HOMOTOPY EXTENSION THEOREM FOR A FIBER–PRESERVING PIECEWISE LINEAR μ–HOMOTOPY

Tran van Thuong Missouri Southern State College

\underline{Abstract}. In this paper, some nice extension properties of a PL R^{∞}–manifold will be investigated. As a consequence, a homotopy extension theorem for a fiber–preserving piecewise linear μ–homotopy will be proved.

1. Introduction and Definitions. R^{∞}–manifolds have been studied by Heisey and Sakai. In 1975, Heisey [2] proved that any R^{∞}–manifold is a countable direct limit of finite dimensional compact metric spaces. In 1981, Heisey [3] introduced the notion of a piecewise linear R^{∞}–structure and he defined the piecewise linear R^{∞}–manifold; then he proved that any separable paracompact piecewise linear R^{∞}–manifold may be regarded as a polyhedron in R^{∞}.

If f : X → Y is a function, we denote the image of f by f(X). If A ⊂ X, then f|A : A → Y will denote the restriction of f to A. The symbol π_{x} : X × Y → X, will represent the projection onto X. Given a homotopy f : X × I → Y, and t ∈ I, the function f_{t} : X → Y is defined by f_{t}(x) = f(x, t). If X is a space and {B_{a}|a ∈ A} is a collection of subspaces of X, then X is said to have the \underline{weak topology} generated by \{B_{a}\} provided a set U ⊂ X is open if and only if U ∩ B_{a} is open in B_{a} for all a ∈ A. Suppose for a sequence of spaces {X_{n}|n ≥ 1}, X_{n} is a subspace of X_{n+1}, we define the direct limit of the sequence, denoted dirlim X_{n}, to be the set ∪X_{n} with the weak topology generated by the collection {X_{n}}. In particular, R^{∞} = dirlim R^{n}, as we consider the line R^{1} ⊂ R^{2} ⊂ ⋯ ⊂ R^{n} ⊂ · · ·. Here, we think of R^{∞} as

{(x_{i}) : all but finitely many x_{i} are 0} ,

and identify R^{n} with R^{n} ×{0}×{0}×⋯ ⊂ R^{∞}. Let a be one–point set in R^{n}, and A, B be subsets of R^{n}, we say that aB is a \underline{cone} with vertex a, and base B (or simply that aB is a cone) if each point not equal to a is expressed uniquely as λa + μb with b ∈ b, λ, μ ≥ 0 and λ + μ = 1. A subset P ⊂ R^{n} is called a \underline{polyhedron in the sense of Rourke − Sanderson} [5] if for every point x ∈ P, there is a cone neighborhood xL, where L is compact. ∂L denotes the boundary of L. For this and other basic definitions and results from piecewise linear topology, see [5].

