Distinguishing locally contractible (in the sense of Borsuk) at $x$ and to $x$ for every $x$.

Question

(As I researched the question further, this became an ask-and-answer, with (I think) a rather basic answer. I hope it may clarify confusion for any future learners. Please note that self-answering is allowed on StackExchange sites, as asserted here.)

[Borsuk, pg. 26] says that a subset $A \subset X$ of a topological space is contractible in $X$ to a subset $B \subset X$ if the inclusion map $A \hookrightarrow X$ is homotopic to a map with values in $B$. And $A$ is contractible in $X$ if $A$ is contractible in $X$ to a singleton. Then [Borsuk, pg. 28] says a space $X$ is locally contractible at a point $x_0 \in X$ if each neighborhood $U$ of $x_0$ contains a neighborhood $U_0$ of $x_0$ which is contractible in $U$ (to some singleton which may not be $\{x_0\}$). And $X$ is ([Borsuk]-)locally contractible (LC) if $X$ is locally contractible at every point. This version of locally contractible appears for instance in [Cappell-Ranicki-Rosenberg, pg. 326], [Bredon, pg. 536], and presently on Wikipedia. (Note that [Borsuk] actually assumes spaces are Hausdorff, but I don't make this restriction for this question.) In [Borsuk], a significant feature of this 'locally contractible' (and related definitions in [Borsuk]) is that it is preserved by taking images of so-called r-maps, which are continuous maps admitting continuous sections. (This generalizes retractions as used in topology because retractions are continuous maps such that an inclusion map is a section, but agrees with 'retraction' as used in category theory).

[Fuchs-Viro, pg. 14] says that a space $X$ is locally contractible to the point $x_0 \in X$ if every neighborhood $U$ of $x_0$ contains a neighborhood $U_0$ of $x_0$ such that the inclusion $U_0 \hookrightarrow U$ is homotopic to the constant map with value $x_0$. And $X$ is ([Fuchs-Viro]-)locally contractible if it is locally contractible to every point. This version of locally contractible is equivalent to what appears in [Rotman, pg. 211] (where it's required that $U_0$ is open), and (I believe is synonymous to what) appears in [Spanier, pg. 57].

It is clear that [Fuchs-Viro]-locally contractible implies [Borsuk]-locally contractible. Question of this thread: Does [Borsuk]-locally contractible space imply [Fuchs-Viro]-locally contractible for arbitrary topological spaces?

For context, I'm a beginner to homotopy theory trying to learn about what distinguishes the versions of 'locally contractible' that appear in authoritative references, in order to determine which if any is the most important.

Borsuk, Karol, Theory of retracts, Monografie Matematyczne. 44. Warszawa: PWN - Polish Scientific Publishers. 251 p. (1967). ZBL0153.52905.

Bredon, Glen E., Topology and geometry, Graduate Texts in Mathematics. 139. New York: Springer-Verlag. xiv, 557 p. (1993). ZBL0791.55001.

Cappell, Sylvain (ed.); Ranicki, Andrew (ed.); Rosenberg, Jonathan (ed.), Surveys on surgery theory. Vol. 1: Papers dedicated to C. T. C. Wall on the occasion to his 60th birthday, Annals of Mathematics Studies. 145. Princeton, NJ: Princeton University Press. vii, 439 p. (2000). ZBL0933.00057.

[Fuchs-Viro] Novikov, S. P. (ed.); Rokhlin, V. A. (ed.); Gamkrelidze, R. V. (ed.), Topology II. Homotopy and homology. Classical manifolds. Transl. from the Russian, Encyclopaedia of Mathematical Sciences 24. Berlin: Springer-Verlag (ISBN 3-540-51996-3/hbk). 257 p. (2004). ZBL1090.55501.

Rotman, Joseph J., An introduction to algebraic topology, Graduate Texts in Mathematics, 119. New York (USA) etc.: Springer-Verlag. xiii, 433 p. DM 108.00 (1988). ZBL0661.55001.

Spanier, Edwin H., Algebraic topology, McGraw-Hill Series in Higher Mathematics. New York etc.:McGraw-Hill Book Company. XIV, 528 p. (1966). ZBL0145.43303.

