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Tautness (topology)

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In mathematics, particularly in algebraic topology, a 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

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For a topological pair in a topological space , a neighborhood of such a pair is defined to be a pair such that and are neighborhoods of and respectively.

If we collect all neighborhoods of , then we can form a directed set which is directed downward by inclusion. Hence its cohomology module is a direct system where is a module over a ring with unity. If we denote its direct limit by

the restriction maps define a natural homomorphism .

The pair is said to be tautly embedded in (or a taut pair in ) if is an isomorphism for all and .[1]

Basic properties

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  • For pair of , if two of the three pairs , and are taut in , so is the third.
  • For pair of , if and have compact triangulation, then in is taut.
  • If varies over the neighborhoods of , there is an isomorphism .
  • If and are closed pairs in a normal space , there is an exact relative Mayer-Vietoris sequence for any coefficient module [2]
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  • Let be any subspace of a topological space which is a neighborhood retract of . Then is a taut subspace of with respect to Alexander-Spanier cohomology.
  • every retract of an arbitrary topological space is a taut subspace of 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

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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

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Example[5]

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Let be the subspace of which is the union of four sets

The first singular cohomology of is and using the Alexander duality theorem on , as varies over neighborhoods of .

Therefore, is not a monomorphism so that is not a taut subspace of with respect to singular cohomology. However, since is closed in , it's taut subspace with respect to Alexander cohomology.[6]

See also

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References

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  1. ^ Spanier, Edwin H. (1966). Algebraic topology. Springer. p. 289. ISBN 978-0387944265.
  2. ^ Spanier, Edwin H. (1966). Algebraic topology. Springer. p. 290-291. ISBN 978-0387944265.
  3. ^ Deo, Satya (197). "On the tautness property of Alexander-Spanier cohomology". Proceedings of the American Mathematical Society. 52 (1): 441–444. doi:10.2307/2040179. JSTOR 2040179.
  4. ^ Dowker, C. H. (1952). "Homology groups of relations". Annals of Mathematics. (2) 56 (1): 84–95. doi:10.2307/1969768. JSTOR 1969768.
  5. ^ Spanier, Edwin H. (1966). Algebraic topology. Springer. p. 317. ISBN 978-0387944265.
  6. ^ Spanier, Edwin H. (1978). "Tautness for Alexander-Spanier cohomology". Pacific Journal of Mathematics. 75 (2): 562. doi:10.2140/pjm.1978.75.561. S2CID 122337937.