Following Heisey [3], we defined: A subset X of R^{∞} is an R^{∞}–\underline{polyhedron} if for each compact polyhedron C in R^{∞}, C∩X is a polyhedron in the usual sense of Rourke–Sanderson [5]. A map f : X → Y between two R^{∞}–polyhedra is an R^{∞}–\underline{piecewise linear} (R^{∞}–PL) if for each compact polyhedron C ⊂ X and any choice of n such that f(C) ⊂ Y ∩ R^{n}, then f|C : C → Y ∩ R^{n} is PL in the usual sense of Rourke–Sanderson [5]. For the convenience of notation, we will denote R^{∞}–piecewise linear by R^{∞}–PL. An R^{∞}–\underline{manifold} X is a Hausdorff topological space such that for each x ∈ X, there exists a homeomorphism f_{x} mapping some open set containing x onto an open set in R^{∞}. A \underline{piecewise linear} R^{∞}–\underline{atlas for a space M} is a collection of pairs {(U_{α}, δ_{α})} where {U_{α}} is an open cover of M by nonempty sets, δ_{α} : U_{α} → δ_{α}(U_{α}) is a homeomorphism onto an open subset of R^{∞}, and where, if U_{α} ∩U_{β} ≠ ∅, ϕ_{β}ϕ_{α}^{−1} : δ_{α}(U_{α} ∩ U_{β}) → ϕ_{β}(U_{α} ∩ U_{β}) is R^{∞}–PL. A \underline{piecewise linear} R^{∞}–\underline{structure for M} is a maximal PL R^{∞}–atlas for M; since any PL R^{∞}–atlas for the space M is contained in a unique maximal PL R^{∞}–atlas, a PL R^{∞}–atlas for M determines a PL R^{∞}–structure for M. A \underline{piecewise linear} R^{∞}–\underline{manifold} (PL R^{∞}–manifold) is a paracompact R^{∞}–manifold with a PL R^{∞}–structure which is defined by Heisey [3]. By Sakai [6], a PL R^{∞}–manifold can be regarded as an open set in R^{∞}. An R^{∞}–\underline{polyhedron in a PL} R^{∞}–\underline{manifold} is an R^{∞}– polyhedron in R^{∞} as we regard it as an open subset in R^{∞}. Let △_{n} be an n–simplex, M and N be PL R^{∞}–manifolds, we define: a map f : △_{n} × M → △_{n} × N is \underline{fiber − preserving} (fp) if π△_{n}∘f = π△_{n}. By Sakai [7], M = dirlimM_{n} where each M_{n} is a compact polyhedral manifold in some R^{n} ⊂ R^{∞}, that is, M_{n} is a compact polyhedron in R^{n} and a PL manifold; then we define a map f : M → N is R^{∞}–\underline{piecewise linear} (R^{∞}–PL) if f|M_{n} is piecewise linear in the sense of Rourke–Sanderson. An fp map f from an R^{∞}–polyhedron P in △_{n} × M → △_{n} × N is called R^{∞}–\underline{piecewise linear} (R^{∞}–PL) if π_{N} ∘ f : P → N is R^{∞}–PL. Two maps f and g : M → N are μ–\underline{close} if given an open cover μ of N, and if for each y ∈ M, there is an open set U ∈ μ such that {f(y), g(y)} ⊂ U. Given any open cover μ of any space X, a homotopy H : Y ×I → X is an μ–\underline{homotopy} if for each element y ∈ Y, the set H(y×I) is contained in some U ∈ μ. A homotopy H is \underline{stationary} on A or relA where A ⊂ Y if we have H(y × I) = {H(y, 0)} for each y ∈ A. A homotopy H : △_{n} × M × I → △_{n} × N is called R^{∞}–\underline{piecewise linear fiber − preserving homotopy} if H_{0} and H_{1} are R^{∞}–PL fp maps where H_{0} : △_{n} × M × 0 → △_{n} × N, H_{1} : △_{n} × M × 1 → △_{n} × N and H_{t} is an R^{∞}–PL for each t ∈ [0, 1]. Let μ, ν be families of subsets of X, U ∈ μ, we define:

\begin{matrix}St(U, ν) = ⋃\{V |V ∈ ν : U ∩ V ≠ ∅\}. \\ St(μ, ν) = ⋃\{St(U, ν) : U ∈ μ\}. \end{matrix}

Inductively, we define:

St^{n+1}(μ, ν) = St(St^{n}(μ, ν)) for n > 1.

μ <^{∗} ν means that St(μ, μ) is a refinement of ν. A closed subspace A of a space X is said to have \underline{the extension property in X} with respect to a space Y, iff every map f : A → Y can be extended over X. The closed subspace A is said to have the \underline{neighborhood extension property in X} with respect to Y, iff every map f : A → Y can be extended over some open subspace U of X which contains A. Here, the open subspace U of X may depend on the given map f.

