{{main|homology manifold}}
{{main|homology manifold}}
An ANR homology manifold of dimension ”n” is a finite-dimensional ANR such that for each point <math>x</math> in <math>X</math>, the homology <math>\operatorname{H}_*(X, X – {x})</math> has <math>\mathbb{Z}</math> at ”n” and zero elsewhere.
An ANR homology manifold of dimension ”n” is a finite-dimensional ANR such that for each point <math>x</math> in <math>X</math>, the homology <math>\operatorname{H}_*(X, X – {x})</math> has <math>\mathbb{Z}</math> at ”n” and zero elsewhere.
== References ==
== References ==
Math concept
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.
Given a class c {\displaystyle {\mathfrak {c}}} of topological spaces, an absolute retract for c {\displaystyle {\mathfrak {c}}} is a topological space X {\displaystyle X} in c {\displaystyle {\mathfrak {c}}} such that for each closed embedding X ↪ Y {\displaystyle X\hookrightarrow Y} into a space Y {\displaystyle Y} in c {\displaystyle {\mathfrak {c}}} , X {\displaystyle X} (that is, the image of X {\displaystyle X} ) is a retract of Y {\displaystyle Y} .[1]
An absolute neighborhood retract or ANR for c {\displaystyle {\mathfrak {c}}} is a topological space X {\displaystyle X} in c {\displaystyle {\mathfrak {c}}} such that for each closed embedding X ↪ Y {\displaystyle X\hookrightarrow Y} into a space Y {\displaystyle Y} in c {\displaystyle {\mathfrak {c}}} , X {\displaystyle X} is a retract of a neighborhood in Y {\displaystyle Y} . In literature, it is the most common to take c {\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 X {\displaystyle X} such that for each Y {\displaystyle Y} in c {\displaystyle {\mathfrak {c}}} and a closed subset A ⊂ Y {\displaystyle A\subset Y} , each continuous map A → X {\displaystyle A\to X} extends to Y → X {\displaystyle Y\to X} . An absolute neighborhood extensor is defined similarly by requiring the existence of an extension only to a neighborhood of A {\displaystyle A} .
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.[3] 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—[4][5] Let Y {\displaystyle Y} be an ANR and α {\displaystyle \alpha } an open cover of Y {\displaystyle Y} . Then there exists a refinement β {\displaystyle \beta } of α {\displaystyle \alpha } with the property: if two maps f , g : X → Y {\displaystyle f,g:X\to Y} from a separable metric space are β {\displaystyle \beta } -near in the sense f − 1 ( U ) ∩ g − 1 ( U ) , U ∈ β {\displaystyle f^{-1}(U)\cap g^{-1}(U),U\in \beta } is an open cover of Y {\displaystyle Y} , then there is an α {\displaystyle \alpha } -homotopy h t {\displaystyle h_{t}} between them; i.e., h 0 , h 1 = f , g {\displaystyle h_{0},h_{1}=f,g} and for each x {\displaystyle x} in X {\displaystyle X} , h I ( x ) ⊂ {\displaystyle h_{I}(x)\subset } some open set in α {\displaystyle \alpha } . Moreover, β {\displaystyle \beta } has the property: if a priori a β {\displaystyle \beta } -homotopy f | A ∼ g | A {\displaystyle f|A\sim g|A} is given for some closed subset A {\displaystyle A} , then the above homotopy can be taken to be an extension of that.
Conversely,[6] a separable metric space Y {\displaystyle Y} is an ANR if there exists an open cover α {\displaystyle \alpha } of Y {\displaystyle Y} with the property: for a pair of α {\displaystyle \alpha } -near maps f , g : X → Y {\displaystyle f,g:X\to Y} , each α {\displaystyle \alpha } -homotopy f | A ∼ g | A {\displaystyle f|A\sim g|A} extends to a homotopy f ∼ g {\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 V {\displaystyle V} of a point, the natural inclusion from some smaller neighborhood of the same point V ↪ U {\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.[7] What we can say is: if X {\displaystyle X} is a locally contractible separable metric space and the homotopy extension theorem holds for it, then X {\displaystyle X} is an ANR.[8]
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.[9]
ANR homology manifold
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An ANR homology manifold of dimension n is a finite-dimensional ANR such that for each point x {\displaystyle x} in X {\displaystyle X} , the homology H ∗ ( X , X − x ) {\displaystyle \operatorname {H} _{*}(X,X-{x})} has Z {\displaystyle \mathbb {Z} } at n and zero elsewhere.
- ^ Mardešić 1999, p. 242 harvnb error: no target: CITEREFMardešić1999 (help)
- ^ Karol Borsuk, Über eine Klasse von lokal zusammenhängenden Räumen, Fund. Math 19 (1932) 220-242
- ^ Hanner 1951, Theorem 3.2.
- ^ Hanner 1951, Theorem 4.1.
- ^ Dugundji 1951, Lemma 7.2.
- ^ Hanner 1951, Theorem 4.2.
- ^ Borsuk, Karol. “Sur un espace compact localement contractile qui n’est pas un rétracte absolu de voisinage.” Fundamenta Mathematicae 35 (1948): 175-180.
- ^ Hanner 1951, Theorem 5.3.
- ^ 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.
