On homotopies of morphisms and admissible mappings

The notion of homotopy in the category of morphisms introduced by Górniewicz and Granas is proved to be equivalence relation which was not clear for years. Some simple properties are proved and a coincidence point index is described.


Introduction
In [9] Górniewicz and Granas defined a convenient category which allows to treat fixed point problems for a broad class of multivalued maps as coincidence problems for single-valued maps. Their notion of morphisms is still the object of interest in various aspects (see, e.g., [1,12,[16][17][18]20,21]). The aim of this paper is to define a homotopy notion and to check its basic properties in their category. In particular, we prove the transitivity property of the homotopy what seems new (comp. the remark before Lemma 3.4. in [1, p. 602]). Note that in some papers, a different notion of homotopy is considered [8,13]. We prove some basic properties of the homotopy and also introduce a homotopy invariant-coincidence point index. This yields some known theorems on coincidence and fixed points (comp., e.g., [5,7,9,14,19,24]).
The index was in fact defined in [5] for a broader class of multivalued maps but then it appeared to be a set of indices. Note that our notion of homotopy appears to be not the usual one in the case of single-valued maps. It seems to be closer to the homology relation. Nevertheless, it is satisfactory from the point of view of index theory. 1. p is onto and closed (an image of a closed set is a closed set), 2. for each y ∈ Y p −1 (y) is a compact acyclic set (i.e., all its reducedČech homology groups with rational coefficients are zero).
We will denote Vietoris maps by a double arrow p : X =⇒ Y . We make use of the following basic properties of them. Observe that Vietoris maps are proper maps (i.e., the preimage of a compact set is compact). We use theČech homology theory with compact carriers here and thereafter.
Given two spaces X and Y , we consider the set of all diagrams of the form X Z Y.
p q Two such diagrams are said to be equivalent if there are maps f : Z → Z and g : Z → Z such that the following two diagrams commute Clearly, this is an equivalence relation (see [9]). Let us note that in [8,13] a different (more restrictive) relation is considered. The maps f, g are required to be homeomorphisms there.

Definition 2.3. ([9]
) An equivalence class of a diagram (p, q) with respect to the above relation is called a morphism and is denoted by Then, we define the composite of ϕ and ψ to be a mor- (p is Vietoris being the pullback of p ). It is easy to check that the above definition is independent of the choice of representatives. The law of composition allows to regard the collection of morphisms on spaces as a category M. The category of usual single-valued maps regarded as morphisms of the form {X X Y} id is a subcategory of M. Górniewicz and Granas [9] established that theČech homology functor extends to the category M.
One easily see that morphisms can be treated as multivalued maps. We define an image of a point as the set ϕ(x) = q p −1 (x) and it is obviously independent of the choice of representatives. We say that a morphism determines a multivalued map ϕ. Such a map ϕ is upper-semicontinuous (see, e.g., [19] for a detailed but easy proof). A broad class of multivalued maps can be viewed in this way. A multivalued map ϕ : X Y is acyclic provided it is u.s.c. and the image of each point ϕ(x) is an acyclic compact set.

Proposition 2.4. A multivalued map ϕ : X Y is determined by a morphism if and only if it is a composition of acyclic maps.
Proof. Every acyclic map is determined by a morphism of the form where Γ ϕ is the graph of ϕ, Γ ϕ = {(x, y) ∈ X × Y ; y ∈ ϕ(x)} and π 1 , π 2 are projections. Therefore, a composition of acyclic maps is determined by a composition of the above morphisms. Now, given any morphism Remark 2.5. In [7], a broader class of admissible maps was considered. Those were maps admitting selections which were determined by morphisms.
Example. A multivalued map may be determined by different morphisms. Consider an acyclic map ϕ : Therefore, the square ϕ 2 = ϕ • ϕ determines a map ψ : S n → S n , ψ(x) = S n . But where f : S n → S n is a map of degree k. It is easily seen that ϕ * : H * (S n ) → H * (S n ) is an identity isomorphism and so is ϕ 2 * . But ψ * is not an identity. Therefore, ϕ 2 and ψ 2 are different morphisms.

Homotopy of morphisms
There are various ways of defining a homotopy relation in the category M.
We have chosen the one which seems to be the most straightforward and has been used by some authors (see, e.g., [1,4,7]), though it was not obvious whether it was an equivalence relation.

