Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), $${\Lambda}$$Λ a k-dimensional topological manifold with topological orientation $${\eta}$$η , and $${f : D \rightarrow X}$$f:D→X a locally compact map, where D is an open subset of $${X \times \Lambda}$$X×Λ . We define Fix(f) as the set of points$${{(x, \lambda) \in D}}$$(x,λ)∈D such that $${x = f(x, \lambda)}$$x=f(x,λ) . For an open pair (U, V) in $${X \times \Lambda}$$X×Λ such that $${{\rm Fix}(f) \cap U \backslash V}$$Fix(f)∩U\V is compact we construct a homomorphism $${\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}$$Σ(f,U,V):Hk(U,V)→R in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a $${C^{\infty}}$$C∞ -manifold $${\Lambda}$$Λ , these properties uniquely determine $${\Sigma}$$Σ . By passing to the direct limit of $${\Sigma_{(f,U,V)}}$$Σ(f,U,V) with respect to the pairs (U, V) such that $${K = {\rm Fix}(f) \cap U \backslash V}$$K=Fix(f)∩U\V , we define a homomorphism $${\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}$$σ(f,K):Hk(Fix(f),Fix(f)\K)→R in the Čech cohomologies. Properties of $${\Sigma}$$Σ and $${\sigma}$$σ are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.


Introduction
Let X and Λ be topological spaces. We consider a continuous map Unless otherwise stated, in the present paper we assume that X is an ANR (absolute neighborhood retract), Λ is a (Hausdorff) topological manifold of The paper is organized as follows. In Section 2 we state Theorems 2.1-2.4, which are the main results here. The remaining part of the paper is devoted to the proofs of Theorem 2.1 (Sections 3-8) and Theorem 2.2 (Sections 9-12), although some results presented there might be of separate interest. In particular, in Section 3 we define Σ in the case X is a finite-dimensional vector space, in Section 4 we state an abstract lemma on Commutativity property as a consequence of other properties and apply it in the proof of the required properties of Σ in the finite-dimensional setting, in Section 5 we state some general results related to the notion of compactness, in Section 6 we extend the definition of Σ to the case of normed spaces and prove some of its properties-proofs of the remaining properties are given in Section 7, and in Section 8 we construct Σ for ANRs and we finish the proof of Theorem 2.1. Section 9 establishes a connection of Σ and σ to the fixed point index theory (Proposition 9.1), in Section 10 we consider mutual relations between homology and cohomology generators and orientations of vector spaces, in Section 11 we establish Proposition 11.1 on determination of σ in the smooth case, and finally, in Section 12 we finish the proof of Theorem 2.2. We use the following notation and terminology. By we denote the projections. I denotes the closed interval [0, 1]. By · we denote the norm of a normed space X and by B(x, ) we denote the closed ball {y ∈ X : x − y ≤ }. A map between topological spaces is called compact provided it is continuous and the closure of its image is compact. It is called locally compact provided its restriction to some neighborhood of each point of its domain is compact. Actually, a locally compact map is compact in some neighborhood of each compact subset of its domain. In order to shorten notation, we call a collection of maps f t : X → Y (where t ∈ I) a homotopy provided f : X × I (x, t) → f t (x) ∈ X is continuous (i.e., the map f is a homotopy in the usual meaning). The homotopy is compact (resp., locally compact) provided f is compact (resp., locally compact). The notation concerning pairs of sets is standard (cf. [D1]); in particular a set A is treated as the pair (A, ∅), for maps g and h, g (A, B)  . Unless otherwise stated, H and H * denote the singular homology and, respectively, the singular cohomology functors with coefficients in R. We treat the direct sum φ ⊕ ψ and the tensor product φ ⊗ ψ of homomorphisms φ : M → R and ψ : N → R of modules over R as the maps (x, y) → φ(x) + ψ(x) and, respectively, (x, y) → φ(x)ψ(y). We regard theČech cohomologiesȞ * as the direct limit of the singular ones; more exactly, for a pair (A, B) of locally compact subspaces of an ANR space X, where the limit is taken over the inverse system of all open neighborhoods (U, V ) of (A, B) and the corresponding inclusions. In consequence, there are natural maps μ is an isomorphism if both A and B are ANRs (therefore we identify H * (A, B) and H * (A, B) in that case). By f * , f * , andf we denote the homomorphism induced by f in singular homologies, singular cohomologies, anď Cech cohomologies, respectively. All nondescribed arrows in the diagrams are induced by inclusions. The image of a cohomology class u under a homomorphism induced by an inclusion is called a restriction of u. By ×, ·, · , , , \ we denote, respectively, both the homology and cohomology cross products, the scalar product, the cup product, the cap product, and the cohomology slant product defined as in [M] and [Sp] (or [D1], but with different sign conventions than given there). By a topological orientation η of Λ over R we mean a concordant family of homology classes (Recall that if Λ is oriented over Z, then it is oriented over an arbitrary R and, in general, Λ is always oriented over Z 2 .) We denote also by η the induced orientation on each open subset of Λ. For a one-point manifold pt we assume that the orientation is given by the (trivial) 0-dimensional singular simplex. If Λ is another manifold with an orientation η over R, by η × η we denote the orientation of Λ × Λ (as well as of each of its open subset) over R determined by for all compact L ⊂ Λ and L ⊂ Λ . If α : Λ → Ξ is a homeomorpism, by α * (η) we denote the induced orientation on Ξ, i.e., the orientation determined by α * (η L ) ∈ H k (Ξ, Ξ \ α(L)). This paper is a revised and extended version of a part of the unpublished preprint [Sr2].