This question is not a duplicate of Definitions of locally contractible spaces. The versions of locally contractible there do not explicitly appear as definitions in published textbooks / peer reviewed literature that I have seen (with very limited experience I should say). I am more broadly interested in distinguishing the definitions from this post to those and others, however. — Geoffrey Sangston, Dec 29 '24 at 18:24
See ANR is locally contractible for some information about distinguishing Borsuk's definition of locally contractible with one about arbitrarily small contractible neighborhoods about each point. — Geoffrey Sangston, Dec 29 '24 at 19:12

score 1 · Answer 1 · answered Dec 29 '24 at 18:20

[Borsuk] defines a range of properties called $LC^n$, $n \in \{0, 1, \ldots, \infty\}$ which are also preserved by r-maps. First he says $X$ is locally homotopically trivial over a space $A$ at a point $x_0 \in X$ if for each neighborhood $U$ of $x_0 \in X$, there is a neighborhood $U_0$ of $x_0$ contained in $U$ such that every map $A \to U$ with image a subset of $U_0$ is homotopic to a constant map. If this is satisfied at every point $x_0 \in X$, then $X$ is called locally homotopically trivial over $A$. (The theorem on line (16.1) of [Borsuk] states that 'locally homotopically trivial over $A$' is preserved by images of r-maps.) Then, $X$ is locally $n$-connected if it is locally homotopically trivial over $S^n$, $X$ is $LC^n$ if it is locally $k$-connected for $k \le n$, and $X$ is $LC^\infty$ if it is locally $k$-connected for every $k$. It is clear that $LC$ implies $LC^\infty$. It is straightforward that $X$ is $LC^0$ if and only if for each $x_0 \in X$ and neighborhood $U$ of $x_0$, there exists a neighborhood $U_0$ of $x_0$ contained in $U$ such that for each $y \in U_0$ there exists a path in $U$ connecting $x_0$ and $y$. From this equivalent version of $LC^0$, this answer straightforwardly demonstrates that $LC^0$ is equivalent to locally path connected (= every point has a neighborhood basis of path connected open sets) in general; also note that Borsuk writes, "Examining these definitions we see that [...] local $0$-connectedness is the same as local arcwise connectedness", and I assume this is archaic terminology for something equivalent to locally path connected.

Claim: If a topological space $X$ is [Borsuk]-locally contractible, then $X$ is [Fuchs-Viro]-locally contractible.

Proof: Let $x_0 \in X$, and let $U$ be a neighborhood of $x_0$. Since $X$ is locally path connected by the answer cited, there exists a path connected neighborhood $U'$ of $x$ which is a subset of $U$. By assumption, $x_0$ has a neighborhood $U_0$ which is a subset of $U'$ and such that $U_0 \hookrightarrow U'$ is homotopic to a constant map with some value $y \in U'$, via some homotopy $F : U_0 \times [0, 1] \to U'$. Since $U'$ is path connected, there exists a continuous map $\sigma : [0, 1] \to U'$ such that $\sigma(0) = y$ and $\sigma(1) = x_0$, hence there exists a homotopy $G : U_0 \times [0, 1] \to U'$, $(u, t) \mapsto \sigma(t)$ from the constant map with value $y$ to the constant map with value $x_0$. Then the concatenation of $F$ and $G$ (e.g., as in [Bredon, Definition 14.11]), $(F \ast G)(x, t) := \begin{cases} F(x, 2t), & t \in [0, \frac{1}{2}] \\ G(x, 2t - 1), & t \in [\frac{1}{2}, 1] \end{cases},$ is a homotopy from the inclusion map $U_0 \hookrightarrow U'$ to the constant map with value $x_0$. Clearly this is also a homotopy from the inclusion map $U_0 \hookrightarrow U$ to the constant map with value $x_0$, so it follows that $X$ is [Fuchs-Viro]-locally contractible.

Distinguishing locally contractible (in the sense of Borsuk) at $x$ and to $x$ for every $x$.

1 Answers1