nLab absolute retract

Definition

A metrisable topological space $Y$ is an absolute neighborhood retract (ANR) if (Borsuk 32, p. 222) for any embedding $Y \subset Z$ as a closed subspace in a metrisable topological space $Z$, $Y$ is a neighborhood retract of $Z$.

A metrisable topological space $Y$ is an absolute retract if for any embedding $Y\subset Z$ as a closed subspace in a metrisable topological space $Z$, $Y$ is a retract of $Z$.

Properties

Proposition

(ANR is a local property for metrizable spaces)
A metrizable topological space which admits an open cover by absolute neighbourhood retracts is itself an absolute neighbourhood retract.

(review in Hu 65, III Thm. 8.1)

Proposition

A metrisable topological space is an absolute retract precisely if it is

1. a contractible topological space,

2. an absolute neighborhood retract.

(Hu 65, Prop. II.7.2)

Examples

Example

Every (finite-dimensional) metrizable locally Euclidean topological space – in particular every topological manifold – is an absolute neighbourhood retract.

By Prop. (see also Hu 65, III Cor. 8.3).

In fact:

Example

Every paracompact Banach manifold is an absolute neighbourhood retract.

By Palais 1966, Cor. to Thm. 5 on p. 3.

Proposition

Let $X$ be an absolute neighbourhood retract (ANR) and $A \xhookrightarrow{i} X$ a closed subspace-inclusion. Then $A$ is an ANR precisely iff the inclusion $i$ is a Hurewicz cofibration.

(Aguilar, Gitler & Prieto 2002, Thm. 4.2.15)

Example

Every finite-dimensional locally finite CW-complex is an absolute neighbourhood retract.

(Dugundji 52, Kodama 56, review in Hu 65, III Cor. 8.4)

Related concepts

• retract,

• neighborhood retract

• deformation retract

References

The notion of absolute neighbourhood retract is due to

• Karol Borsuk, Über eine Klasse von lokal zusammenhängenden Räumen, Fund. Math 19 (1932) 220-242 (dml:212574, pdf)

Further development:

• Olof Hanner, Some theorems on absolute neighbourhood retracts, Arkiv För Matematik Band 1 nr 30 (1950) (doi:10.1007/BF02591376)

• James Dugundji, Note on CW polytopes, Portugaliae mathematica (1952) 11 1 (1952) 7-10-b (dml:114693)

• Yukihiro Kodama, Note on an absolute neighborhood extensor for metric spaces, Journal of the Mathematical Society of Japan 8 3 (1956) 206-215 (doi:10.2969/jmsj/00830206)

• Karol Borsuk, Concerning the classification of topological spaces from the stand point of the theory of retracts, Fundamenta Mathematicae 46 (3) (1959) 321-330 (dml:213516)

Discussion for infinite-dimensional manifolds:

• Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

Textbook accounts and review:

• Karol Borsuk, Theory of retracts, Vol. 44. Państwowe Wydawn. Naukowe, 1967

• Sze-Tsen Hu, Theory of Retracts, Wayne State University Press (1965) (google-books)

• Sibe Mardešić, Absolute Neighborhood Retracts and Shape Theory (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Def. 4.2.10 in: Algebraic topology from a homotopical viewpoint, Springer (2002) (doi:10.1007/b97586, toc pdf)

See also:

• eom, Absolute retract for normal spaces

• eom, Retract of a topological space

• Wikipedia, Absolute neighbourhood retract