A GENERALIZED BORSUK-ULAM THEOREM IN A REAL CLOSED FIELD

IKUMITSU NAGASAKI, TOMOHIRO KAWAKAMI, YASUHIRO HARA AND FUMIHIRO USHITAKI

Abstract. Let C_{k} be the cyclic group of order k and N = (R, +, ·, <, …) an o-minimal expansion of a real closed field R. Let X be a definably connected definable set with a free definable C_{k}-action. Assume that there exists a positive integer n such that H_{q}(X; ℤ/kℤ) = 0 for 1 ≤ q ≤ n. If Y is a definable set with a free definable C_{k}-action such that H_{n+1}(Y/C_{k}, ℤ/kℤ) = 0, then there is no definable C_{k}-map from X to Y. We also prove the topological version of this definable version.

1. Introduction

Let C_{k} be the cyclic group of order k. Let 𝕊^{n} be the n-dimensional unit sphere of the (n + 1)-dimensional Euclidean space ℝ^{n+1} with the antipodal C_{2}-action. From the viewpoint of transformation groups, the classical Borsuk-Ulam theorem states that if there exists a continuous C_{2}-map from 𝕊^{n} to 𝕊^{m}, then n ≤ m. There are several equivalent statements of it and many related generalizations (e.g. [2], [12], [13], [14], [16]).

The classical Borsuk-Ulam theorem is generalized to topological spaces by several authors. For example, J.W. Walker [20], Pedro L. Q Pergher, Denise de Mattos and Edivaldo L. dos Santos [17].

Several C_{k}-versions of the classical Borsuk-Ulam theorem are discussed in [10] and [7]. The following two theorems are C_{k}-versions for topological spaces which are generalizations of [20], [17], [10] and [7].

Theorem 1.1. Let X be an arcwise connected free C_{k}-space and Y a Hausdorff free C_{k}- space. If there exists a positive integer n such that H_{q}(X; ℤ/kℤ) = 0 for 1 ≤ q ≤ n and H_{n+1}(Y/C_{k}; ℤ/kℤ) = 0, then there is no continuous C_{k}-map from X to Y. Here this homology means the singular homology.

Let k be a prime. For a topological space Y, let D = {(y_{1}, …, y_{k}) ∈ Y × ⋯ × Y |y_{1} = ⋯ = y_{k}} be the diagonal and write Y^{∗} = Y × ⋯ × Y − D admitting the free C_{k}-action defined by g(y_{1}, y_{2}, …, y_{k}) = (y_{k}, y_{1}, …, y_{k−1}), where g generates C_{k}.

2 IKUMITSU NAGASAKI, TOMOHIRO KAWAKAMI, YASUHIRO HARA AND FUMIHIRO USHITAKI

Theorem 1.2. Let k be a prime and X an arcwise connected free C_{k}-space. If there exists a positive integer n such that H_{q}(X; ℤ/kℤ) = 0 for 1 ≤ q ≤ n and Y is a Hausdorff free C_{k}-space with H_{n+1}(Y^{∗}/C_{k}; ℤ/kℤ) = 0, then every continuous map f : X → Y has a C_{k}-coincidence point, that is, a point x such that f(x) = f(gx), where g is a generator of C_{k}.

2000 Mathematics Subject Classification. 57S10, 57S17, 55M20, 55M35, 03C64.

Keywords and P hrases. The Borsuk-Ulam theorem, o-minimal, real closed fields, finite groups, definable C_{k}-maps, continuous C_{k}-maps.

The fourth author is partially supported by Kyoto Sangyo University Research Grants.

The purpose of this paper is to consider the definable versions of Theorem 1.1 and Theorem 1.2

Let N = (R, +, ·, <, …) be an o-minimal expansion of a real closed field R. Any definable category is a generalization of the semialgebraic category. Many results in the semialgebraic geometry hold in the o-minimal setting and there exist uncountably many ominimal expansions of the standard structure of the field ℝ of real numbers ([18]). See also [4], [6], [11] for examples and constructions of o-minimal structures. General references on them are [3], [5], [19]. In this paper “definable” means “definable with parameters in N ”, everything is considered in N and each definable map is continuous unless otherwise stated.

The singular definable homology is introduced in [21].

Theorem 1.3 (Definable Borsuk-Ulam Theorem). Let X be a definably connected definable set with a free definable C_{k}-action. If there exists a positive integer n such that H_{q}(X; ℤ/kℤ) = 0 for 1 ≤ q ≤ n and Y is a definable set with a free definable C_{k}-action such that H_{n+1}(Y/C_{k}; ℤ/kℤ) = 0, then there is no definable C_{k}-map from X to Y. Here this homology means the singular definable homology.