The homomorphisms and their properties
The main result of the current paper is the following.
commutes. (III) Homotopy Invariance. If f t : U → X is a locally compact homotopy and commutes. (V) Multiplicativity. Let Λ be a manifold of dimension k and let η be its orientation over R. Assume that X is an ANR, (U , V ) is an open pair in X × Λ , and f : U → X is locally compact. Let commutes.

494
R. Srzednicki J FPTA (VI) Normalization. Let x 0 ∈ X and let c : (VII) Orientation Invariance. If Λ is a k-dimensional manifold with an orientation η , α : Λ → Λ is a continuous injection (hence a homeomorphism onto α(Λ) which is open in Λ by Domain Invariance Theorem), and the induced orientation α * (η) coincides with η on α(Λ), then the diagram commutes. (VIII) Commutativity. Let X be another ANR, let D be an open subset of X × Λ, let D be open in X × Λ, and let g : D → X and g : D → X be continuous. Assume that one of the maps g or g is locally compact.
Then (a) G and G induce mutually inverse homeomorphisms is an open pair in X × Λ, and g : U → Y is locally compact, then Vol. 13 (2013) Fixed point homomorphisms 495 commutes. (XI) Ring Naturality. Let ρ : R → R be a homomorphism of rings-with-unit and let ρ * denote both the natural map between the homologies and between the cohomologies with coefficients in R and R induced by it. Then the diagram Actually, the properties of Contraction and Topological Invariance are direct consequences of Commutativity. Moreover, we have the following theorem.
In Sections 3-8, in several steps we provide a construction of the homomorphism Σ satisfying Theorem 2.1. The proof of Theorem 2.2 is postponed to Section 12.
Let Σ be given by Theorem 2.1. Assume that K is a compact subset of Fix(f ). The set of open pairs (U, V ) ⊃ (Fix(f ), Fix(f )\K), and the inclusions J FPTA among them, is an inverse system, hence by (II), Σ (f,U,V ) form a direct system of homomorphisms. Define Theorem 2.3. The homomorphism σ has the following properties.
commutes. (III σ ) Homotopy Invariance. Let f t : D → X be a locally compact homotopy and let
The counterpart of Solvability for σ is redundant: if K = ∅, then the cohomologies of the pair (Fix(f ), Fix(f ) \ K) are equal to 0. The properties of σ follow the corresponding properties of Σ by passing to the limit. However, we do not treat Theorem 2.3 as a corollary of Theorem 2.1 since at some stage of the construction of Σ, our proof of the property (VIII σ ) predeceases the proof of (VIII) (see Step 3 in the proof of Lemma 4.1).
hence each of the properties (II)-(XI) is equivalent to the corresponding properties among (I σ )-(XI σ ).
By Remark 2.1, Theorem 2.2 has the following equivalent interpretation for σ.

