Tautness (topology) explained
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
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
\bar{H}q(A,B;G)=\varinjlimHq(U,V;G)
the
restriction maps
define a natural homomorphism
i:\bar{H}q(A,B;G)\toHq(A,B;G)
.
The pair
is said to be
tautly embedded in
(or a
taut pair in
) if
is an isomorphism for all
and
.
[1] Basic properties
of
, if two of the three pairs
(B,\emptyset),(A,\emptyset)
, and
are taut in
, so is the third.
of
, if
and
have
compact triangulation, then
in
is taut.
varies over the neighborhoods of
, there is an isomorphism
\varinjlim\bar{H}q(U;G)\simeq\bar{H}q(A;G)
.
and
are closed pairs in a normal space
, there is an exact relative
Mayer-Vietoris sequence for any coefficient module
[2] … \to\bar{H}q(A\cupA',B\cupB')\to\bar{H}q(A,B) ⊕ \bar{H}q(A',B')\to\bar{H}q(A\capA',B\capB')\to …
Properties related to cohomology theory
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
A1=\{(x,y)\midx=0,-2\leqy\leq1\}
A2=\{(x,y)\mid0\leqx\leq1,y=-2\}
A3=\{(x,y)\midx=1,-2\leqy\leq0\}
A4=\{(x,y)\mid0<x\leq1,y=\sin2\pi/x\}
The first singular cohomology of
is
and using the
Alexander duality theorem on
,
as
varies over neighborhoods of
.
Therefore,
\varinjlim\{Hq(U;Z)\}\toH1(X;Z)
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
Notes and References
- Book: Spanier . Edwin H. . Algebraic topology . 1966 . 289. Springer . 978-0387944265.
- Book: Spanier . Edwin H. . Algebraic topology . 1966 . 290-291. Springer . 978-0387944265.
- Deo . Satya . On the tautness property of Alexander-Spanier cohomology . Proceedings of the American Mathematical Society. 197 . 52 . 1 . 441–444 . 10.2307/2040179. 2040179 . free .
- Dowker . C. H. . Homology groups of relations . Annals of Mathematics . 1952 . (2) 56 . 1 . 84–95 . 10.2307/1969768 . 1969768 .
- Book: Spanier . Edwin H. . Algebraic topology . 1966 . 317. Springer . 978-0387944265.
- Spanier . Edwin H. . Tautness for Alexander-Spanier cohomology . Pacific Journal of Mathematics . 1978 . 75 . 2 . 562. 10.2140/pjm.1978.75.561 . 122337937 . free .