If Y is a definable set with a definable C_{k}-action, then by 10.2.18 [3], Y/C_{k} is a definable set and the orbit map π : Y → Y/C_{k} is definable. If dim Y ≤ n, then by 4.1.6 [3] dim Y/C_{r} ≤ n. Thus if dim Y ≤ n, then H_{n+1}(Y/C_{k}; ℤ/kℤ) = 0.

Let S^{n} denote the n-dimensional unit sphere of R^{n+1}.

Corollary 1.4. (1) Suppose that k ≥ 3 and that C_{k} acts on S^{2m+1} and S^{2n+1} definably and freely. If there exists a definable C_{k}-map f : S^{2m+1} → S^{2n+1}, then m ≤ n.

(2) If S^{m} and S^{n} have free definable C_{2}-actions and there exists a definable C_{2}-map f : S^{m} → S^{n}, then m ≤ n.

Corollary 1.4 is a generalization of 1.1 [15].

Theorem 1.5. Let k be a prime and X a definably connected definable set with a free definable C_{k}-action. Assume that there exists a positive integer n such that H_{q}(X; ℤ/kℤ) = 0 for 1 ≤ q ≤ n. If Y is a definable set with H_{n+1}(Y^{∗}/C_{k}; ℤ/kℤ) = 0, then every definable map f : X → Y has a C_{k}-coincidence point, that is, a point x such that f(x) = f(gx), where g is a generator of C_{k}.

2. Proof of results

We first prove Theorem 1.3. Let ℤ/kℤ[C_{k}] denote the group ring of C_{k} over ℤ/kℤ. For any q ∈ ℕ ∪ {0}, the q-dimensional chain group C_{q}(X; ℤ/kℤ) has the standard C_{k}-action. Then this action induces ℤ/kℤ[C_{k}]-action on C_{q}(X; ℤ/kℤ).

Let g be a generator of C_{k}, α = 1 + g + ⋯ + g^{k−1}, and β = 1 − g. Then by definition αβ = βα = 0, for every q, αC_{q}(X; ℤ/kℤ) and βC_{q}(X; ℤ/kℤ) are ℤ/kℤ[C_{k}]- submodules of C_{q}(X; ℤ/kℤ) and α∂ = ∂α, β∂ = ∂β, where ∂ is the boundary operator of {C_{q}(X; ℤ/kℤ)}. Therefore {αC_{q}(X; ℤ/kℤ)} and {βC_{q}(X; ℤ/kℤ)} are subchain complexes of {C_{q}(X; ℤ/kℤ)}.

Proposition 2.1. For every q, the following two sequences are exact.

0 → αC_{q}(X; ℤ/kℤ) →^{i} C_{q}(X; ℤ/kℤ) →^{β} βC_{q}(X; ℤ/kℤ) → 0,

0 → βC_{q}(X; ℤ/kℤ) →^{j} C_{q}(X; ℤ/kℤ) →^{α} αC_{q}(X; ℤ/kℤ) → 0,

where i, j denote the inclusions and α (resp. β) stands for the multiplication of α (resp. β).

P roof. Since β ∘ i = 0, α ∘ j = 0, Im i ⊂ Ker β, Im j ⊂ Ker α.

Let s = ∑_{j} ∑_{i=0}^{k−1} n_{ji}g^{i}σ_{j} ∈ Ker β, where g is a generator of C_{k}. If l ≠ l′ and 0 ≤ i ≤ k − 1, then g^{i}σ_{l} ≠ σ_{l′} . Since βs = 0, for any j, ∑_{i=0}^{k−1} n_{ji}g^{i}(1 − g)σ_{j} = 0. Thus for every j, ∑_{i=1}^{k−1}(n_{ji} − n_{j(i−1)})g^{i}σ_{i} + (n_{j0} − n_{j(k−1)})σ_{j} = 0. Hence for each j, n_{j0} = n_{j1} = ⋯ = n_{jk−1}. We set n_{j} = n_{j0}(= n_{j1} = ⋯ = n_{jk−1}). Then We have s = ∑_{j} n_{j}(1 + g + ⋯ + g^{k−1})σ_{j} = α ∑_{j} n_{j}σ_{j} ∈ Im i. Therefore Ker β = Im i.