Construction of Σ in finite-dimensional vector spaces
Let (U, V ) be an open pair in X × Λ and let K : Our aim is to construct Σ (f,U,V ) . Unless otherwise stated, in what follows we will assume that U is contained in the domain D of f , since in the general case the composition will satisfy all requirements. It follows, in particular, that Assume first X = R n . Let o n be a generator of H n (R n , R n \ 0) and let s n ∈ H n (R n , R n \ 0) be the dual generator to o n , i.e., s n , o n = 1. The generator o n determines the orientation of R n denoted by o. Define Actually, if the products are defined as in [D1], the above composition of homomorphisms should be multiplied by (−1) nk .
Remark 3.1. In the considered case Σ (f,U,V ) can be alternatively defined in the spirit of the fixed point transfer from [D2], as the composition Indeed, by formulas on products given in [D1] (taking into account the fact that the definitions of products in [D1] differ by a sign from the definitions in [M] and [Sp]) one has We prefer the present definition of Σ (f,U,V ) rather than Σ since it seams to have a simpler geometric meaning and directly generalizes a standard homology approach to the fixed point index (like in [D1]).
It is easy to see that the above-defined Σ (f,U,V ) does not depend on the choice of o n . Using a linear isomorphism we extend that definition to the case of an n-dimensional vector space X. Properties of homologies and cohomologies, and their products, imply immediately the following lemma.

Commutativity property for locally compact maps
We begin with an observation that (a) in (VIII) does not require any assumption on compactness of g or g , and it is easy to verify. Therefore (b) is the essential part of Commutativity. The following lemma reduces the proof of (b) for locally compact maps to verification of other properties. Let E denote a subclass of the class of all normed spaces, closed with respect to the cartesian products. Proof. We follow an idea from [D1,Subsection VII.5.9] or [G,Section 8]. Set and K := G −1 (K ). Denote by q and q the projections and a map It follows that and q : Fix( G) → Fix(g • G) and q : Fix( G) → Fix(g • G ) are homeomorphisms, hence Fix( G) ∩ U \ V is compact. By assumptions, G is a locally compact map and K := q −1 (K ) is compact.
Step 1. The diagram

commutes.
In order to prove the claim, define the following subsets of X × X × Λ: Vol. 13 (2013) Fixed point homomorphisms 501 and the locally compact homotopies: In particular, a 1 = G. One has It follows by (III) that commutes by (II). Let P denote a one-point space (hence a 0-dimensional connected manifold) and let an orientation θ of P be determined by the cycle equal to the (unique) 0-dimensional singular simplex on P . Let p and α be the projections α is a homeomorphism such that In the diagram u u l l l l l l l l l l l l l l l l l l The diagram commutes. Indeed, the upper left triangle commutes by the units property of the cohomology cross product, the upper right triangle commutes since X is path connected and therefore the homomorphism sends the generator 1 (X ×P ) to 1 by (VI), the lower triangle above the diagonal commutes by (V), and the lower triangle below the diagonal commutes by (VII). Since q = p • π • id X×X ×α −1 , by the commutativity of the above two diagrams, we get hence the claim is proved.
Step 2. The diagramȞ u u l l l l l l l l l l ľ q ∼ = / / σ ( G, K) 5 5 l l l l l l l l l l l l l l l l l l Indeed, the upper triangle commutes by Step 1 and the limit passage. Define Since G restricted to the set of fixed points is a homeomorphism (i.e., (a) in (VIII) holds), Vol. 13 (2013) Fixed point homomorphisms 503 By Step 1, Σ ( G, U , V ) • q * = Σ (g•G ,U ,V ) .

Since the diagram
hence by passing to the limit we get the commutativity of the lower triangle and Step 2 is proved.
Step 3. The conclusion of the lemma holds.
Indeed, since Step 2, by the commutativity of the diagram

Compactness in normed spaces
In this section we assume that X is a normed space and Λ is a first countable Hausdorff space. We provide sufficient conditions of the compactness of the set Fix(f ) ∩ U \ V . They modify well-known criteria for unparameterized maps.

