Abstracts
A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing <img src="http:/img/fbpe/aabc/v73n2/fo1.gif" alt="fo1.gif (867 bytes)"> given by (alpha, beta) <img SRC="http:/img/fbpe/aabc/v73n2/m1img7.gif"> (alpha * beta) [X] induces isomorphisms <img src="http:/img/fbpe/aabc/v73n2/fo2.gif" alt="fo2.gif (1110 bytes)"> <img src="http:/img/fbpe/aabc/v73n2/fo3.gif" alt="fo3.gif (1086 bytes)"> onto the smooth Pontrjagin duals. In particular, <img SRC="http:/img/fbpe/aabc/v73n2/m1img13.gif"> and <img SRC="http:/img/fbpe/aabc/v73n2/m1img13a.gif"> are injective with dense range in the group of all continuous homomorphisms into the circle. A coboundary map is introduced which yields a long sequence for the character groups associated to the pair (X, <img SRC="http:/img/fbpe/aabc/v73n2/m1img14.gif">X). The relation of the sequence to the duality mappings is analyzed.
Differential characters; Lefschetz duality; deRham theory
Uma teoria de caracteres diferenciais é aqui desenvolvida para variedades com bordo. Isto é feito tanto do ponto de vista de Cheeger-Simons como do deRham-Federer. O resultado central deste artigo é a formulação e a prova de um teorema da dualidade de Lefschetz-Pontrjagin, que afirma que o pareamento <img src="http:/img/fbpe/aabc/v73n2/fo1.gif" alt="fo1.gif (867 bytes)"> dado por (alfa,beta) <img SRC="http:/img/fbpe/aabc/v73n2/m1img7.gif"> (alfa * beta) [X] induz isomorfismos <img src="http:/img/fbpe/aabc/v73n2/fo2.gif" alt="fo2.gif (1110 bytes)"> <img src="http:/img/fbpe/aabc/v73n2/fo3.gif" alt="fo3.gif (1086 bytes)"> sobre os duais diferenciáveis de Pontrjagin. Em particular, <img SRC="http:/img/fbpe/aabc/v73n2/m1img13.gif"> e <img SRC="http:/img/fbpe/aabc/v73n2/m1img13a.gif"> são injetivos com domínios densos no grupo de todos os homeomorfismos contínuos no círculo. Uma aplicação de cobordo é introduzida, a qual fornece uma sequência longa para os grupos de caracteres associados ao par ( X, <img SRC="http:/img/fbpe/aabc/v73n2/m1img14.gif">X). A relação desta sequência com as aplicações de dualidade é analisada.
caracteres diferenciais; dualidade de Lefschetz; teoria de deRham
Lefschetz-Pontrjagin Duality for Differential Characters* * Invited paper **Foreign Member of Academia Brasileira de Ciências Correspondence to: Blaine Lawson E-mail: blaine@math.sunysb.edu / harvey@math.rice.edu
REESE HARVEY1 and BLAINE LAWSON2
1The Department of Mathematics, P.O. Box 1892, Rice University, Houston, TX 77251-1892
2The Department of Mathematics, The University at Stony Brook, Stony Brook, NY 11794
Manuscript received on February 5, 2001; accepted for publication on February 12, 2001;contributed by BLAINE LAWSON ** * Invited paper **Foreign Member of Academia Brasileira de Ciências Correspondence to: Blaine Lawson E-mail: blaine@math.sunysb.edu / harvey@math.rice.edu
ABSTRACT
A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing
given by (a, b) (a * b) [X] induces isomorphisms
onto the smooth Pontrjagin duals. In particular, and
are injective with dense range in the group of all continuous homomorphisms into the circle. A coboundary map is introduced which yields a long sequence for the character groups associated to the pair (X,
X). The relation of the sequence to the duality mappings is analyzed.
Key words: Differential characters, Lefschetz duality, deRham theory.
INTRODUCTION
The theory of differential characters, introduced by Jim Simons and Jeff Cheeger in 1973, is of basic importance in geometry. It provides a wealth of invariants for bundles with connection starting with the classical one of Chern-Simons in dimension 3 and including large families of invariants for flat bundles and foliations. Its cardinal property is that it forms the natural receiving space for a refined Chern-Weil theory. This theory subsumes integral characteristic classes and the classical Chern-Weil characteristic forms. It also tracks certain "transgression terms'' which give cohomologies between smooth and singular cocycles and lead to interesting secondary invariants.
Each standard characteristic class has a refinement in the group of differential characters. Thus for a complex bundle with unitary connection, refined Chern classes are defined and the total class gives a natural transformation
= 1 +
+
+ ... :
(X)
(X)
from the K-theory of bundles with connection to differential characters which satisfies the Whitney sum formula: (E
F) =
(E)*
(F). This last property leads to non-conformal immersion theorems in riemannian geometry.