Let s = ∑_{j} ∑_{i=0}^{k−1} n_{ji}g^{i}σ_{j} ∈ Ker α. Since αs = ∑_{j}(n_{j0}+⋯+n_{j(k−1)})(1+⋯+g^{k−1})σ_{j} = 0, n_{j0} + ⋯ + n_{j(k−1)} = 0.

Thus s = ∑_{j}(n_{j0}(1 − g) + (n_{j0} + n_{j1})g(1 − g) + (n_{j0} + n_{j1} + n_{j2})g^{2}(1 − g) + ⋯ + (n_{j0} + n_{j1} + ⋯ + n_{j(k−2)})g^{k−2}(1 − g))σ_{j} ∈ Im j. Therefore Ker α = Im j. □

Let H_{q}^{α}(X, ℤ/kℤ) (resp. H_{q}^{β}(X, ℤ/kℤ)) denote the homology group induced from the chain complex {αC_{q}(X; ℤ/kℤ)} (resp. {βC_{q}(X; ℤ/kℤ)}).

By Proposition 2.1, we have the following theorem.

Theorem 2.2. The following two sequences are exact.

⋯ → H_{q}^{α}(X; ℤ/kℤ) →^{i_{∗}} H_{q}(X; ℤ/kℤ) →^{β_{∗}} H_{q}^{β}(X; ℤ/kℤ) →^{∂_{∗}} H_{q−1}^{α}(X; ℤ/kℤ) → …

⋯ → H_{q}^{β}(X; ℤ/kℤ) →^{j_{∗}} H_{q}(X; ℤ/kℤ) →^{α_{∗}} H_{q}^{α}(X; ℤ/kℤ) →^{∂′_{∗}} H_{q−1}^{β}(X; ℤ/kℤ) → ….

In particular, if p = 2, then α = β and

⋯ → H_{q}^{α}(X; ℤ/kℤ) →^{i_{∗}} H_{q}(X; ℤ/kℤ) →^{α_{∗}} H_{q}^{α}(X; ℤ/kℤ) →^{∂_{∗}} H_{q−1}^{α}(X; ℤ/kℤ) → …

is exact.

Definable fiber bundles are introduced in [8].

Proposition 2.3. Let X be a definable set with a free definable C_{k}-action. Then (X, π, X/C_{k}, C_{k}) is a principal definable C_{k}-fiber bundle, where π : X → X/C_{k} denotes the orbit map. In particular π : X → X/C_{k} is a definable covering map.

Let p : E → X be a definable map. We say that p has the definable homotopy lifting property if for any definable set Y, each definable homotopy h : Y × [0, 1] → X and a definable map F : Y → E such that p ∘ F (y) = h(y, 0) for all y ∈ Y, there exists a definable homotopy H : Y × [0, 1] → E such that p ∘ H = h and H(y, 0) = F (y) for all y ∈ Y.

Theorem 2.4 (4.10 [1]). Every definable covering map has the definable homotopy lifting property.

Corollary 2.5. Let X be a definable set with a free definable C_{k}-action. Then the orbit map π : X → X/C_{k} has the definable homotopy lifting property.

Proposition 2.6. Under the assumptions in Theorem 1.3, for every q, H_{q}^{α}(Y, ℤ/kℤ) ≅ H_{q}(Y/C_{k}, ℤ/kℤ).

P roof. We first show that the map α : C(Y; ℤ/kℤ) → C(Y; ℤ/kℤ) and the map π_{∗} : C(Y; ℤ/kℤ) → C(Y/C_{k}; ℤ/kℤ) induced from the orbit map π : Y → Y/C_{k} have the same kernel. Let σ be a singular s-simplex of Y. We need only to consider elements of C(C_{k}σ), since C(Y) ≅ ⊕_{[σ]∈Δ(s)/C_{k}}C(C_{k}σ), where Δ(s) is the set of singular s-simplexes of Y and Δ(s)/C_{k} is its orbit set under the induced action.

Since α(∑ n_{i}g^{i}σ) = (∑ n_{i})α(σ), α(∑ n_{i}g^{i}σ) = 0 if and only if ∑ n_{i} = 0, and similarly π_{∗}(∑ n_{i}g^{i}σ) = (∑ n_{i})π ∘ σ = 0 if and only if ∑ n_{i} = 0; therefore, both kernels coincide.

