Tautness (topology)

Comment: I have added the ISBN parameter so the book can be identified, but this is insufficient to get the draft accepted. We need references from other sources as well.
We require references from significant coverage about the topic of the article, and independent of it, in multiple secondary sources which are WP:RS please. See WP:42. Please also see WP:PRIMARY which details the limited permitted usage of primary sources and WP:SELFPUB which has clear limitations on self published sources. Providing sufficient references, ideally one per fact referred to, that meet these tough criteria is likely to allow this article to remain. Lack of them or an inability to find them is likely to mean that the topic is not suitable for inclusion, certainly today.
This can be from peer reviewed papers, academic journals etc. AT present we just have Spanier's word for this being a 'thing'. Fiddle^Timtrent Faddle^Talk to me 06:58, 1 August 2021 (UTC)

Comment: Seems to be good but needed more references. Fade258 (talk) 04:44, 1 August 2021 (UTC)

In mathematics, particularly in algebraic topology, taut pair is a topological pair whose direct limit of cohomology module of open neighborhood of that pair which is directed downward by inclusion is isomorphic to the cohomology module of original pair.

Definition

For topological pair $(A,B)$ in a topological space $X$ , a neighborhood $(U,V)$ of such pair is defined to be a pair such that $U$ and $V$ are neighborhoods of $A$ and $B$ respectively.

If we collect all neighborhoods of $(A,B)$ , then we can form a directed set which is directed downward by inclusion. Hence its cohomology module $H^{q}(U,V;G)$ is a direct system where $G$ is a module over a ring with unity. If we denote its direct limit by

{\bar {H}}^{q}(A,B;G)=\varinjlim H^{q}(U,V;G)

the restriction maps $H^{q}(U,V;G)\to H^{q}(A,B;G)$ define a natural homomorphism $i:{\bar {H}}^{q}(A,B;G)\to H^{q}(A,B;G)$ .

The pair $(A,B)$ is said to be tautly embedded in $X$ (or a taut pair in $X$ ) if $i$ is an isomorphism for all $q$ and $G$ .^[1]

Basic properties

For pair $(A,B)$ of $X$ , if two of the three pairs $(B,\emptyset ),(A,\emptyset )$ , and $(A,B)$ are taut in $X$ , so is the third.
For pair $(A,B)$ of $X$ , if $A,B$ and $X$ have compact triangulation, then $(A,B)$ in $X$ is taut.
If $U$ varies over the neighborhoods of $A$ , there is an isomorphism $\varinjlim {\bar {H}}^{q}(U;G)\simeq {\bar {H}}^{q}(A;G)$ .
If $(A,B)$ and $(A',B')$ are closed pairs in a normal space $X$ , there is an exact relative Mayer-Vietoris sequence for any coefficient module $G$ ^[2]

\cdots \to {\bar {H}}^{q}(A\cup A',B\cup B')\to {\bar {H}}^{q}(A,B)\oplus {\bar {H}}^{q}(A',B')\to {\bar {H}}^{q}(A\cap A',B\cap B')\to \cdots

Properties related to cohomology theory

Let $A$ be any subspace of a topological space $X$ which is a neighborhood retract of $X$ . Then $A$ is a taut subspace of $X$ with respect to Alexander-Spanier cohomology.
every retract of an arbitrary topological space is a taut subspace of $X$ with respect to Alexander-Spanier cohomology.
A closed subspace of a paracompactt Hausdorff space is a taut subspace of relative to the Alexander cohomology theory^[3].

Note

Since the Čech cohomology and the Alexander-Spanier cohomology are naturally isomorphic on the category of all topological pairs^[4], all of the above properties are valid for Čech cohomology. However, it's not true for singular cohomology (see Example)

Dependence of cohomology theory

Example^[5]

Let $X$ be the subspace of ${\mathbb {R}}^{2}\subset S^{2}$ which is the union of four sets

A_{1}=\{(x,y)\mid x=0,-2\leq y\leq 1\}

A_{2}=\{(x,y)\mid 0\leq x\leq 1,y=-2\}

A_{3}=\{(x,y)\mid x=1,-2\leq y\leq 0\}

A_{4}=\{(x,y)\mid 0<x\leq 1,y=\sin 2\pi /x\}

The first singular cohomology of $X$ is $H^{1}(X;Z)=0$ and using the Alexander duality theorem on $S^{2}-X$ , $\varinjlim \{H^{q}(U;{\mathbb {Z}})\}={\mathbb {Z}}$ as $U$ varies over neighborhoods of $X$ .

Therefore, $\varinjlim \{H^{q}(U;{\mathbb {Z}})\}\to H^{1}(X;{\mathbb {Z}})$ is not a monomorphism so that $X$ is not a taut subspace of ${\mathbb {R}}^{2}$ with respect to singular cohomology. However, since $X$ is closed in ${\mathbb {R}}^{2}$ , it's taut subspace with respect to Alexander cohomology^[6].

References

^ Spanier, Edwin H. (1966). Algebraic topology. p. 289. ISBN 978-0387944265.
^ Spanier, Edwin H. (1966). Algebraic topology. p. 290-291. ISBN 978-0387944265.
^ Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American mathematical society. 52: 441-442.
^ Dowker, C. H. (1952). "Homology groups of relations". Annals of Mathematics. (2) 56: 84-95.
^ Spanier, Edwin H. (1966). Algebraic topology. p. 317. ISBN 978-0387944265.
^ Spanier, Edwin H. (1978). "Tautness for Alexander-Spanier cohomology". Pacific Journal of Mathematics. 75: 562.

[auto-1] Spanier, Edwin H. (1966). Algebraic topology. p. 289. ISBN 978-0387944265.

[2] Spanier, Edwin H. (1966). Algebraic topology. p. 290-291. ISBN 978-0387944265.

[3] Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". American mathematical society. 52: 441-442.

[4] Dowker, C. H. (1952). "Homology groups of relations". Annals of Mathematics. (2) 56: 84-95.

[5] Spanier, Edwin H. (1966). Algebraic topology. p. 317. ISBN 978-0387944265.

[6] Spanier, Edwin H. (1978). "Tautness for Alexander-Spanier cohomology". Pacific Journal of Mathematics. 75: 562.

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