Lemma 5.1. Let A be a closed subset of X × Λ such that p(A) is compact. If
Proof. In order to prove (a) assume that B is a closed subset of A, (x n , λ n ) ∈ B, and x n − f (x n , λ n ) → y ∈ X. Without loss of generality, we can assume that there exist z ∈ X and λ ∈ Λ such that f (x n , λ n ) → z, λ n → λ. Then (x n , λ n ) → (y +z, λ) ∈ B, since B is closed. It follows that f (y +z, λ) = z, hence y ∈ (j − f )(B). The conclusion (b) follows directly form (a), and (c) is a straightforward consequence of (a) and (b).

Lemma 5.2. Let A be as in Lemma
Let (U, V ) be an open pair in X × Λ. We apply the above lemmas to sets related to the homomorphism Σ.

Lemma 5.3. Assume that p(U ) is compact.
(a) If f : U → X is compact and It follows by assumptions that The right-hand set is compact by Lemma 5.1(b), hence (a) is proved. Similarly, hence (b) follows by Lemma 5.2. We end this section by a simple observation on an inverse of Lemma 5.3.

Construction of Σ in normed spaces
Throughout this section X denotes a normed space. At the beginning, assume , Fix(f ) ∩ U \ V is compact and by Lemma 5.1(c), Let g : U → X be a finite-dimensional -approximation of f | U (by Schauder Approximation Theorem, see [GD] or [G, (4.1)]), where 0 < < ζ. By Lemma 5.3(a), the set Fix(g)∩U \V is compact. Let Y be a finite-dimensional subspace of X which contains the image of g and let g U Y : U ∩ Y → Y be the restriction of g.
Proof. The independence of the choice of Y follows by property (IX) stated in Lemma 4.2. Let g : U → X be another -approximation of f | U with the image contained in a finite-dimensional subspace Y of X. We can assume Y = Y . Define a homotopy For every (x, λ) ∈ U and t ∈ I, g t (x, λ) is contained in the closed ball B(f (x, λ), ). That ball does not contain x provided (x, λ) ∈ U \ V \ U , hence by Lemma 5.3(b), the set such that U ⊂ U , f | U is a compact map, and p(U ) is compact (because Λ is a locally compact space). Set V := U ∩V . By Lemma 5.4, Fix(f )∩U \ V \U is empty. We define One can check that (II) in the finite-dimensional case implies that the above definition is independent of the choice of U . Proof. We apply Lemma 3.1. The properties (II), (IV)-(VII), (X), and (XI) follow directly from the corresponding properties in the finite-dimensional case.

Proof of (I).
Let Σ (f,U,V ) = 0. We can assume that U ⊂ D, p(U ) is compact, and f | U is a compact map. Let g n be a finite-dimensional 1/n-approximation of f in U . Let x n = g n (x n , λ n ) for some (x n , λ n ) ∈ U , for sufficiently large n (by the finite-dimensional case of (I)). We can assume λ n → λ. By the compactness of f , we can assume f (x n , λ n ) → y. Since f (x n , λ n ) − x n < 1/n, x n → y, and thus f (y, λ) = y.

Proof of (III). Choose an open set U such that
t∈I is compact, and f t | U is a compact homotopy. It follows that for each > 0 there exists a finite-dimensional homotopy g t : U → X which is an -approximation of f t . Set V := U \ V . If is small enough, by Lemmas 5.4 and 5.3(b), and by the finite-dimensional case of (III), it follows that hence the result follows.

Commutativity property in normed spaces
We extend Lemma 4.1 to the following general case.
Lemma 7.1. If X and X are normed spaces, then property (VIII) holds.
Proof. By Lemmas 4.1 and 6.2, property (VIII) holds provided both g and g are locally compact. Since (VIII) holds if and only if (VIII σ ) holds, and the role of g and g is symmetric in (VIII σ ) by (a) in (VIII), in order to finish the proof it suffices to assume that only g is locally compact. We adapt an argument from [G,Section 8].

