Factorization of certain maps up to homotopy

Abstract

If/: X->Y is a map of a space X into a space Y, we say that/ is a /oca/ connection in dimension n, provided that for every point yE.Y and every neighborhood N of y there is a neighborhood FCA7' of y such that for 0 g & ^ «-1 any map g : Sk->/-1 F extends to a map g': Bk+1-^f~1N and for any map g: 5"-¡-/"^F the map fg: 5n->F extends to a map fe: Bn+1->N. Using star-refinements of open covers and a standard approximation technique we establish the following theorem (a slightly weaker form of which has been announced by Price [2]).

Cell-like mappings of 𝐴𝑁𝑅’𝑠

“…A crucial step in the proof of (2.1) T. Price independently and G. Kozlowski obtained versions of (2.2) (see [7] and [9]). Added in proof.…”

Approximate fibrations and a movability condition for maps

“…Approximate fibrations and approximate lifting were introduced in [3] as an abstraction of the useful lifting properties possessed by t/V^-maps [9], [11], [12]. It is shown in [3] that approximate fibrations have shape theoretic properties analogous to the homotopy theoretic properties of Hurewicz fibrations.…”

“…)], so pfe(a?, t) = PSί(Xf V) which is α)-close to g(x 9 y) = λ(a?, j/), which is α>-close to h(x, t) by our choice of g(a ), so ph(x, t) is ε-close to h(x, t). If t > q{x) 9 ph(x 9 t) is ω-close to g(χ 9 t) = λ(a?, ί) Proof. Given ε, let δ = δ k < δ fc _ x < < δ 0 = ε be a collection of covers so that δ t plays the role of δ in 1.1 for ε = δ t^l9 i > 0.…”

“…Necessity is clear. For the converse, observe that approximate fibrations can be described in terms of approximate path lifting functions [3], whose existence is equivalent to the AHLP for Δ 9 .…”

Mappings between ANRs that are fine homotopy equivalences

“…Various versions of Lemma 3 have been proven by Smale [8], Armentrout and Price [2], Kozlowski [5] and Lacher [6], The difference in this lemma is that K is not required to be a finite dimensional complex.…”