Differential characters form a highly structured theory with certain aspects of cohomology: contravariant functoriality, ring structure, and a pairing to cycles. There are deRham-Federer formulations of the theory (Gillet and Soulé 1989), (Harris 1989), (Harvey et al. 2001), analogous to those given for cohomology, which are useful for example in the theory of singular connections (Harvey and Lawson 1993, 1995). Furthermore, the groups (X) of differential characters carry a natural topology. The connected component of 0 in this group consists of the smooth characters, those which can be represented by smooth differential forms.
In (Harvey et al. 2001), where the deRham-Federer appoach is developed in detail, the authors showed that differential characters satisfy Poincaré-Pontrjagin duality: On an oriented n dimensional manifold X the pairing
(X) x
(X)
S1
given by
(a, b) (a
b)[X]
(where denotes characters with compact support) induces injective maps
(X)
Hom
(X), S1
and
(X)
Hom
(X), S1
with dense range in the groups of continuous homomorphisms into the circle. Moreover this range consists exactly of the smooth homomorphisms. These are defined precisely in §4 but can be thought of roughly as follows. The connected component of 0 in (X) consists essentially (i.e., up to a finite-dimensional torus factor) of the exact (k + 1)-forms d
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In this paper we formulate the theory of differential characters for compact manifolds with boundary (X,X) and prove a Lefschetz-Pontrjagin Duality Theorem analogous to the one above. To do this we introduce the relative groups
(X,
X) and develop the theory from (Harvey et al. 2001) for this case. The main theorem asserts the existence of a pairing
(X) x
(X,
X)
S1
given by (a, b) (a
b)[X] and inducing injective maps with dense range as above.
The two pairings above have a formal similarity but are far from the same. The delicate part of these dualities comes from the differential form component of characters. In the first pairing (on possibly non-compact manifolds) we contrast forms having no growth restrictions at infinity with forms with compact support. The second dualtiy (on compact manfiolds with boundary) opposes forms smooth up to the boundary with forms which restrict to zero on the boundary.
In cohomology theory there are long exact sequences for the pair (X,X) which interlace the Pontrjagin and Lefschetz Duality mappings. In the last sections of this paper the parallel structure for differential characters is studied. We introduce coboundary maps
:
(X)
(X,
X), yielding long sequences which intertwine the duality mappings and reduce to the standard picture under the natural transformation to integral cohomology.
1. DIFFERENTIAL CHARACTERS ON MANIFOLDS WITH BOUNDARY
Let X be a compact oriented differentiable n-manifold with boundary X. Let
*(X) denote the de Rham complex of differential forms which are smooth up to the boundary, and set
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The cohomology of this complex is naturally isomorphic to H*(X,X;
). Let C*(X) denote the complex of C
-singular chains on X and C*(X,
X)
C*(X)/C*(
X) the relative complex.
Denote by
Z*(X,X)
{c
C*(X,
X) :
c = 0}
the cycles in this complex. We begin with definitions of differential characters in the spirit of Cheeger-Simons.
DEFINITION 1.1. The group of differential characters of degree k on X is the set of homomorphisms
(X;
/
)
{a
Hom(Zk(X), S1) :
(a)
where denotes the coboundary. Similarly the group of relative differential characters of degree k on (X,
X) is defined to be
(X,
X;
/
)
{a
Hom(Zk(X,
X), S1) :
(a)
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Inclusion and restriction give maps (X,
X)
(X)
(
X). with
o j = 0.
There is an alternative de Rham-Federer approach to these groups. Set
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DEFINITION 1.2. An element a
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-
(1.3)
Denote by
k(X) the group of all such sparks and by![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f5fc619cf7cdfb11809ccdfb62e822869178dd9f.gif)
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(X)
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Given a spark a
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We define relative sparks and relative deRham-Federer characters on (X,X) by
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The decomposition (1.3) is unique. In fact we have the following. Recall that a current T is said to be integrally flat if it can be written as T = R + dS where R and S are rectifiable. Then from §1.5 in (Harvey et al. 2001) one concludes:
PROPOSITION 1.4. Let a be any current of degree k on X such that da = - R where
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d = 0 and dR
= 0
and has integral periods on cycles in X. In the case that
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c53abbe089de429c25d403f21808f6235621916c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
Set
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/fb8ce5043f000b75ffc66e1f6aa4b70129c84043.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d437ec7c6c6a07ca1791f7f6e44993154d4e5818.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/fb8ce5043f000b75ffc66e1f6aa4b70129c84043.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d437ec7c6c6a07ca1791f7f6e44993154d4e5818.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
(1.5)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a1f2a167fd8b2b8f7c218fa7c7be5fe06e73e79e.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/22bb0dc9fba7bc39e8b4f0cee3f352e86073bbef.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d437ec7c6c6a07ca1791f7f6e44993154d4e5818.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/53076c028ef781a0092cf015be703fd16a020c4e.gif)
Z
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a1f2a167fd8b2b8f7c218fa7c7be5fe06e73e79e.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/17b29b319e26484bdf70b4de0bbb12f4c735292c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
COROLLARY 1.6. Taking d1a = and d2a = R from the decomposition (1.3) gives well-defined mappings
d1 : Sk(X)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/544e6707b7dcd0bf697eef2db9d46b1edc5dd2ee.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6c9aa531e1baf9df15070dcf53604ab8c3e80765.gif)
and
d1 : Sk(X, X)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/544e6707b7dcd0bf697eef2db9d46b1edc5dd2ee.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6c9aa531e1baf9df15070dcf53604ab8c3e80765.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
PROPOSITION 1.7. There are natural isomorphisms
:
(X)
(X;
/
) and
:
(X,
X)
(X,
X;
/
)
induced by integration.