Fixed point homomorphisms 507
As in the proof of Lemma 4.1, denote By assumptions, K and K = G(K) are compact and contained in D ∩ U and D ∩ U , respectively. Let E be an open subset of D such that hence, by (II), it suffices to prove that the diagram Vol. 13 (2013) Fixed point homomorphisms 509 commutes. Since the left-hand side triangle also commutes and h As a consequence of Lemma 7.1 we get the Contraction property for normed spaces, hence, by Lemma 6.2, we have the following lemma.
Lemma 7.2. If X, X , and Y are normed spaces, then the homomorphism Σ satisfies properties (I)-(XI).

Construction of Σ for ANRs
If X is an open subset of a normed space E, we define where i : X → E is the inclusion. Lemma 7.2 directly generalizes to the following one. Assume that X is an ANR. By [G, (9.3)], there exist an open subset Y of a normed space and continuous mappings r : Y → X and s : X → Y such that r • s = id X . Define By a similar argument as in [G,Section 10], one can check that the Commutativity property stated in Lemma 8.1 implies the independence of that definition from the choice of Y , r, and s. Finally, Theorem 2.1 follows by Lemma 8.1.

Relation of the homomorphisms to the fixed point index
In this section we assume R = Z. Let U be an open subset of an ANR X and let g : U → X be a locally compact map with a compact set of fixed points K, hence Fix(g • j) = K × Λ.

Orientations and (co-)homology generators
In order to avoid confusions with topological terminology, the term "vector orientation" refers to an orientation of a vector space (over R) in the linear algebra meaning. Let W be an n-dimensional vector space; n ≥ 1.
One has Vol. 13 (2013) Fixed point homomorphisms 511 Let ψ : W → W be a linear isomorphism. It transforms a basis w into the basis ψ(w) of Z, hence it induces a bijection ψ * : Or(W ) → Or(W ) given by ψ * (o(w)) := o(ψ(w)). Now we set up some notation concerning homology and cohomology. Below we concentrate on properties of Ω.

Lemma 10.2. The diagram of bijections
Proof. Let w be a basis of W . Then Proof. Let p : W × 0 → W be the projection. Since p * (s) = s × 1 0 , the first equation follows by Lemma 10.2.
Lemma 10.4. If s ∈ Gcoh(W ) and t ∈ Gcoh(Y ), then Proof. Let w and y be the bases of W and Y , respectively, and let Ω(s) = o(w) and Ω(t) = o(y). The cartesian product φ w × φ y sends the canonical basis of R n+m to w × y, hence Assume that Z = W + Y is a direct sum of subspaces W and Y . Let Moreover, are isomorphisms because the projection Z → W , parallel to Y , and the projection Z → Y , parallel to W , are deformation retractions.
As a corollary of Lemma 10.5 we have the following lemma.
Lemma 10.6. Let W , Y , Z, t, and κ be the same as in Lemma 10.5. Assume that Y is another subspace of Z such that W + Y = Z is a direct sum. Let be the isomorphism induced by the inclusion κ : Y → Z. If ω is a vector orientation of W , then ω ∧ Ω(κ * (κ * ) −1 (t)) = ω ∧ Ω(t).

Determining Σ in the regular-value case
In this section we assume that X = R n and Λ = R k , f is of C ∞ class, and 0 is a regular value of j − f (hence Fix(f ) is a manifold and we do not distinguish between singular andČech cohomologies of pairs of its open subsets). Let o be a (topological) orientation of R k . Our purpose is to determine Σ (o,f,U,V ) under the assumption that Σ is an arbitrary homomorphism satisfying properties (I)-(XI). We restrict ourselves to the case of a single-point set K, i.e., In the next section we describe how to reduce the case of an arbitrary compact K to that case by an application of Mayer-Vietoris sequences. Moreover, using suitable embeddings and applying properties (II), (VII), and (X), it is easy to pass to the case where X and Λ are arbitrary C ∞ -manifolds. At first we consider the linear case and we assume R = Z. Let f : R n × R k → R n be a linear map and let j : R n ×R k → R n be the projection. Assume that j − f is an epimorphism. Set Y := ker(j − f ) = Fix(f ).