In
mathematics, Lefschetz duality is a version of
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
in
geometric topology, applying to a
manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
. Such a formulation was introduced by , at the same time introducing
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
, for application to the
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
. There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.
Formulations
Let ''M'' be an
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
manifold of dimension ''n'', with boundary
, and let
be the
fundamental class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...
of the manifold ''M''. Then
cap product
In algebraic topology the cap product is a method of adjoining a chain of degree ''p'' with a cochain of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by Eduard Čech in 1936, ...
with ''z'' (or its dual class in cohomology) induces a pairing of the (co)
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s of ''M'' and the relative (co)homology of the pair
. Furthermore, this gives rise to isomorphisms of
with
, and of
with
for all
.
Here
can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality.
There is a version for triples. Let
decompose into subspaces ''A'' and ''B'', themselves compact orientable manifolds with common boundary ''Z'', which is the intersection of ''A'' and ''B''. Then, for each
, there is an isomorphism
:
Notes
References
*
*{{Citation , last=Lefschetz , first=Solomon , author-link=Solomon Lefschetz, title=Transformations of Manifolds with a Boundary , jstor=84764 , publisher=National Academy of Sciences , year=1926 , journal=
, issn=0027-8424 , volume=12 , issue=12 , pages=737–739 , doi=10.1073/pnas.12.12.737, pmc=1084792 , pmid=16587146, doi-access=free
Duality theories
Manifolds