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

\partial(M)

, and let

z\in H_n(M,\partial(M); \Z)

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

(M,\partial(M))

. Furthermore, this gives rise to isomorphisms of

H^k(M,\partial(M); \Z)

with

H_(M; \Z)

, and of

H_k(M,\partial(M); \Z)

with

H^(M; \Z)

for all

k

. Here

\partial(M)

can in fact be empty, so Poincaré duality appears as a special case of Lefschetz duality. There is a version for triples. Let

\partial(M)

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

k

, there is an isomorphism :

D_M\colon H^k(M,A; \Z)\to H_(M,B; \Z).

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=

Proceedings of the National Academy of Sciences of the United States of America ''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sc ...

, 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