariant gradient lo- cal maps. In consequence, in the case of a finite group action the product formula holds also for deg^∇_G.

Appendix A.

Assume that p:E →M is a vector bundle over a manifold M such that dimM >0,rankE= dimM andE is orientable as a manifold.

Lemma A.1. For anyq ∈M, any disc neighbourhood D of q and any α∈ {−1,1}there is a local cross sections:D⊂M →Esuch thats⁻¹(M) ={q}

and I(s) =α.

Proof. Let B = {x ∈ Rⁿ | |x|< 1} and T B =B ×Rⁿ. Consider a local cross section sA:B →T B given by sA(x) = (x, Ax), where A is linear and detA=α. Observe that I(s_A) =α.

Let us note that we can identify T D withE |_D. Take a diffeomorphism ϕ : B → D such that ϕ(0) = q. It induces a tangent map T ϕ: T B → T D. Finally, define a local cross section s: D → E by formula s(x) = T ϕ(sA(ϕ⁻¹(x))). It is easy to see that s⁻¹(M) = q and I(s) = I(sA) =

α.

An immediate consequence of the above lemma is the following result.

Corollary A.2. Letl∈Z\{0}. For any setQof|l|distinct points inM and any set of disjoint disc neighbourhoods of these points there is a local cross sectionsdefined on the union of these neighbourhoods such thats⁻¹(M) =Q and I(s) =l.

Acknowledgements. The authors wish to express their thanks to the ref- eree for helpful comments concerning the paper.

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(P. Bartłomiejczyk)Faculty of Applied Physics and Mathematics, Gdańsk Uni- versity of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland [email protected]

(B. Kamedulski) Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Po- land, and Gdynia Maritime University, Department of Mathematics, Morska 81-87, 81-225 Gdynia, Poland

[email protected]

(P. Nowak-Przygodzki) Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Po- land

[email protected]

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