Proposition 3.2. The homotopy relation is an equivalence relation in the set
Proof. The only non-obvious property is the transitivity. Assume we have two commutative diagrams: We have the mapping cylinders: Now, we define a space K = Z f1 ∪ Z1 Z f2 which can be seen schematically in Fig. 1.
Define a mapp : , and a mapq : It follows immediately from the definition of K thatp,q are correctly defined continuous maps. Moreover,p is a Vietoris map since the preimages of points are exactly the same as for p 1 whenever −1 < t < 1, and for t = 1, t = −1 the maps π 1 • p and π 1 •p, which are Vietoris maps, respectively.
Proof. It is sufficient to write a commutative diagram where the maps are defined by formulas: It is easily seen thatp,q are well defined.

Definition 3.4. Two morphisms
pq and maps f 0 , f 1 such that the following diagram commutes: The following proposition is an immediate consequence of Propositions 3.2 and 3.3: The following property is an immediate consequence of the definition of homotopy: Proposition 3.6. Homotopic morphisms induce the same homology homomorphism.
Given three spaces X, Y, Z and maps α : Proposition 3.7. Compositions of homotopic morphisms are also homotopic.
Proof. The proof is a diagram game. First, assume that and Here f (x, y) = ((x, 0), f 0 (y)), g(x, y) = ((x, 1), f 1 (y)), where f i are the maps from the diagram defining a homotopy between ϕ 0 and ϕ 1 (see Def. 3.4), and π 1 (x, y) = x, π 2 (x, y) = y. One easily checks that the above diagram satisfies the conditions of 3.1 and thus the morphisms ϕ 0 •ψ and ϕ 1 •ψ are homotopic. Now let ϕ 0 , ϕ 1 be as above and ψ = {Y Z W}. p q Consider the following diagram: where π i are projections and f i (x, y) = (f i (x), y). The above diagram defines a homotopy joining morphisms ψ • ϕ 0 and ψ • ϕ 1 . Now we use the transitivity property of homotopy to end the proof.

Acyclic maps and homotopy of morphisms
As the Example 2 shows, different morphisms may determine the same multivalued map. In view of 3.6 these morphisms may even be not homotopic in general. The reason for it was the fact that the determined map in 2 was not acyclic.
Proposition 4.1. If ϕ : X Y is an acyclic map then every two morphisms which determine a selection of ϕ, must be homotopic.

Corollary 4.2.
If two morphisms ϕ 0 , ϕ 1 : X → Y have a common acyclic support (i.e., an acyclic map such that ϕ 0 , ϕ 1 determine its selections) then, the induced homology homomorphisms are the same. Proof. It is enough to consider a space X which is acyclic but not contractible. Then, the identity map id : X → X is not homotopic to any constant map y 0 (x) = y 0 . But they are homotopic as morphisms since they are selections of the acyclic map ϕ : X X, ϕ(x) = X.

A local coincidence index for morphisms
In [5], an index theory for a broad class of multi-valued maps including admissible maps as well some maps with non-connected images of points has been constructed (comp. [7], Ch. IV). It has been proved that such maps induce arbitrarily fine chain approximations on nerves of coverings in the sense of [22] (see [5,Thm 4.3]). Regarding a representative of a morphism ϕ = {X Y X} p q as a sequence of two acyclic maps p −1 and q there are chain maps induced by it. If X is a compact polyhedron and U is a polyhedral open subset of X such that x ∈ q(p −1 (x)) for x ∈ ∂U , an index of (p, q) can be defined to be a 'local Lefschetz number' (comp. Vol. 22 (2020) On homotopies of morphisms Page 9 of 17 91 [5,22]). It has been proved to be independent of the choice of approximation and representatives (since they are all homotopic). Therefore, the index i(X, ϕ, U ) of a morphism ϕ is defined (see [5,Def. 5.6]) to be a rational number (more exactly, it is an element of the coefficient field). The definition can be extended in quite a routine way to the case of a morphism , where X is a metric ANR, ϕ(X) compact and U ⊂ X an open set with x ∈ ϕ(x) for x ∈ ∂U (equivalently p(x) = q(x) for x ∈ p −1 (∂U )). The detailed proofs can be found in [5].
Let (K, τ ) be a compact polyhedron with a fixed triangulation τ . Its n-th barycentric subdivision is denoted by τ n . A subset U ⊂ K is polyhedral provided there is an integer l such that τ l induces a triangulation of the closure U of U in K.
Let l be a natural number and F a field. We denote by C * (K, τ l ) the oriented chain complex C * (K, τ l ; F ). The carrier of c ∈ C * (K, τ ) (carr c) is the smallest polyhedral subset X ⊂ K such that c ∈ C * (X, τ ). By b : C * (K, τ ) → C * (K, τ l ) we denote the subdivision chain mapping which maps each chain into its l-th barycentric subdivision.
By χ : C * (K, τ l ) → C * (K, τ ), we denote a chain mapping which is induced by a simplicial approximation of the identity map id : (K, τ l ) → (K, τ ). It is known that b and χ are chain homotopy inverses to each other. In the context of approximations the notion of a closed star is useful. Let Let Φ : (K, τ ) (L, μ) be an u.s.c. multivalued mapping.