We next show that π_{∗} is surjective; namely, there is a definable lift τ̃ : Δ^{s} → Y of τ : Δ^{s} → Y/C_{k}, where Δ^{s} denotes the affine span of (s + 1)-points which are affine independent. Since Δ^{s} is definably contractible, there is a definable homotopy H′ : Δ^{s} × [0, 1] → Δ^{s} such that H′(−, 0) = c_{e_{0}} and H′(−, 1) = id_{Δ^{s}}, where c_{e_{0}} denotes the constant map whose value is e_{0} ∈ Δ^{s}. Then the composition H = τ ∘ H′ is a definable homotopy from the constant map c_{τ(e_{0})} to τ. Let y_{0} be a point of Y such that π(y_{0}) = τ(e_{0}), and c_{y_{0}} : Δ^{s} → Y the constant map whose value is y_{0}. Since H(−, 0) = π ∘ c_{y_{0}}, it follows from Corollary 2.5 that there exists a definable lift H̃ : Δ^{s} × [0, 1] → Y of H such that H̃(−, 0) = c_{y_{0}}. Then τ̃ := H̃(−, 1) is a definable lift of τ = H(−, 1).

Since π_{∗} is surjective, αC(Y; ℤ/kℤ) and C(Y/C_{p}; ℤ/kℤ) are isomorphic as chain complexes. Accordingly their homology groups are also isomorphic. □

The topological version of Proposition 2.6 is studied in 5.33 [9].

P roof of T heorem 1.3. Assume that there exists a definable C_{k}-map f : X → Y under the conditions of Theorem 1.3. Since X is definably connected, f(X) is definable connected. Hence f(X) is contained in a definably connected component of Y. Therefore it is sufficient to prove the case where Y is definably connected.

We first prove the case where k = 2. Since f is a definable C_{2}-map, αf_{♯} = f_{♯}α.

For simplicity, we abbreviate the coefficient ℤ/2ℤ in the definable homology. By Theorem 2.2, we have a commutative diagram

→ H_{n+1}^{α}(X) →^{∂_{∗}^{X}} H_{n}^{α}(X) →^{i_{∗}^{X}} H_{n}(X) →^{α_{∗}^{X}} H_{n}^{α}(X) →^{∂_{∗}^{X}} H_{n−1}^{α}(X) → …
f_{∗}^{α} ↓ f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗}^{α} ↓
→ H_{n+1}^{α}(Y) →^{∂_{∗}^{Y}} H_{n}^{α}(Y) →^{i_{∗}^{Y}} H_{n}(Y) →^{α_{∗}^{Y}} H_{n}^{β}(Y) →^{∂_{∗}^{Y}} H_{n−1}^{α}(Y) → …
→ H_{1}^{α}(X) →^{i_{∗}^{X}} H_{1}(X) →^{α_{∗}^{X}} H_{1}^{α}(X) →^{∂_{∗}^{X}} H_{0}^{α}(X) →^{i_{∗}^{X}} H_{0}(X) →^{α_{∗}^{X}} H_{0}^{α}(X) → 0
f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗} ↓
→ H_{1}^{α}(Y) →^{i_{∗}^{Y}} H_{1}(Y) →^{α_{∗}^{Y}} H_{1}^{α}(Y) →^{∂_{∗}^{Y}} H_{0}^{α}(Y) →^{i_{∗}^{Y}} H_{0}(Y) →^{α_{∗}^{Y}} H_{0}^{α}(Y) → 0

with exact rows.

By definition, (i_{∗}^{X})_{0} = 0 and (i_{∗}^{Y})_{0} = 0. Thus (α_{∗}^{X})_{0} : H_{0}(X) → H_{0}^{α}(X) and (α_{∗}^{Y})_{0} : H_{0}(Y) → H_{0}^{α}(Y) are isomorphisms. By assumption, H_{0}(X) ≅ ℤ/2ℤ. Hence H_{0}(X) ≅ H_{0}^{α}(X) ≅ ℤ/2ℤ. Similarly, H_{0}(Y) ≅ H_{0}^{α}(Y) ≅ ℤ/2ℤ. Since (f_{∗})_{0} : H_{0}(X) → H_{0}(Y) is an isomorphism and (α_{∗}^{Y})_{0} ∘ (f_{∗})_{0} = (f_{∗}^{α})_{0} ∘ (α_{∗}^{X})_{0}, (f_{∗}^{α})_{0} : H_{0}^{α}(X) → H_{0}^{α}(Y) is an isomorphism. Since (i_{∗}^{X})_{0} = 0, we have Im (∂_{∗}^{X})_{1} = Ker (i_{∗}^{X})_{0} = H_{0}^{α}(X). Thus we see that (∂_{∗}^{Y})_{1}∘(f_{∗}^{α})_{1} = (f_{∗}^{α})_{0}∘(∂_{∗}^{X})_{1} : H_{1}^{α}(X) → H_{0}^{α}(Y) is a non-zero homomorphism. Hence (f_{∗}^{α})_{1} : H_{1}^{α}(X) → H_{1}^{α}(Y) is a non-zero homomorphism. Using the assumptions on X, we see that (∂_{∗}^{X})_{q} : H_{q}^{α}(X) → H_{q−1}^{α}(X) is an isomorphism for each 1 ≤ q ≤ n. Using this fact and by induction, we have the claim that (f_{∗}^{α})_{q} : H_{q}^{α}(X) → H_{q}^{α}(Y) is a non-zero homomorphism for each 0 ≤ q ≤ n.