Let C be any topological class of spaces, by an \underline{absolute extensor for the class C}, (AE for the class C), we mean a space Y such that every closed subspace A of any space X in the class C has the extension property in X with respect to Y. Similarly, by an \underline{absolute neighborhood extensor} for the class C (ANE for the class C), we mean a space Y such that every closed subspace A of any space X in the class C has the neighborhood extension property in X with respect to Y.

2. Extension of PL maps.

\underline{Proposition 2.1}. Let M, N be PL R^{∞}–manifolds. Let A ⊂ B be two compact subpolyhedra of △_{n} × N. Let μ be an open cover of △_{n} × M and g : B → △_{n} × M be a fp continuous map where g|A is a fp R^{∞}–PL map, then f is μ–homotopic to g (fp) rel A.

\underline{Proof}. By Sakai [6], we may regard M as an open set in R^{∞}. From compactness of g(B), we have g(B) ⊂ △_{n} × (M ∩ R^{p}) for some p > 0. Consider a finite subcover of g(B): {V_{i} × U_{i}}_{i=1,2,...,q}. where V_{i} is a basis open set in △_{n}, U_{i} is a basis open set in M and V_{i} × U_{i} is contained in some open set of μ. Since each U_{i} is a basis open set in R^{∞}, we may assume that

U_{j} = (−ϵ_{1}^{j}, ϵ_{1}^{j}) × ⋯ × (−ϵ_{p}^{j}, ϵ_{p}^{j}) × · · · .

For each j > 0, let ϵ_{p+j} = min{ϵ_{p+j}^{1}, …, ϵ_{p+j}^{q}}. Observe that M ∩ R^{p} is a subset of (M ∩ R^{p}) × (−ϵ_{p+1}, ϵ_{p+1}) · · ·. Now let m = 2(n + dimB) and

ϵ = min{ϵ_{p+1}, ϵ_{p+2}, …, ϵ_{p+m}}.

Consider g : B → △_{n} × (M ∩ R^{p}) ⊂ △_{n} × (M ∩ R^{p}) × (−ϵ, ϵ)^{m} ⊂ △_{n} × M. Let h be an R^{∞}–PL embedding from B into ∂(−\frac{ϵ}{2}, \frac{ϵ}{2})^{m}. Let V be an open set containing A so that Cl V ⊂ B, then by Heisey [3] and Henderson [1], there is an R^{∞}–PL map Ψ : B → [0, 1] so that Ψ^{−1}(0) = A and Ψ(B\V) = 1. First, we write h(x) = (h_{1}(x), h_{2}(x), …), then we define f : B → △_{n} × M by the following way: we write f(x) = (f_{1}(x), f_{2}(x), …) where f_{i}(x) = max{−Ψ(x), min{Ψ(x), h_{i}(x)}} for x ∈ B; then f is the desired fp R^{∞}–PL map. Note that by the way we construct the embedding h, we have {f(x), g(x)} ⊂ V_{i} × U_{i} for some i = 1, 2, 3, …, q. Therefore, by the convexity of V_{i} × U_{i}, we can define a fp R^{∞}–PL homotopy H(x, t) = tf(x) + (1 − t)g(x), and H is the desired homotopy.

\underline{Proposition 2.2}. Let f : △_{n} × M → △_{n} × N be a fp map, M and N are PL R^{∞}– manifolds, A a closed R^{∞}–subpolyhedron of △_{N} × M and f|A : A → △_{N} × N is a fp R^{∞}–PL map, then there is a fp R^{∞}–PL map g : △_{n} × M → △_{n} × N such that g|A = f|A.

