Tautness (topology)
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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 in a topological space , a neighborhood of such 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[]
- 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]
[]
- 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[]
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 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[]
- Alexander-Spanier cohomology
- Čech cohomology
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.
- Algebraic topology