• Source: Lefschetz duality

In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz (1926), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. There are now numerous formulations of Lefschetz duality or Poincaré–Lefschetz duality, or Alexander–Lefschetz duality.




Formulations


Let M be an orientable compact manifold of dimension n, with boundary



∂
(
M
)


{\displaystyle \partial (M)}

, and let



z
∈

H

n


(
M
,
∂
(
M
)
;

Z

)


{\displaystyle z\in H_{n}(M,\partial (M);\mathbb {Z} )}

be the fundamental class of the manifold M. Then cap product with z (or its dual class in cohomology) induces a pairing of the (co)homology groups of M and the relative (co)homology of the pair



(
M
,
∂
(
M
)
)


{\displaystyle (M,\partial (M))}

. Furthermore, this gives rise to isomorphisms of




H

k


(
M
,
∂
(
M
)
;

Z

)


{\displaystyle H^{k}(M,\partial (M);\mathbb {Z} )}

with




H

n
−
k


(
M
;

Z

)


{\displaystyle H_{n-k}(M;\mathbb {Z} )}

, and of




H

k


(
M
,
∂
(
M
)
;

Z

)


{\displaystyle H_{k}(M,\partial (M);\mathbb {Z} )}

with




H

n
−
k


(
M
;

Z

)


{\displaystyle H^{n-k}(M;\mathbb {Z} )}

for all



k


{\displaystyle k}

.
Here



∂
(
M
)


{\displaystyle \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



∂
(
M
)


{\displaystyle \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


{\displaystyle k}

, there is an isomorphism





D

M


:

H

k


(
M
,
A
;

Z

)
→

H

n
−
k


(
M
,
B
;

Z

)
.


{\displaystyle D_{M}\colon H^{k}(M,A;\mathbb {Z} )\to H_{n-k}(M,B;\mathbb {Z} ).}





Notes






References


"Lefschetz_duality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Lefschetz, Solomon (1926), "Transformations of Manifolds with a Boundary", Proceedings of the National Academy of Sciences of the United States of America, 12 (12), National Academy of Sciences: 737–739, Bibcode:1926PNAS...12..737L, doi:10.1073/pnas.12.12.737, ISSN 0027-8424, JSTOR 84764, PMC 1084792, PMID 16587146

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