\underline{Proof}. By Heisey [3], we regard M as a closed PL submanifold of R^{∞}. By Sakai [7], we write M = dirlimM_{n} where each M_{n} is a compact polyhedral manifold in some R^{n} ⊂ R^{∞}, and M_{n} ⊂ M_{n+1}. Similarly, write N = dirlimN_{n} where each N_{n} is a compact polyhedral manifold in some R^{n} ⊂ R^{∞}. It follows that △_{n} × M = dirlim(△_{n} × M_{n}) and △_{n} × N = dirlim(△_{n} × N_{n}). Let A_{i} = A ∩ (△_{n} × M_{i}) for i = 1, 2, 3, …, then A_{i} is a compact R^{∞}–polyhedron and A = dirlimA_{i}. Now, we consider f|△_{n} × M_{1} where f|△_{n} × M_{1} : △_{n} × M_{1} → △_{n} × N and f|A_{1} is R^{∞}–PL, we may apply Proposition 2.1 to obtain a fp R^{∞}–PL map g_{1} : △_{n} × M_{1} → g(△_{n} × M_{1}) ⊂ △_{n} × N_{j} such that g_{1}|A_{1} = f|A_{1}.

Now, extend g_{1} to a fp R^{∞}–PL map g̃_{1} as follows:

\begin{aligned}g̃_{1}(x) & = f(x) if x ∈ A_{2} . \\ & = g_{1}(x) if x ∈ △_{n} × M_{1} . \end{aligned}

By Hu [4], there is an extension of g̃_{1} to g̃_{2} where g̃_{2} : △_{n} × M_{2} → △_{n} × N such that \overline{g}_{2}|(△_{n} × M_{1}) ∪ A_{2} = g̃_{1}. Applying Proposition 2.1 again, we may obtain a fp R^{∞}–PL map g_{2} : △_{n} × M_{2} → g_{2}(△_{n} × M_{2}) ⊂ △_{n} × N_{k} ⊂ △_{n} × N such that g_{2}|(△_{n} × M_{1}) ∪ A_{2} = g̃_{1}. Continuing this process, we obtain a family of fp R^{∞}–PL maps {g_{n}} where g_{n} : △_{n} ×M_{n} → △_{n} × N such that g_{n}|(△_{n} × M_{n−1}) ∪ A_{n} = g̃_{n−1}. Therefore {g_{n}} induces a fp R^{∞}–PL map g where g : △_{n} × M → △_{n} × N such that g|A = f|A.

\underline{Proposition 2.3}. Let M, N be PL R^{∞}–manifolds, A a closed R^{∞}–polyhedron of M, μ an open cover of N. Let h be a homotopy of M to N such that h|A × I : A × I → N, is a μ–homotopy; then there is a PL map Ψ : M → [0, 1] such that Ψ^{−1}(1) ⊃ A and {h(m × [0, Ψ(m)])} < μ.

\underline{Proof}. Consider M as an open set in R^{∞}. Now, for each t ∈ I, x ∈ A, we have an open neighborhood V_{t} of x such that h(V_{t} × W_{t}) ⊂ U_{x}. But {W_{t} : T ∈ I} is an open cover of I and I is compact; there are finite numbers t_{1}, …, t_{n} ∈ I such that

I = ⋃_{i=1}^{n} W_{t_{i}} .

Now, put

V_{x} = ⋂_{i=1}^{n} V_{t_{i}} , and W = ⋃_{a∈A} V_{a} ,

then h|W × I is a μ–homotopy. By Heisey [3], there is a PL map Ψ : M → I such that Ψ|A = 1 and Ψ|M − W = 0; then Ψ satisfies the conditions of the Proposition.

\underline{Theorem (Homotopy Extension Theorem)}. Let M and N be PL R^{∞}–manifolds and μ an open cover of △_{n} × M. Let A be a closed R^{∞}–subpolyhedron of △_{n} × N. Let h : A × I → △_{n} × M be an R^{∞}–PL fp μ–homotopy such that h_{0} : A → △_{n} × M, has a fp R^{∞}–PL extension f : △_{n} × N → △_{n} × M. Then h extends to h̃ : △_{n} × N × I → △_{n} × M which is an R^{∞}–PL fp μ–homotopy with h̃_{0} = f. Moreover, if Γ is a compact subpolyhedron of △_{n} and if h is stationary on A ∩ (Γ × N), then h̃ can be chosen to be stationary on Γ × N.

