Absolute neighborhood retract

Summarize Timeline Fact Check

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

An absolute neighborhood retract or ANR for ${\displaystyle {\mathfrak {c}}}$ is a topological space ${\displaystyle X}$ in ${\displaystyle {\mathfrak {c}}}$ such that for each closed embedding ${\displaystyle X\hookrightarrow Y}$ into a space ${\displaystyle Y}$ in ${\displaystyle {\mathfrak {c}}}$, ${\displaystyle X}$ is a retract of a neighborhood in ${\displaystyle Y}$. In literature, it is the most common to take ${\displaystyle {\mathfrak {c}}}$ 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 ${\displaystyle X}$ such that for each ${\displaystyle Y}$ in ${\displaystyle {\mathfrak {c}}}$ and a closed subset ${\displaystyle A\subset Y}$, each continuous map ${\displaystyle A\to X}$ extends to ${\displaystyle Y\to X}$. An absolute neighborhood extensor is defined similarly by requiring the existence of an extension only to a neighborhood of ${\displaystyle A}$.

Summarize Timeline Top Qs Fact Check

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 ${\displaystyle 1\Rightarrow 2}$. Assuming ${\displaystyle Y\subset U\subset K=}$ the convex hull of ${\displaystyle Y}$ and a retraction ${\displaystyle r:U\to Y}$ exists, by Dugundji's extension theorem, each ${\displaystyle A\to Y\subset K}$ extends to ${\displaystyle g:X\to K}$. Then ${\displaystyle g^{-1}(U){\overset {g}{\to }}U{\overset {r}{\to }}Y}$ is a required extension. Finally, ${\displaystyle 3\Rightarrow 1}$ holds by taking ${\displaystyle A=Y}$. ${\displaystyle \square }$

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

The theorem in particular implies that an ANR is locally contractible in the geometric topology sense; i.e., given a neighborhood ${\displaystyle V}$ of a point, the natural inclusion from some smaller neighborhood of the same point ${\displaystyle V\hookrightarrow U}$ 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 ${\displaystyle X}$ is a locally contractible separable metric space and the homotopy extension theorem holds for it, then ${\displaystyle X}$ 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]

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