# are acyclic fibrations of nice spaces absolute extensors for perfectly normal spaces?

A space $$Y$$ is called an absolute extensor for normal spaces (also sometimes solid) if, for any normal space $$X$$, closed subset $$A$$ of $$X$$, and map $$f:A\to Y$$, there exists a map $$f′:X\to Y$$ such that $$f′|A=f$$, i.e. $$A\to X$$ has the left lifting property with respect to the map $$Y\to pt$$ from $$Y$$ to a singleton $$pt$$ $$A\to B \perp Y\to pt$$

What is the analogous notion for a map $$g:Y_1\to Y_2$$ instead of a space $$Y$$? The analogous notion for maps instead of spaces is called being soft with respect to a pair of spaces $$(A,B)$$ where $$A$$ is a closed subset of a normal space $$B$$. (E. V. Shchepin, Soft maps of manifolds, Uspekhi Mat. Nauk, 1984, Volume 39, Issue 5(239), 209–224; \S2, Def.). Are there any references ? Obviously a necessary condition is being a Serre acyclic fibration.

In more details:

Scchepin [ibid] calls a map $$g:Y_1\to Y_2$$ soft with respect to a pair of spaces $$X,A$$ where $$A\subset X$$ iff the inclusion map $$i:A\to X$$ has the left lifting property with respect to the map $$Y\to pt$$ from $$Y$$ to a singleton $$pt$$ $$A\xrightarrow{i} B \perp Y_1\xrightarrow{g} Y_2$$

Is a Serre fibration of sufficiently nice spaces (say, a cellular map of finite CW complexes) necessarily soft for any pair $$X,A$$ where $$A$$ is a closed subset of hereditary perfectly normal space $$X$$ ?
A map $$p:|Y|\to Y$$ from the geometric realisation of a finite simplicial complex to the simplicial complex viewed as a finite topological space, does indeed have this property, by the lifting theorem for finite spaces (Thm. 3.5, Eric Wofsey), see details of the statement in this question.
• Let $C\mathbb{N}$ be the cone over a countably infinite discrete complex (this is a contractible 1-dimensional polyhedron). van Douwen and Pol have constructed a countable regular $T_2$ space $X$ (which is thus perfectly normal) and a function $A\rightarrow C\mathbb{N}$, defined on a certain closed $A\subseteq X$, which does not extend over any neighboourhood in $X$. In particular, the map of countable complexes $C\mathbb{N}\rightarrow\ast$ is both a Hurewicz fibration and a homotopy equivalence, but is not soft wrt all perfectly normal pairs. Nov 20, 2021 at 17:55
• Following up the names you gave, found a maybe useful result in (van Douwen, E. K.; Lutzer, D. J.; Przymusi\'nski, T. C. Some extensions of the Tietze-Urysohn theorem. Amer. Math. Monthly 84 (1977), no. 6, 435--441. (Reviewer: J. Dugundji) 54E40 (54E35)): for A closed in X normal, there is a continuous map $C^{bounded}(A)\to C^{bounded}(X)$ where $C^{bounded}(X)$ denotes the space of bounded continous functions on $X$.. Does this result belong a result to "selection theory" you refer to ? Nov 21, 2021 at 9:36