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.
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.
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 .
The next theorem characterizes an ANR in terms of the extension property.
This is a consequence of Dugundji's extension theorem and the EilenbergâÂÂWojdysà Âawski 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. 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.
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.
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.
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.