\underline{Proof}. Let \overline{H} : ([△_{n} × N × 0] ∪ [A ∪ Γ × N]) × I → △_{n} × M be the map defined as follows:

\overline{H}(λ, x, t) =h(λ, x, t) if (λ, x) ∈ A
f(λ, x) if λ ∈ Γ .
f(λ, x) if t = 0 .

Observe that R^{∞} is an AE for the countable direct limits of compact (metric) spaces, and M can be embedded in R^{∞} as an open set. It is easy to see that M is ANE for the countable direct limits of compacta. Note that △_{n} × N = dirlim{△_{n} × N_{g}} where each N_{g} is a PL finite dimensional compact submanifold of N_{g+1} and each △_{n} × N_{g} is compact; it follows that M is ANE for the class △_{n} × N where N is a PL R^{∞}–manifold. Now we consider π_{M} ∘ \overline{H}|[A ∪ Γ × N] × I : [A ∪ Γ × N] × I → M. Since M is ANE for the direct limits of towers of compacta, there is an open set W in △_{n} × N × I containing [A ∪ Γ × N] × I such that π_{M} ∘ \overline{H} extends to W. Since I is compact, we may assume that W = V × I where V ⊃ [A ∪ Γ × N] then [A ∪ Γ × N] and (△_{n} × N)\V are disjoint closed subsets of the normal space △_{n} × N. By Urysohn’s Lemma, there is a map g : △_{n} × N → I such that:

\begin{matrix}g(λ, x) = & 1 if (λ, x) ∈ [A ∪ Γ × N] \\ 0 if (λ, x) ∈ (△_{n} × N)\V . \end{matrix}

Now we extend π_{M} ∘ \overline{H} by H(λ, x, t) = π_{M} ∘ \overline{H}(λ, x, tg(λ, x)). As a consequence of Proposition 2.2, there is an R^{∞}–PL map G : △_{n} × N × I → M such that G|[△_{n} × N × 0] ∪ [A ∪ Γ × N] × I = H. Now, we define: \overline{h}(λ, x, t) = (λ, G(λ, x, t)) for (λ, x, t) ∈ △_{n} × N × I; then \overline{h} is an R^{∞}–PL fp extension of h. Applying Proposition 2.3, there is a PL map ϕ : △_{n} × N → I such that ϕ^{−1}(1) ⊃ [A ∪ Γ × N] and the family \{\overline{h}(z × [0, ϕ(z)] : z ∈ △_{n} × N\} < μ. Now defining h̃(λ, x, t) = \overline{h}(λ, x, tϕ(λ, x)) for (λ, x, t) ∈ △_{n} ×N ×I; then h̃ is the desired R^{∞}–PL fp μ–homotopy.

\underline{References}

1. D. W. Henderson, “A Simplicial Complex Whose Product with any ARN is a Simplicial Complex,” General Topology and its Applications, (1973), 81–83.

2. R. E. Heisey, “Manifolds Modelled on R^{∞} or Bounded Weak Topology,” Trans. Amer. Math. Soc., 206 (1975), 295–312.

3. R. E. Heisey, “Embedding Piecewise R^{∞}–manifolds into R^{∞},” Topology Proc., 6 (1981), 317–328.

4. S. Hu, \underline{Theory of Retract} Detroit–Wayne State University Press, 1965.

5. C. P. Rourke and B. J. Sanderson, \underline{Introduction to Piecewise Linear Topology}, Ergebnisse Math, Grenzgebiete, Band 69, Springer–Verlag, NY, 1972.

6. K. Sakai, “Each R^{∞}–manifold has a Unique Piecewise Linear R^{∞}–structure,” Proc. Amer. Math. Soc., 90 (1984), 616–618.

7. K. Sakai, \underline{Studies on Continua and Infinite − Dimensional Manifolds}, RIMS Kokyuroku 509, Kyoto University, Kyoto, Japan, 51–57.