() @ Appl. Gen. Topol. 16, no. 1(2015), 19-30doi:10.4995/agt.2015.2057 c© AGT, UPV, 2015 On some topological invariants for morphisms defined in homological spheres Nasreddine Mohamed Benkafadar a and Boris Danielovitch Gel’man b a Faculty of Sciences, Department of Mathematics, Constantine Algeria. (kafadar@gmx.com) b Faculty of Mathematics, Department of Functional Theory and Geometry, Voronezh, Russia. (gelman@math.vsu.ru) Abstract In the paper one defines topological invariants of type degree for mor- phisms in the category T op(2) of topological pairs of spaces and contin- uous single valued maps, which admit homological n-spheres as target and arbitrary topological pairs of spaces as source. The different des- cribed degrees are acquired by means homological methods, and are a powerful tool in the root theory. Several existence theorems are ob- tained for equations with multivalued transformations. 2010 MSC: 55N10; 54H25; 54C60. Keywords: Homology; homotopy; topological degree; fixed points. 0. Introduction The concept of topological degree deg is well know for maps of homological n-spheres and oriented n-dimensional manifolds (see for example, [4] and [2]). Recall this concept: if X and Y are both homological n-spheres and f : X → Y is a continuous single valued map by fixing some generated elements z1 and z2 of homology groups Hn(X) and Hn(Y ) respectively, one obtains an equality fn∗(z1) = k ·z2. This number k is called the degree of f and is denoted by deg f. Topological degree theory plays a preponderant role in topology fixed points theory and non linear analysis. The different degrees can be considered as a generalization of the Winding number, Kronecker’s characteristic and others Received 23 December 2013 – Accepted 17 October 2014 http://dx.doi.org/10.4995/agt.2015.2057 N. M. Benkafadar and B. D. Gel’man topological invariants. Different generalizations of the topological degree has been studied for multivalued transformations (see for example [7] - [10]). 1. Homological invariants for some classes of morphisms in the categories T op and T op(2) 1.1. Notations and definitions. In the present section one introduces some basic topics which play an important role in the sequel. A pair (X, A) of topological spaces such that A ⊆ X is called a pair of topological spaces, in this context, a topological space X is conceived as the pair (X, ∅). Let (X, A) and (S, T ) be some pairs of topological spaces and f ∈ MorT op(X, S) such that f(A) ⊆ T, then f is named a continuous single valued map of pairs of topological spaces and denoted f : (X, A) → (S, T ). The collections of pairs of topological spaces and continuous single valued maps of pairs of topological maps with the composition of maps define a category denoted T op(2), it admits as a full subcategory the category T op of topological spaces and continuous single valued maps. Two morphisms f, g ∈ MorT op(2) ((X, A), (S, T )) are called homotopic if and only if there exists a morphism Φ ∈ MorT op(2) ((X, A) × [0, 1], (S, T )) such that Φ(x, 0) = f(x) and Φ(x, 1) = g(x) for every x ∈ X. By H one denotes the covariant functor H of singular homology with co- efficients in the abelian ring of integer Z, defined from the category T op(2) in the category Gd of graded groups and homomorphisms of degree zero, where G the category of abelian groups and homomorphisms of groups is a full subcategory. Thus, for a given object (X, A) ∈ Obj(T op(2)) and a mor- phism f ∈ MorT op(2) ((X, A), (Y, B)) the functor H assigns a graded group {Hi(X, A)}i≥0 and a homomorphism of degree zero {Hi(f)}i≥0 ∈ MorGd({Hi(X, A)}i≥0,{Hi(Y, B)}i≥0. 