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 of topological spaces, an absolute retract for is a topological space in such that for each closed embedding into a space in , (that is, the image of ) is a retract of .
An absolute neighborhood retract or ANR for is a topological space in such that for each closed embedding into a space in , is a retract of a neighborhood in . In literature, it is the most common to take to be the class of metric spaces or separable metric spaces. The notion of ANRs is due to Borsuk.Karol Borsuk, Über eine Klasse von lokal zusammenhängenden Räumen, Fund. Math 19 (1932) 220-242
A closely related notion is that of an absolute extensor; namely, an absolute extensor is a topological space such that for each in and a closed subset , each continuous map extends to . An absolute neighborhood extensor is defined similarly by requiring the existence of an extension only to a neighborhood of .
Results The next theorem characterizes an ANR in terms of the extension property.
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 . Assuming the convex hull of and a retraction exists, by Dugundji's extension theorem, each extends to . Then is a required extension. Finally, holds by taking .
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. 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
The theorem in particular implies that an ANR is locally contractible in the geometric topology sense; i.e., given a neighborhood of a point, the natural inclusion from some smaller neighborhood of the same point is nullhomotopic. On the other hand, Borsuk has given an example of a locally contractible space that is not an ANR.Borsuk, Karol. "Sur un espace compact localement contractile qui n'est pas un rétracte absolu de voisinage." Fundamenta Mathematicae 35 (1948): 175-180. What we can say is: if is a locally contractible separable metric space and the homotopy extension theorem holds for it, then is an ANR.
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.The separable case is due to Yukihiro Kodama. "On LCn metric spaces." Proc. Japan Acad. 33 (2) 79 - 83, 1957. https://doi.org/10.3792/pja/1195525144
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.Theorem 1 in
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.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.
ANR homology manifold
An ANR homology manifold of dimension n is a finite-dimensional ANR such that for each point in , the homology has at n and zero elsewhere.
Notes
References
Works
Sakai, Katsuro. Geometric aspects of general topology. Tokyo: Springer, 2013.
Further reading https://math.stackexchange.com/questions/1082601/anr-is-locally-contractible https://mathoverflow.net/questions/102700/does-local-strict-contractibility-imply-anr https://mathoverflow.net/questions/123513/the-role-of-anr-in-modern-topology?r=SearchResults&s=5%7C50.6943 https://ncatlab.org/nlab/show/absolute+retract