By Proposition 2.6, H_{n+1}^{α}(Y) ≅ H_{n+1}(Y/C_{p}). Thus H_{n+1}^{α}(Y) = 0. Hence (i_{∗}^{Y})_{n} : H_{n}^{α}(Y) → H_{n}(Y) is injective and (i_{∗}^{Y})_{n} ∘ (f_{∗}^{α})_{n} : H_{n}^{α}(X) → H_{n}(Y) is a non-zero homomorphism.

On the other hand, since H_{n}(X) = 0, (i_{∗}^{Y})_{n} ∘ (f_{∗}^{α})_{n} = (f_{∗})_{n} ∘ (i_{∗}^{X})_{n} = 0. This contradiction proves the theorem in this case.

Next we prove the case where k > 2. For simplicity, we abbreviate the coefficient ℤ/kℤ in the definable homology. By Theorem 2.2, we have two commutative diagrams

→ H_{n}^{α}(X) →^{i_{∗}^{X}} H_{n}(X) →^{β_{∗}^{X}} H_{n}^{β}(X) →^{∂_{∗}^{X}} H_{n−1}^{α}(X) → …
f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{β} ↓ f_{∗}^{α} ↓
→ H_{n}^{α}(Y) →^{i_{∗}^{Y}} H_{n}(Y) →^{β_{∗}^{Y}} H_{n}^{β}(Y) →^{∂_{∗}^{Y}} H_{n−1}^{α}(Y) → …
→ H_{1}^{α}(X) →^{i_{∗}^{X}} H_{1}(X) →^{β_{∗}^{X}} H_{1}^{β}(X) →^{∂_{∗}^{X}} H_{0}^{α}(X) →^{i_{∗}^{X}} H_{0}(X) →^{β_{∗}^{X}} H_{0}^{β}(X) → 0
f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{β} ↓ f_{∗}^{α} ↓ f_{∗} ↓ f_{∗}^{β} ↓ f_{∗} ↓
→ H_{1}^{α}(Y) →^{i_{∗}^{Y}} H_{1}(Y) →^{β_{∗}^{Y}} H_{1}^{β}(Y) →^{∂_{∗}^{Y}} H_{0}^{α}(Y) →^{i_{∗}^{Y}} H_{0}(Y) →^{β_{∗}^{Y}} H_{0}^{β}(Y) → 0

and

→ H_{n+1}^{α}(X) →^{∂_{∗}^{′X}} H_{n}^{β}(X) →^{j_{∗}^{X}} H_{n}(X) →^{α_{∗}^{X}} H_{n}^{α}(X) →^{∂_{∗}^{′X}} H_{n−1}^{β}(X) → …
f_{∗}^{α} ↓ f_{∗}^{β} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗}^{β} ↓
→ H_{n+1}^{α}(Y) →^{∂_{∗}^{′Y}} H_{n}^{β}(Y) →^{j_{∗}^{Y}} H_{n}(Y) →^{α_{∗}^{Y}} H_{n}^{α}(Y) →^{∂_{∗}^{′Y}} H_{n−1}^{β}(Y) → …
→ H_{1}^{β}(X) →^{j_{∗}^{X}} H_{1}(X) →^{α_{∗}^{X}} H_{1}^{α}(X) →^{∂_{∗}^{′X}} H_{0}^{β}(X) →^{j_{∗}^{X}} H_{0}(X) →^{α_{∗}^{X}} H_{0}^{α}(X) → 0
f_{∗}^{β} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗}^{β} ↓ f_{∗} ↓ f_{∗}^{α} ↓ f_{∗} ↓
→ H_{1}^{β}(Y) →^{j_{∗}^{Y}} H_{1}(Y) →^{α_{∗}^{Y}} H_{1}^{α}(Y) →^{∂_{∗}^{′Y}} H_{0}^{β}(Y) →^{j_{∗}^{Y}} H_{0}(Y) →^{α_{∗}^{Y}} H_{0}^{α}(Y) → 0

with exact rows.

