# 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.

• There is a notion of soft map by Eugene Schepin , e.g. see E. V. Shchepin, Soft maps of manifolds, Uspekhi Mat. Nauk, 1984, Volume 39, Issue 5(239), 209–224, or A. Chigogidze, Inverse Spectra, North Holland, Amsterdam, 1996, and reference therein. I haven't yet understood whether they are related to this particular question. Nov 19, 2021 at 18:26
• 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
• On the other hand, if you restrict to (suitably nice) acyclic Serre fibrations between (suitably nice) finite complexes, then something should be true. How far you get with this will probably depend on your knowledge of selection theory. Nov 20, 2021 at 18:01
• Many thanks, interesting! Though I was not able to find online their paper, presumably van Douwen, Eric K.; Pol, Roman Countable spaces without extension properties. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), no. 10, 987--991. (Reviewer: D. J. Lutzer) 54C20. Frankly, I do find it rather surprising if the statement isn't known (either proof or some standard counterexample, for finite complexes...) Nov 21, 2021 at 9:29
• 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