Absolute neighborhood retract

In mathematics, especially algebraic topology, an absolute neighborhood retract (or ANR) is a "nice" topological space that is considered in homotopy theory; more specifically, in the theory of retracts.

For a more general introduction to ANRs, see also Retraction (topology)#Absolute neighborhood retract (ANR). This article focuses more on results on ANRs.

Definitions

Given a class ${\mathfrak {c}}$ of topological spaces, an absolute retract for ${\mathfrak {c}}$ is a topological space $X$ in ${\mathfrak {c}}$ such that for each closed embedding $X\hookrightarrow Y$ into a space $Y$ in ${\mathfrak {c}}$ , $X$ (that is, the image of $X$ ) is a retract of $Y$ .^[1]

An absolute neighborhood retract or ANR for ${\mathfrak {c}}$ is a topological space $X$ in ${\mathfrak {c}}$ such that for each closed embedding $X\hookrightarrow Y$ into a space $Y$ in ${\mathfrak {c}}$ , $X$ is a retract of a neighborhood in $Y$ . In literature, it is the most common to take ${\mathfrak {c}}$ to be the class of metric spaces or separable metric spaces. The notion of ANRs is due to Borsuk.^[2]

A closely related notion is that of an absolute extensor; namely, an absolute extensor is a topological space $X$ such that for each $Y$ in ${\mathfrak {c}}$ and a closed subset $A\subset Y$ , each continuous map $A\to X$ extends to $Y\to X$ . An absolute neighborhood extensor is defined similarly by requiring the existence of an extension only to a neighborhood of $A$ .

Results

The next theorem characterizes an ANR in terms of the extension property.

Theorem—^[3] The following are equivalent for a metric space $Y$ :

$Y$ is an ANR for metric spaces.
There is an embedding $Y\hookrightarrow E$ into a normed linear space such that $Y$ is a retract of a neighborhood in the convex hull of the image of $Y$ .
For each metric space $X$ and closed subset $A\subset X$ , each $A\to Y$ extends to a neighborhood of $A$ ; in short, $Y$ is an absolute neighborhood extensor.

This is a consequence of Dugundji's extension theorem and the Eilenberg–Wojdysławski theorem. Indeed, the latter theorem says every metric space embeds into a normed space as a closed subset of the convex hull of the image. This gives $1\Rightarrow 2$ . Assuming $Y\subset U\subset K=$ the convex hull of $Y$ and a retraction $r:U\to Y$ exists, by Dugundji's extension theorem, each $A\to Y\subset K$ extends to $g:X\to K$ . Then $g^{-1}(U){\overset {g}{\to }}U{\overset {r}{\to }}Y$ is a required extension. Finally, $3\Rightarrow 1$ holds by taking $A=Y$ . $\square$

There is also the notion of a local ANR, a metric space in which each point has a neighborhood that is an ANR. But as it turns out, the two notions ANR and local ANR coincide.^[4] In particular, a topological manifold is an ANR (even strongly it is a Euclidean neighborhood retract.)

There is also the following type of the approximation theorem

Theorem—^[5]^[6] Let $Y$ be an ANR and $\alpha$ an open cover of $Y$ . Then there exists a refinement $\beta$ of $\alpha$ with the property: if two maps $f,g:X\to Y$ from a separable metric space are $\beta$ -near in the sense $f^{-1}(U)\cap g^{-1}(U),U\in \beta$ is an open cover of $Y$ , then there is an $\alpha$ -homotopy $h_{t}$ between them; i.e., $h_{0},h_{1}=f,g$ and for each $x$ in $X$ , $h_{I}(x)\subset$ some open set in $\alpha$ . Moreover, $\beta$ has the property: if a priori a $\beta$ -homotopy $f|A\sim g|A$ is given for some closed subset $A$ , then the above homotopy can be taken to be an extension of that.

Conversely,^[7] a separable metric space $Y$ is an ANR if there exists an open cover $\alpha$ of $Y$ with the property: for a pair of $\alpha$ -near maps $f,g:X\to Y$ , each $\alpha$ -homotopy $f|A\sim g|A$ extends to a homotopy $f\sim g$ .

The theorem in particular implies that an ANR is locally contractible in the geometric topology sense; i.e., given a neighborhood $V$ of a point, the natural inclusion from some smaller neighborhood of the same point $V\hookrightarrow U$ is nullhomotopic. On the other hand, Borsuk has given an example of a locally contractible space that is not an ANR.^[8] What we can say is: if $X$ is a locally contractible separable metric space and the homotopy extension theorem holds for it, then $X$ is an ANR.^[9]

An n-dimensional metric space is an ANR if and only if it is locally connected up to dimension n in the sense of Lefschetz.^[10]^[11]

A topological space has the homotopy type of a countable CW-complex if and only if it has the homotopy type of an absolute neighborhood retract for separable metric spaces.^[12]

An open subset of a CW-complex may not be a CW-complex (due to Cauty). However, Cauty showed that a metric space is an ANR if and only if each open subset has the homotopy type of an ANR or equivalently the homtopy type of a CW-complex.^[13]

ANR homology manifold

An ANR homology manifold of dimension n is a finite-dimensional ANR such that for each point $x$ in $X$ , the homology $\operatorname {H} _{*}(X,X-{x})$ has $\mathbb {Z}$ at n and zero elsewhere.^[14]

References

Notes

^ Mardešić 1999, p. 242
^ Karol Borsuk, Über eine Klasse von lokal zusammenhängenden Räumen, Fund. Math 19 (1932) 220-242
^ Dugundji 1951, Theorem 7.1.
^ Hanner 1951, Theorem 3.2.
^ Hanner 1951, Theorem 4.1.
^ Dugundji 1951, Lemma 7.2.
^ Hanner 1951, Theorem 4.2.
^ Borsuk, Karol. "Sur un espace compact localement contractile qui n'est pas un rétracte absolu de voisinage." Fundamenta Mathematicae 35 (1948): 175-180.
^ Hanner 1951, Theorem 5.3.
^ The separable case is due to Kuratowski, Casimir (1935). "Sur les espaces localement connexes et péaniens en dimensions n". Fundamenta Mathematicae. 24 (1): 269–287. ISSN 0016-2736.
^ Yukihiro Kodama. "On LCⁿ metric spaces." Proc. Japan Acad. 33 (2) 79 - 83, 1957. https://doi.org/10.3792/pja/1195525144
^ Theorem 1 in Milnor, John (1959). "On spaces having the homotopy type of CW-complex". Transactions of the American Mathematical Society. 90 (2): 272–280. doi:10.2307/1993204. JSTOR 1993204.
^ Cauty, Robert (1994), "Une caractérisation des rétractes absolus de voisinage", Fundamenta Mathematicae, 144: 11–22, doi:10.4064/fm-144-1-11-22, MR 1271475.
^ Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S. (1996). "Topology of Homology Manifolds". Annals of Mathematics. 143 (3): 435–467. doi:10.2307/2118532. ISSN 0003-486X.

Works

Hanner, O. (1951). Some theorems on absolute neighborhood retracts, Arkiv Mat. 1. p. 389-408.
Dugundji, James (1 September 1951). "An extension of Tietze's theorem". Pacific Journal of Mathematics. 1 (3): 353–367. ISSN 0030-8730.
Mardešić, Sibe (1999). "Absolute Neighborhood Retracts and Shape Theory" (PDF). maths.ed.ac.uk.
Hu, Sze-Tsen (1965), Theory of Retracts, Wayne State University Press, MR 0181977
Sakai, Katsuro. Geometric aspects of general topology. Tokyo: Springer, 2013.