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Formulations [ edit ]
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}
.[2]
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[3]
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} ).}
^ Biographical Memoirs By National Research Council Staff (1992), p. 297.
^ Vick, James W. (1994). Homology Theory: An Introduction to Algebraic Topology . p. 171.
^ Hatcher, Allen (2002). Algebraic topology . Cambridge: Cambridge University Press . p. 254. ISBN 0-521-79160-X .
References [ edit ]
"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, doi :10.1073/pnas.12.12.737 , ISSN 0027-8424 , JSTOR 84764 , PMC 1084792 , PMID 16587146
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Lefschetz_duality&oldid=1230441157 "
C a t e g o r i e s :
● D u a l i t y t h e o r i e s
● M a n i f o l d s
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● T e m p l a t e S p r i n g e r E O M w i t h b r o k e n r e f
● T h i s p a g e w a s l a s t e d i t e d o n 2 2 J u n e 2 0 2 4 , a t 1 8 : 3 0 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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