Absolute neighborhood retract

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

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 $𝔠$ of topological spaces, an absolute retract for $𝔠$ is a topological space $X$ in $𝔠$ such that for each closed embedding $X\hookrightarrow Y$ into a space $Y$ in $𝔠$, $X$ (that is, the image of $X$) is a retract of $Y$.

An absolute neighborhood retract or ANR for $𝔠$ is a topological space $X$ in $𝔠$ such that for each closed embedding $X\hookrightarrow Y$ into a space $Y$ in $𝔠$, $X$ is a retract of a neighborhood in $Y$. In literature, it is the most common to take $𝔠$ 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 $X$ such that for each $Y$ in $𝔠$ 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.

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 $g^{-1}(U)\stackrel{g}{\to }U\stackrel{r}{\to }Y$ is a required extension. Finally, $3\Rightarrow 1$ holds by taking $A=Y$. $◻$

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 $V$ 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. 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.

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.

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 ${\mathrm{H}}_{\ast }(X,X-x)$ has $\mathbb{Z}$ at n and zero elsewhere.

References

Works

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• 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