1.2. Degree for a class of morphisms in the category T op. An object Y in the category T op will be called a homological n-sphere if the topological space Y admits the same homological groups of the n-sphere. Consider f ∈ MorT op(X, Y ) a morphism with source an arbitrary topo- logical space X and target Y a homological n-sphere. Let Hn(f) := fn∗ ∈ MorG(Hn(X), Hn(Y )) be the induced homomorphism of f and e be a genera- tor of Hn(Y ). Definition 1.1. The degree of a morphism f ∈ MorT op(X, Y ) is the integer denoted and defined as dg(f, X, Y ) =| k |, where k ∈ Z verifies a = k · e and Imfn∗ =< a >⊆ Hn(Sn) =< e > . Example 1.2. Consider the open subset: U = {(x1, x2) ∈ R2 | (x1 −1)2 + x22 < 1}∪{(x1, x2) ∈ R2 | (x1 + 1)2 + x22 < 1} of R2 and let X = ∂U be the boundary of U. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 20 On some topological invariants for morphisms defined in homological spheres If x0 is a fixed element of R 2\X one can define the morphism f ∈ MorT op(X, R2�{0}) given by the rule f(x) = x−x0 for every x ∈ X. Then the next equalities are satisfied: dg(f, X, R2�{0}) = { 1, if x0 ∈ U 0, if x0 /∈ U Let us give some properties of this topological invariant. One will begin by giving the relation between the Winding number deg f of a morphisms f ∈ MorT op(Sn, Sn) defined on the sphere and its topological invariant dg(f, Sn, Sn). Proposition 1.3. Let f ∈ MorT op(Sn, Sn) then dg(f, Sn, Sn) =| deg f | . Proof. Consider Hn(S n) =< e > then Imfn∗ =< fn∗(e) >, moreover fn∗(e) = deg f · e, hence dg(f, Sn, Sn) =| deg f | . � Definition 1.4. A morphism f ∈ MorT op(2) ((X, A), (Y, B)) is called h - sec- tional if f admits a right inverse homotopy. It is not difficult to check the next properties of this degree: Proposition 1.5. The Topological degree satisfies the following assertions: (1) let f ∈ MorT op(X, Y ) be a constant map then dg(f, X, Y ) = 0; (2) let f ∈ MorT op(X, Y ) be h sectional morphism then dg(f, X, Y ) = 1; (3) Let Z be is a topological space and (f, g) ∈ MorT op(X, Z)×MorT op(Z, Y ) then dg(g ◦f, X, Y ) is a multiple of dg(g, Z, Y ); (4) let X0 be a subset of X an object in the category T op and fX0 ∈ MorT op(X0, Z) be the restriction of a morphism f ∈ MorT op(X, Y ) then there exists a natural number n ∈ N such that dg(fX0, X0, Y ) = n ·dg(f, X, Y ). Proposition 1.6. Let Y1 and Y2 be both some homological n-spheres in the cat- egory T op and (f, g) ∈ MorT op(X, Y1)×MorT op(Y1, Y2) be a pair of morphisms then dg(g ◦f, X, Y2) = dg(f, X, Y1) ·dg(g, Y1, Y2) Proof. This is a consequence of the definition 1.1, of the topological degree. � The topological degree is invariant for homotopic morphisms: Proposition 1.7. Let (f, g) ∈ MorT op(X, Y ) × MorT op(X, Y ) then if f and g are homotopic dg(f, X, Y ) = dg(g, X, Y ). Let us consider some aspects of the degree dg(f, X, Y ) in that case where the target Y = Sn. Proposition 1.8. Let f ∈ MorT op(X, Sn) such that dg(f, X, Sn) 6= 0 then f is an epimorphism in the category T op. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 21 N. M. Benkafadar and B. D. Gel’man Proof. Suppose that f ∈ MorT op(X, Sn) is not an epimorphism so f(X) ⊂ Sn. Let y ∈ Sn�f(X), then one can diagramed: X f→ Sn f̃ ց ↑ i Sn�{y} where f̃ is the submap of f and i is the canonical injection. One can conclude by remarking that Hn(S n�{y}) is a trivial group. � Proposition 1.9. Let f, g ∈ MorT op(X, Sn) then if dg(f) 6= dg(g) the mor- phisms f and g admit at least a coincidence point in X. Proof. Indeed, if f(x) 6= g(x) for every element x ∈ X then for (x, t) ∈ X×[0, 1] the vector field v(x, t) = (1− t) ·f(x) + t ·(−g)(x) ∈ Rn+1 is free of zero. This finding offers the opportunity to get the morphism F ∈ MorT op(X × [0, 1], Sn) where, F(x, t) = v(x,t ‖v(x,t)‖ for all element (x, t) from the source X × [0, 1]. The morphism F defines a homotopy between f and (−g) . Hereafter, from propo- sitions 1.7 and the definition 1.1, one takes dg(f, X, Sn) = dg(−g, X, Sn) = dg(g, X, Sn). � 1.3. Degree for a class of morphisms in the category T op(2). An object (Y, B) ∈ Obj(T op(2)) is called a homological n-sphere if H0(Y, B) = Hn(Y, B) isomorphic to the abelian ring of integers Z and Hi(Y, B) = {0} for all other indices. For more notions one this topics see [9]. For instance, the pairs of spaces (Rn, Rn�{0}); (Bn, Sn−1) where Bn is the closed ball in Rn and Sn−1 = ∂Bn, are some n-spheres in that category. Definition 1.10. The degree of a morphism f ∈ MorT op(2) ((X, A), (Y, B)) where (Y, B) is a homological n-sphere with Hn(Y, B) =< η > is denoted and defined by dgr(f, (X, A), (Y, B)) =| k |, where Imfn∗ =< b > and b = k · η. The next properties are obvious. Proposition 1.11. The following assertions are satisfied: (1) if a morphism f ∈ MorT op(2) ((X, A), (Y, B)) is a constant map then dgr(f, (X, A), (Y, B)) = 0; (2) if (f, g) ∈ MorT op(2) ((X, A), (X′, A′))×MorT op(2) ((X′, A′), (Y, B)) then there exits an integer k ∈ N such that : dgr(g ◦f, (X, A), (Y, B)) = k · dgr(g, (X′, A′), (Y, B)); (3) let (X0, A0) ⊆ (X, A) and f0 ∈ MorT op(2) ((X0, A0), (Y, B)) be the submap of the morphism f ∈ MorT op(2) ((X, A), (Y, B)) then there ex- ists a natural number k ∈ N such that : dgr(f0, (X0, A0), (Y, B)) = k ·dgr(f, (X, A), (Y, B)); c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 22 On some topological invariants for morphisms defined in homological spheres (4) let (Y1, B1) and (Y2, B2) be some n-spheres in the category T op(2) and (f, g) ∈ MorT op(2) ((X, A), (Y1, B1)) × MorT op(2)((Y1, B1), (Y2, B2)) be a pair of morphisms then : dgr(g ◦f, (X, A), (Y2, B2)) = dgr(f, (X, A), (Y1, B1)) · dgr(g, (Y1, B1), (Y2, B2)) (5) if f, g ∈ MorT op(2) ((X, A), (Y, B)) are some homotopic morphisms then dgr(f, (X, A), (Y, B)) = dgr(g, (X, A), (Y, B)). This homological invariant satisfies some more specific properties. Let us describe some of them. Proposition 1.12. Let Z ⊂ Int(A) ⊆ A ⊆ A ⊂ X and f ∈ MorT op(2) ((X, A), (Y, B)) then dgr(f̃, (X�Z, A�Z), (Y, B)) = dgr(f, (X, A), (Y, B)) where f̃ ∈ MorT op(2) ((X�Z, A�Z), (Y, B)) is the submap of the morphism f on the pair (X�Z, A�Z). Proof. This is a consequence of the following commutative diagram: Hn(X, A) ց fn∗ in∗ ↑ Hn(Y, B) Hn(X�Z, A�Z) ր f̃n∗ where i ∈ MorT op(2) ((X�Z, A�Z), (X, A)) is the natural injection. From excision theorem one infers that in∗ is an isomorphism and concludes the proof. � Proposition 1.13. Let f ∈ MorT op(2) ((X, A), (Y, B)) be a morphism such that dgr(f, (X, A), (Y, B)) 6= 0 then there exist x ∈ X�A and y ∈ Y �B such that f(x) = y. Proof. Indeed, if f(x) /∈ Y �B for every element x ∈ X�A on can get the following commutative diagram: Hn(X, A) fn∗→ Hn(Y, B) f̃n∗ ↓ ր in∗ Hn(B, B) where i ∈ MorT op(2) ((B, B), (Y, B)) is the natural injection and f̃ = f. One concludes by observing that Hn(B, B) is a trivial group. � Corollary 1.14. Let f ∈ MorT op(X, Rn) be a morphism in the category T op and A be a closed subset of X such that f(a) 6= 0 for every x ∈ A, then if the degree of the morphism f ∈ MorT op(2) ((X, A), (Rn, Rn�{0})) is not zero, there exists x0 ∈ X�A such that f(x0) = 0. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 23 N. M. Benkafadar and B. D. Gel’man Corollary 1.15. Let f ∈ MorT op(Bn, Rn) such that f(x) 6= 0 for every ele- ment x ∈ ∂Bn = Sn−1 then if the degree of the morphism f ∈ MorT op(2) ((B n, Sn−1), (Rn, Rn�{0})) is not zero there exists at least an element x in the interior of the ball such that f(x) = 0. 2. Homological invariant for a class of multivalued transformations Let (X, A), (S, T ) ∈ Obj(T op(2)) a correspondence F : (X, A) → (S, T ) which assigns for each element x ∈ X a subset F(x) ⊆ S, and F(A) = ∪ a∈A F(a) ⊆ T is named a multivalued transformation, the graph of F denoted ΓF is the pair (Γ X F , Γ A F ) ∈ Obj(T op(2)), where ΓXF = {(x, s) ∈ X×S | s ∈ F(x)} and ΓAF = {(a, t) ∈ A ×T | t ∈ F(a)}. A representation of a multivalued transformation F : (X, A) → (S, T ) is a quintuple Q = [(X, A), (S, T ), (M, N), p, q] where (p, q) ∈ MorT op(2) ((M, N), (X, A)) ×MorT op(2) ((M, N), (S, T )) and q(p−1(x)) = F(x) for every element x ∈ X. In the case when p := tF ∈ MorT op(2) ((Γ X F , Γ A F ), (X, A)) and q := rF ∈ MorT op(2)((ΓXF , ΓAF ), (S, T )) are the natural projections the quintuple Q̃ = [(X, A), (S, T ), (ΓXF , Γ A F ), tF , rF ] is named the canonical representation of F. 2.1. Degree for multivalued transformations defined in homological n- spheres. Let Y1 and Y2 be both some homological n-spheres and F : Y1 → Y2 be a multivalued transformation with a representation Q = [Y1, Y2, X, p, q]. Definition 2.1. The degree of a multivalued transformation F : Y1 → Y2 rela- tive to the representation Q is denoted and defined by Dg(F, Q) = dg(p, X, Y1)· dg(q, X, Y2). The degree Dg(F, Q̃) of F relative to the canonical representation Q̃ will be called the degree of the multivalued transformation F and will denoted by Dg(F). Let us give some properties of this homological invariant. Proposition 2.2. Let Q = [Y1, Y2, X, p, q] be a representation of F, then Dg(F, Q) is a multiple of Dg(F). Proof. One can consider the following commutative diagram: ΓF tF ւ ↑ λ ցrF Y1 ←− p X → q Y2 where λ(x) = (p(x), q(x)) for every element x ∈ X. One concludes by using assertion 3 of proposition 1.5. � c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 24 On some topological invariants for morphisms defined in homological spheres Proposition 2.3. Let F, G : Y1 → Y2 be two multivalued transformations such that G(x) ⊆ F(x) for every element x ∈ Y1 then Dg(G) = k · Dg(F) for some integer k ∈ N. Proof. Under the hypothesis, one obtains that ΓG ⊆ ΓF . One concludes the proof with the following commutative diagram: tF ւ ΓF ցrF Y1 ↑ i Y2 tG տ ΓG րrG and by referring to the assertion 3 of proposition 1.5. � What happened if one gets a morphism f ∈ MorT op(Y1, Y2) and considers it as a multivalued morphism in the following sense F(x) = {f(x)} for every element x ∈ Y1. In such situation, one has which follows: Proposition 2.4. Let f ∈ MorT op(Y1, Y2) and F : Y1 → Y2 be the multivalued transformation given by the rule F(x) = {f(x)} := f(x) for every element x ∈ Y1, then Dg(F) = dg(f, Y1, Y2). Proof. It is a consequence of the following commutative diagram: Γf rf =rF→ Y2 tf = tF ↓ Y1 ր f where the morphism tf ∈ MorT op(Γf , Y1) realizes a homeomorphism. � Corollary 2.5. Let G : Y1 → Y2 be a multivalued mapping which admits a selector f ∈ MorT op(Y1, Y2) then dg(f, Y1, Y2) = k · Dg(G) for some natural number k ∈ N. Proof. Indeed F(x) := {f(x)} ⊆ G(x) for every x ∈ Y1 and thus one can conclude by referring to the propositions 2.3 and 2.4. � Proposition 2.6. Let Y1, Y2 and Y3 be some homological n-spheres, F : Y1 → Y2 be