Definition 5.2. A graded set
where A(Φ) j ⊂ hom(C * (K, τ j ), C * (L, μ j )) is called an approximation system (A-system) for Φ provided there is an integer n = n(A) such that: where ϕ 1 is an (n, j)-approximation of Φ.
(ii) For every j ∈ N, there exists j 1 ∈ N such that for m ≥ l ≥ j 1 and for L be an u.s.c. homotopy joining Φ 1 and Φ 2 . Let A(Φ 1 ) and A(Φ 2 ) be A-systems for Φ 1 and Φ 2 , respectively. They are H-homotopic provided there is an integer m ∈ N such that the following condition holds: One can define an index for an A-system as follows: Let K be a compact polyhedron with a given triangulation τ and let U be a polyhedral open subset of K (i.e., U is open andŪ is a subpolyhedron of K). Let Φ :Ū K be such that x / ∈ Φ(x) for x ∈ ∂U , and let A = A(Φ) be an A-system for Φ. Denote by p U : C * (K, τ k ) → C * (Ū, τ k ) the natural linear projection and let ϕ ∈ A(Φ) k . We define a 'local Lefschetz number' It has been proved in [22] that there exists sufficiently large k 0 such that the above defined number does not depend on the choice of ϕ ∈ A(Φ) k , k ≥ k 0 .
Thus, we can define the index properly.
The existence of an A-system for maps determined by morphisms was proved in fact in [5], where more general maps were considered. Recall that we can associate an abstract simplicial complex with a locally finite covering α of a given space X, called its' nerve N (α). Simplices of N (α) are the nonempty intersections of the sets V i ∈ α. We denote the support of a simplex s ∈ N (α) as the union of the forming sets (vertices). Similarly we define the star of a set as the union of all simplices intersecting it. We denote by N n (α) the ndimensional skeleton of the nerve. Let X, Y be compact spaces. By CovX, CovY we denote the sets of finite open coverings of these spaces, respectively. Definition 5.7. Let α,ᾱ ∈ CovX, β,β ∈ CovY and let Φ : X Y . A chain map ϕ : C * (N n (ᾱ)) → C * (N n (β)) is called an (α, β)approximation of Φ provided: for every simplex s ∈ N n (ᾱ) there exists a point p(s) ∈ X such that supp s ⊂ St(p(s), α), supp ϕs ⊂ St (Φ(p(s)), β). The following basic approximation result is a special case of Theorem 4.3 in [5]: Y be an acyclic-valued u.s.c. map of compact spaces, and α ∈ CovX, β ∈ CovY . Then for every n ∈ N, there exist: a refinementᾱ of α and an (α, β)approximation ϕ : C * (N n (ᾱ)) → C * (N n (β)) of Φ.
Recall that given a morphism ϕ = {X Z Y} p q , the map p −1 : X Z is u.s.c. and acyclic, and Theorem 5.8 applies. Thus, we can formulate Therefore, the compositions of the form λ • ψ 1 • ψ 2 • θ • b form an Asystem for the given morphism ϕ. The map b is the standard subdivision map (see [23]) and λ, θ are the above isomorphisms for barycentric subdivisions τ k , μ k .
Let now U be an open polyhedral subset of a compact polyhedron (K, τ ) (i.e. U is a subpolyhedron) and let ϕ : U → K be a morphism such that x / ∈ ϕ(x) (equivalently p(y) = q(y) for y ∈ p −1 (∂U ) for the representatives). Then, we define By localization property, we do not need to assume that U is polyhedral. Now let X be a compact metric ANR, U ⊂ X open and let ϕ : U → X be a morphism such that Fix ϕ ∩ ∂U = ∅. Then, there exists a finite polyhedron K ε-dominating X with ε > 0 smaller than the distance of Fixϕ from ∂U (see [3, p. 41]). Denoting the maps r : K → X, s : X → K (r • s = id X , d(s • r(x), x) < ε), we define the index as follows: i(X, ϕ, U ) := i(K, s • ϕ • r, r −1 (U )).