We easily see that (i_{∗}^{X})_{0} = 0 and (i_{∗}^{Y})_{0} = 0. Thus (β_{∗}^{X})_{0} : H_{0}(X) → H_{0}^{β}(X) and (β_{∗}^{Y})_{0} : H_{0}(Y) → H_{0}^{β}(Y) are isomorphisms. Since (f_{∗})_{0} : H_{0}(X) → H_{0}(Y) is an isomorphism, we have the claim that (f_{∗}^{β})_{0} : H_{0}^{β}(X) → H_{0}^{β}(Y) is an isomorphism. Similarly we see that (f_{∗}^{α})_{0} : H_{0}^{α}(X) → H_{0}^{α}(Y) is an isomorphism from the second diagram. Since H_{1}(X) = 0 and (i_{∗}^{X})_{0} = 0, (∂_{∗}^{X})_{1} : H_{1}^{β}(X) → H_{0}^{α}(X) is an isomorphism. Similarly (∂_{∗}^{′X})_{1} : H_{1}^{α}(X) → H_{0}^{β}(X) is an isomorphism. Since (∂_{∗}^{Y})_{1} ∘ (f_{∗}^{β})_{1} = (f_{∗}^{α})_{0} ∘ (∂_{∗}^{X})_{1} and (∂_{∗}^{′Y})_{1} ∘ (f_{∗}^{α})_{1} = (f_{∗}^{β})_{0} ∘ (∂_{∗}^{′X})_{1}, (f_{∗}^{α})_{1} : H_{1}^{α}(X) → H_{1}^{α}(Y) and (f_{∗}^{β})_{1} : H_{1}^{α}(X) → H_{1}^{β}(Y) are non-zero homomorphisms. By induction, we have the claim that (f_{∗}^{α})_{q} : H_{q}^{α}(X) → H_{q}^{α}(Y) and (f_{∗}^{β})_{q} : H_{1}^{α}(X) → H_{q}^{β}(Y) are non-zero homomorphism for each 0 ≤ q ≤ n. By Proposition 2.6, H_{n+1}^{α}(Y) ≅ H_{n+1}(Y/C_{p}). Hence H_{n+1}^{α}(Y/C_{p}) = 0 and (j_{∗}^{Y})_{n} : H_{n}^{β}(Y) → H_{n}(Y) is injective. Therefore (j_{∗}^{Y})_{n} ∘ (f_{∗}^{β})_{n} is a non-zero homomorphism.

On the other hand, (j_{∗}^{Y})_{n} ∘ (f_{∗}^{β})_{n} = (f_{∗})_{n} ∘ (j_{∗}^{X})_{n} = 0 because H_{n}(X) = 0. This is a contradiction. Therefore the proof is complete. □

P roof of T heorem 1.5. Suppose that f(x) ≠ f(gx) for any x ∈ X. Then the map F : X → Y^{∗} defined by F (x) = (f(x), f(gx), …, f(g^{k−1}x)) is a definable C_{k}-map. This contradicts Theorem 1.3. □

P roof of T heorem 1.1 and T heorem 1.2. Similar proofs of Theorem 1.3 and Theorem 1.5 prove Theorem 1.1 and Theorem 1.2, respectively. □

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[21] A. Worheide, O-minimal homology, Ph.D. thesis (1996), University of Illinois at Urbana-Champaign.

Department of Mathematics, Kyoto Prefectural University of Medicine, 13 Nishi-Takatsukaso-Cho, Taishogun Kita-ku, Kyoto 603-8334, Japan

Department of Mathematics, Faculty of Education, Wakayama University, Sakaedani Wakayama 640-8510, Japan

Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan

Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Kamigamo Motoyama, Kita-ku, Kyoto 603-8555, Japan

E-mail address: nagasaki@koto.kpu-m.ac.jp
E-mail address: kawa@center.wakayama-u.ac.jp
E-mail address: hara@math.sci.osaka-u.ac.jp
E-mail address: ushitaki@ksuvx0.kyoto-su.ac.jp