a multivalued transformation and f ∈ MorT op(Y2, Y3) then dg(f, Y1, Y2) · Dg(F) = k ·Dg(f ◦F) for some k ∈ N. Proof. Of course, the quintuple Q = [Y1, Y2, ΓF , tF , f◦rF ] is a representation of the multivalued morphism f ◦F : Y1 → Y2 therefore, from proposition 1.6 one obtains Dg(f◦F, Q) = dg(f, Y1, Y2)·Dg(F) one concludes thanks to proposition 2.2. � Proposition 2.7. Let Y be a homological n-sphere and F : Y → Sn be a multivalued transformation such that Dg(F) is different from zero then F(Y ) = Sn. Proof. Of course, in this case dg(rF , ΓF , S n) 6= 0 one concludes by referring to the proposition 1.8. � c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 25 N. M. Benkafadar and B. D. Gel’man Definition 2.8. Two multivalued transformations F0, F1 : Y1 → Y2 defined on some homological n-spheres Y1 and Y2 are called homotopic if there exists a quintuple [Y1, Y2, X × [0, 1], Φ, Ψ] such that Q0 = [Y1, Y2, X, Φ0, Ψ0] and Q1 = [Y1, Y2, X, Φ1, Ψ1] realize some representations of F0 and F1 respectively, where Φt : X → Sn and Ψt : X → Sn are defined by the rules Φt(x) = Φ(x, t), Ψt(x) = Ψ(x, t) for every element (x, t) ∈ X ×{0, 1}. Proposition 2.9. Let F0, F1 : Y1 → Y2 be some multivalued transformations defined on some homological n-spheres Y1 and Y2 then if F0 and F1 are homo- topic there exist some natural numbers k0, k1 ∈ N such that k0 · Dg(F0) = k1 ·Dg(F1). Proof. For this purpose one refers to propositions 1.7 and 2.2. � 2.2. Degree for multivalued transformations with images in homo- logical n-spheres in the category T op(2). In this section one displays a homological invariant for multivalued transformations acting between homolog- ical n-spheres of the category T op(2) Let (Y0, B0) and (Y1, B1) be some homological n-spheres and F : (Y0, B0) → (Y1, B1) be a multivalued transformation that admits a quintuple Q = [(Y0, B0), (Y1, B1), (M, N), p, q] as a representation. Definition 2.10. The degree of a multivalued transformation F : (Y0, B0) → (Y1, B1) relative to the representation Q is denoted and defined by Dgr(F, Q) = dgr(p, (M, N), (Y0, B0)) · dgr(q, (M, N), (Y1, B1)). The degree of F relative to the canonical representation Q̃ = [(Y0, B0), (Y1, B1), (Γ Y0 F , ΓB0 F ), tF , rF ] will be denoted by Dgr(F) := Dgr(F, Q̃). In the sequel, one describes some properties of this homological invariant. Proposition 2.11. Let Q = [(Y0, B0), (Y1, B1), (X, X ′), p, q] be a representa- tion of a multivalued transformation F : (Y0, B0) → (Y1, B1) then there exists a natural number k ∈ N such that Dgr(F, Q) = k · Dgr(F). Proof. Of course, one can consider the next commutative diagram: (Y0, B0) p←− (X, X′) q→ (Y1, B1) t տ F ↓ λ ր rF (ΓY0 F , ΓB0 F ) where λ(x) = (p(x), q(x)) for every element x ∈ X. One concludes by using the assertion 2 of the proposition 1.11. � c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 26 On some topological invariants for morphisms defined in homological spheres Corollary 2.12. Let F : (Y0, B0) → (Y1, B1) be a multivalued transformation Q = [Y0, Y1, X, p, q] be a representation of F : Y0 → Y1 then the quintuple Q = [(Y0, B0), (Y1, B1), (X, p −1(B0)),p, q] is a representation of F : (Y0, B0) → (Y1, B1) and there exists a natural number k ∈ N such that Dgr(F, Q) = k · Dgr(F). Proof. This is a consequence of the definition 2.10 and the proposition 2.11. � Example 2.13. Let B1(0) be the unit ball of the complex plane C and S1(0) = ∂B1(0) be the boundary of B1(0) and let F : (B1(0), S1(0)) → (C, C�{0}) be the multivalued transformation defined by the rule F(z) = n √ z. The quintuple Q = [(B1(0), S1(0)), (C, C�{0}), (B1(0), S1(0)), p, q] where p(w) = wn and q(w) = w for every element w ∈ B1(0), is a representation of the multivalued mapping F. Moreover, dgr(p, (B1(0), S1(0)), (B1(0), S1(0))) = dg(p) = n and dgr(q, (B1(0), S1(0)), (C, C�{0})) = dg(q) = 1 so Dgr(F, Q) = n. On the other hand, the single valued map λ : (B1(0), S1(0)) → (ΓB1(0)F , Γ S1(0) F ) where λ(w) = (p(w), q(w)) is an isomorphism in the category T op(2) this implies that the induced homomorphism in homology is an isomorphism in the category G of groups and homomorphisms of groups and thus Dgr(F) = Dgr(F, Q) = n. Proposition 2.14. Let F, G : (Y0, B0) → (Y1, B1) be both some multival- ued transformations such that G(x) ⊆ F(x) for every element x ∈ Y0 then Dgr(G) = k ·Dgr(F) for some natural number k ∈ N. Proof. Under the hypothesis, one obtains that (ΓY0 G , ΓB0 G ) ⊆ (ΓY0 F , ΓB0 F ). There- fore one infers the assertion from, the following commutative diagram: tF ւ (ΓY0 F , ΓB0 F ) ցrF (Y0, B0) ↑ i (Y1, B1) tG տ (ΓY0G , Γ B0 G ) րrG and by referring to the assertion 2 of the proposition 1.11 � Proposition 2.15. Let f ∈ MorT op(2) ((Y0, B0), (Y1, B1)) and F : (Y0, B0) → (Y1, B1) be the multivalued transformation given by the rule F(x) = {f(x)} := f(x) for every element x ∈ Y0, then Dgr(F) = dgr(f, (Y0, B0), (Y1, B1)). Proof. For this purpose one can consider the next diagram: (Y0, B0) tf =tF←− (ΓY0 f , ΓB0 f ) rf =rF→ (Y1, B1) � Corollary 2.16. Let G : (Y0, B0) → (Y1, B1) be a multivalued transformation which admits a selector f ∈ MorT op(2)((Y0, B0), (Y1, B1)) then: dgr(f, (Y0, B0), (Y1, B1)) = k ·Dgr(G) for some natural number k ∈ N. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 27 N. M. Benkafadar and B. D. Gel’man Proof. This is a consequence of the propositions 2.14 and 2.15. � Proposition 2.17. Let (Y0, B0), (Y1, B1) and (Y2, B2) be some homological n-spheres, F : (Y0, B0) → (Y1, B1) be a a multivalued transformation and f ∈ MorT op(2) ((Y1, B1), (Y2, B2)) then dgr(f, (Y1, B1), (Y2, B2)) ·Dgr(F) = k · Dgr(f ◦F) for some natural number k ∈ N. Proof. Of course, the quintuple Q = [(Y0, B0), (Y2, B2), (Γ Y0 F , ΓB0 F ), tF , f ◦rF ] is a representation of the multivalued transformation f ◦F : (Y0, B0) → (Y2, B2) therefore, from the assertion 4 of proposition 1.11 one obtains the next equality: Dgr(f ◦F, Q) = dgr(f, (Y1, B1), (Y2, B2)) · Dgr(F), one concludes due to proposition 2.11 � Definition 2.18. Let F0, F1 : (X, A) → (S, T ) be some multivalued transfor- mations, F0 and F1 are called homotopic if there exists a quintuple Q = [(X, A), (S, T ), (M, N)× [0, 1], Φ, Ψ] such that the following quintuples: Q0 = [(X, A), (S, T ), (M, N), Φ0, Ψ0] and Q1 = [(X, A), (S, T ), (M, N), Φ1, Ψ1] are some representations of F0 and F1 respectively and where Φt : (M, N) → (X, A) and Ψt : (M, N) → (X, A) are defined by the rules Φt(m) = Φ(m, t), Ψt(m) = Ψ(m, t) for every element (m, t) ∈ M ×{0, 1}. Proposition 2.19. Let F0, F1 : (Y0, B0) → (Y1, B1) be some multivalued trans- formations then if F0 and F1 are homotopic there exist some representations Q0 and Q1 of F0 and F1 respectively such that Dgr(F0, Q0) = Dgr(F1, Q1). Proof. It is a consequence of assertion 5 of proposition 1.11. � Proposition 2.20. Let F : (Y0,B0) → (Rn, Rn�{0}) be a multivalued trans- formation such that Dgr(F) 6= 0 then there exists an element y ∈ Y0�B0 such that 0 ∈ F(y). Proof. Indeed, Dgr(F) 6= 0 so dgr(rF , (ΓY0F , Γ B0 F ), (Rn, Rn�{0})) 6= 0 after which one can conclude due to the proposition 1.13. � Let S be the boundary of a closed ball B of Rn and F : B → Rn be a multivalued