On homotopies of morphisms and admissible mappings
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Journal of Fixed Point Theory and Applications
On homotopies of morphisms and admissible mappings Zdzislaw Dzedzej
and Tomasz Gzella To the memory of A. Granas.
Abstract. The notion of homotopy in the category of morphisms introduced by G´ orniewicz 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. Mathematics Subject Classification. Primary 55M20; Secondary 47H04, 54C60. Keywords. Homotopy, morphisms, coincidence index, admissible mapping.
1. Introduction In [9] G´ orniewicz 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., [1,12,16–18,20,21]). 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. 0123456789().: V,-vol
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2. The G´ orniewicz–Granas category of morphisms By a space, we understand a Hausdorff topological space and by a map—a continuous transformation. Definition 2.1. A map p : X → Y is a Vietoris map provided 1. p is onto and closed (an image of a closed set is a closed set), ˇ 2. for each y ∈ Y p−1 (y) is a compact acyclic set (i.e., all its reduced Cech 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). Theorem 2.2. (see [11, Ch. VI §19 Thm 3.2]) If p : X =⇒ Y then the induced linear map p∗ : H∗ (X) → H∗ (Y ) is an isomorphism. The composite of Vietoris maps is also a Vietoris map. The pullback of a Vietoris map is a Vietoris map. ˇ We use the Cech homology theory with compact carriers here and thereafter. Given two spaces X and Y , we consider the set of all diagrams of the p q form X Z Y. 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 Z p
Z p
q
X
Y
f p
Z
Y .
g
X
q
q
p
q
Z
Clearly, this is an equivalence relation (see [9]). Let us note
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