Independent of the choice of K, r, s is detailly proved in [5], and [8, Sect. IV.52]. The next step is to extend the definition of the index to the case of arbitrary ANRs. By the Arens-Eells embedding Theorem (see, e.g., [8,Thm 1.6]) we can assume that X is a closed subset of a normed space E. Since X is an ANR, there is an open subset V ⊂ E and a retraction r : V → X. Consider such that q(Z) is compact, and p(z) = q(z) for p(z) ∈ ∂U . Denoting by i : X → V the inclusion map we define a morphism ψ = i • ϕ • r : U = r −1 (U ) → V . One easily checks that Fixψ ∩ r −1 (∂U ) = ∅. By the result of J. Girolo [6] there exists a compact ANR K such that ψ(U ) ⊂ K ⊂ V . We let W = K ∩ r −1 (U ) and then the index i(K, ψ K , W ) is well defined as above.
Corollary 5.11. If i(X, ϕ, U ) = 0 then the determined map has a fixed point in U (or equivalently any representative X Z X p q has a coincidence in p −1 (U )).
The first corollary is a consequence of additivity and the last one of commutativity (see. Thm 5.10). Now, we prove a generalization of the well-known nonlinear alternative (comp. [11, II.6, Thm 5.2]) from which one can derive many fixed point or coincidence results. We omit the corollaries since they are routine (comp. [4,7,11]). (i) ϕ has a fixed point in U , (ii) for some x ∈ ∂U and 0 < t < 1 there is y ∈ p −1 (x) such that x = t·q(y).
Proof. Consider the following commutative diagram wherep(x, t) = (p(x), t),q(x, t) = t · q(x), 0(x) = 0, f(x) = (x, 0) and g(x) = (x, 1). Denoting by ψ the morphism defined by (p, 0) we observe that i(C, ψ, U ) = 1 since 0 ∈ U and it is a constant map x → 0. If ϕ has a fixed point in U then we are done. If not and if moreover (ii) is not satisfied that means exactly that assumptions of homotopy invariance (Thm 5.10) are fulfilled. Therefore, i(C, ϕ, U ) = 1 and ϕ has a fixed point in U by Cor. 5.11.
Remark 5.14. We assumed in this section that the considered morphisms determined compact maps, but one can extend all the results to some noncompact maps. We touched the example of compact attraction maps in [5]. The classes of condensing morphisms mentioned in [15] (comp. [24]) may be also included into the theory.
Remark 5. 15. The index is a priori a rational number but there are no examples where it is not an integer. Moreover, in [2], it was proved that in the case of acyclic maps of ENR-s, the index is an integer and the theory is unique.

Extending morphisms from a sphere
In this short section, we prove one extremely simple geometric result which analog for single-valued maps is an exercise (see, e.g., [11, Thm II.5.1]). We say a morphism is nullhomotopic provided it is homotopic to a morphism where y 0 (x) = y 0 is any constant map. Proof. Assume ϕ is nullhomotopic, i.e., we have a diagram  It is easily seen that p is a Vietoris map, and moreover Hence a morphism Φ = {D n+1 K Y} pq is an extension of ϕ.
Conversely, assume we have a morphism {D n+1Z Y}.
Combining Propositions 6.1 together with 4.3, we obtain: Corollary 6.2. If Y is an acyclic space with π n (Y ) = 0 then there exists a single-valued map f : S n → Y such that f is not extendable to D n+1 as a map but it is extendable as a morphism (there is a multivalued extension).
The results of this section suggest that one can extend theorems concerning essential maps of [11,Ch. II]. A version of this approach can be found in [16][17][18].