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.^{[ jargon ]}

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 ${\displaystyle {\mathfrak {c}}}$ 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 ${\displaystyle A}$.

Results

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

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

1. ${\displaystyle Y}$ is an ANR for metric spaces.
2. There is an embedding ${\displaystyle Y\hookrightarrow E}$ into a normed linear space such that ${\displaystyle Y}$ is a retract of a neighborhood in the convex hull of the image of ${\displaystyle Y}$.
3. For each metric space ${\displaystyle X}$ and closed subset ${\displaystyle A\subset X}$, each ${\displaystyle A\to Y}$ extends to a neighborhood of ${\displaystyle A}$; in short, ${\displaystyle Y}$ 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 ${\displaystyle 1\Rightarrow 2}$. 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

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

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

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]

ANR homology manifold

Main article: homology manifold

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]

References

Notes

1. Mardešić 1999 , p. 242
2. Karol Borsuk, Über eine Klasse von lokal zusammenhängenden Räumen, Fund. Math 19 (1932) 220-242
3. Dugundji 1951, Theorem 7.1.
4. Hanner 1951 , Theorem 3.2.
5. Hanner 1951, Theorem 4.1.
6. Dugundji 1951, Lemma 7.2.
7. Hanner 1951, Theorem 4.2.
8. Borsuk, Karol. "Sur un espace compact localement contractile qui n'est pas un rétracte absolu de voisinage." Fundamenta Mathematicae 35 (1948): 175-180.
9. Hanner 1951 , Theorem 5.3.
10. 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.
11. Yukihiro Kodama. "On LCⁿ metric spaces." Proc. Japan Acad. 33 (2) 79 - 83, 1957. https://doi.org/10.3792/pja/1195525144
12. 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.
13. 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.
14. 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.

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