transformation. The multivalued vector field induced by F noted by Φ is the multivalued transformation given by the rule Φ(x) = x − F(x) for every element x ∈ B. It is obvious that if Q = [B, Rn, ΓF , p, q] is the canonical representation of F and the multivalued vector field induced by F is such that Φ : (B, S) → (Rn, Rn�{0}) then the quintuple Q̂ = c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 28 On some topological invariants for morphisms defined in homological spheres [(B, S), (Rn, Rn�{0}), (ΓBF , ΓSF ), p, p−q)] is a representation of the multivalued vector field Φ. In the sequel the degrees: dgr(p, (ΓBF , Γ S F ), (R n, Rn�{0})) and dgr(p −q, (ΓBF , ΓSF ), (Rn, Rn�{0})) will be denoted by dgr(p) and dgr(p −q) respectively. Proposition 2.21. Let F : B → Rn be a multivalued transformation which is free of fixed point on the boundary S of a closed ball B then if the topological degree Dgr(Φ) of the multivalued vector field induced by F is not zero i.e. Dgr(Φ) 6= 0, the multivalued transformation F : B → Rn admits a fixed point in the interior of the ball B. Proof. This is a consequence of proposition 2.20. � Proposition 2.22. Let F : B → Rn be a multivalued transformation free of fixed point on the boundary S of a ball B and F(S) ⊆ B then the following equivalence is satisfied: Dgr(Φ, Q̂) 6= 0 if and only if dgr(p) 6= 0. Proof. Consider the morphisms p, p − q ∈ MorT op(2) ((ΓBF , ΓSF ), (Rn, Rn\{0})) and let Ψ ∈ MorT op(ΓBF × [0, 1]), Rn) be a morphism given by the rule: Ψ((x, y), λ) = p(x, y) −λ · q(x, y) for every element ((x, y), λ) ∈ ΓBF × [0, 1]. It follows that the morphism: Ψ ∈ MorT op(2) ((Γ B F , Γ S F ) × [0, 1]), (Rn, Rn\{0})) is a homotopy between the morphisms: p, p−q ∈ MorT op(2) ((Γ B F , Γ S F ), (R n, Rn\{0})) therefore from assertions 5 of proposition 1.11 one deduces what follows : dgr(p) = dgr(p −q) and thus one obtains the next equality: Dgr(Φ, Q̂) = (dgr(p)) 2 . Hence, Dgr(Φ, Q̂) 6= 0 if and only if dgr(p) 6= 0. � The last proposition 2.22, permits to obtain a generalization of the theorems due to Eilenberg-Montgomery [3] and Kakutani [8]. c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 1 29 N. M. Benkafadar and B. D. Gel’man Theorem 2.23. Let F : B → Rn be a multivalued transformation which sat- isfies the following conditions: (1) dgr(p) 6= 0, (2) F(S) ⊆ B. Then the multivalued transformation F admits in the ball a fixed point. Proof. Of course, if F has a fixed point on S then the conclusion of the theorem is satisfied. Otherwise, if F is free of fixed point on S then Dgr(Φ) 6= 0. One concludes the proof from proposition 2.21. � Acknowledgements. Project supported by CNEPRU No B00920110092 and PNR / ATRST No 8/u250/713. Laboratory M.M.E.R.E. U.M.1 Constantine and Russian FBR grant 11-01 -00382-a. References [1] Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis and V. V. Obukhovskii, Topological methods in the fixed point theory of multivalued maps, Uspehi Mat. Nauk 35, no. 1 (1980), 59–126 (in Russian); English translation: Russian Math. Surveys 35 (1980), 65–143. [2] A. Dold , Lectures on Algebraic Topology, Springer-Verlag, Berlin-Heidelberg -New York, 1972. [3] S. Eilenberg and D. Montgomery, Fixed point theorems for multivalued transformations, Amer. J. Math 68 (1946), 214–222. [4] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton university Press, Princeton, New Jersey, 1952. [5] L. Górniewicz, Homological methods in fixed point theory of multi-valued maps, Disser- tationes Math. Warsaw 129 (1976), 1–66. [6] L. 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