PROOF. The first is proved in (Harvey et al. 2001). The argument for the second is exactly the same.
REMARK 1.8. In (Harvey et al. 2001) we showed that there are many different (but equivalent) deRham-Federer definitions of differential characters on a manifold without boundary. Each of these different presentations has obvious analogues for (X) and
(X,
X). The proof of the equivalence of these definitions closely follows the arguments in §2 of (Harvey et al. 2001) and will not be given here. However, this flexibility in definitions is important in our treatment of the
-product.
To illustrate the point we give one example. Recall that a current R on X is called integrally flat if R = S + dT where S and T are rectifiable. Denote by k(X)
{
n - k(X)}' the space of currents of degree k on X. Let
(X,
X) denote the set of a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1fd8cc33d086526154daec46d8fd912d6b743b7f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/70128bd1066299ec62f5aab865ae8ce01a649391.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
and R is integrally flat. Let (X,
X) denote the subgroup of elements of the form db + S where b is smooth near
X, b
= 0, and S is integrally flat. Then the inclusion
k (X,
X)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/4c76b96d97100b438ff1811150012619d79fb6a5.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
(X,
X)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/822b87aee7e0dd63c587c70bd04933f3c7753891.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/8d6559c026f87095427ff2c4488d782b76a17557.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/822b87aee7e0dd63c587c70bd04933f3c7753891.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
2. THE EXACT SEQUENCES
The fundamental exact sequences established by Cheeger and Simons in (Cheeger and Simons 1985) carry over to the relative case.
DEFINITION 2.1. A character a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c7123987ed4f69412c87e3cfd86dffa026ab40dd.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1577ac97abf30d5721697a5c1d616ebd99719d2f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/867597bb6e6e47dbc66bfd89b21261cec942b780.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/dd94a2d4e27800d9406e34690ec915459d7ac51d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
PROPOSITION 2.2. The mappings d1 and d2 induce functorial short exact sequences:
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/dcfd34ac7545e23a335eb345d6436b9787b9d767.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d388c1fe40e6200bcf833fc51bf3a27cfc375743.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/01d8c69d00dfd6fb475289a4695a6b4d892a335f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/867597bb6e6e47dbc66bfd89b21261cec942b780.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/544e6707b7dcd0bf697eef2db9d46b1edc5dd2ee.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f000f6ae097d8b64755663043e76d915f8a5dee2.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bf4f19ab548803c0341fdfea3695b54e2c7dc773.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d388c1fe40e6200bcf833fc51bf3a27cfc375743.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/61ae11d7f1070164bb9f91335b5789f05f701d46.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
PROOF. Note that
X has a cofinal system of tubular neighborhoods each of which is diffeomorphic to![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
LEMMA 2.3. For any a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/70128bd1066299ec62f5aab865ae8ce01a649391.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/70128bd1066299ec62f5aab865ae8ce01a649391.gif)
PROOF. Write a = a1 + dt a2 where a1 and a2 are forms on X whose coefficients depend smoothly on t
[0, 1), or in other words, a1(t), a2(t) are smooth curves in
k(X) and
k - 1(X) respectively with a1(0) = 0. Now da = dxa1 + dt
- dt
dxa2 = 0. We conclude that dxa1 = 0 and dxa2 =
. Since a1(0) = 0 we have
a1(t) = (s) ds =
dxa2(s) ds = dx
We shall also need the following result. On any manifold Y let
k(Y)![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a2c69f3bdea65ae3a26f272ff2ab22f455b57cd8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/810012f6a3c8f5e8eadcea6b625f6f9752712d1d.gif)
denote flat currents and (Y) those with compact support. Note that d
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c1f23e602fa8237d911f91d971642066897b1601.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/cb0f7730a4d00e088d9dcc1613674f59eb5776c8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
LEMMA 2.4.
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/8512bcd2246bd8fdb74d39ea811c1c9322004660.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/7013fe6749ed0bfee61238a69cf2913a8a4f5e00.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/07755cd0b759058cce3dd7a67553116f047cbbb3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/7013fe6749ed0bfee61238a69cf2913a8a4f5e00.gif)
PROOF. Fix f
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c1f23e602fa8237d911f91d971642066897b1601.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
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Observe now that f - is d-closed and has compact support in Y. Since H*(
(Y))
H*(
(Y)) we conclude that there exist a smooth form
and a flat form g, both having compact support on Y such that f -
=
+ dg. Now by the paragraph above we can write g = b + de where b is L
with compact support. Hence f =
+
+ db.
We first prove the surjectivity of . Fix
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We now construct the map j1. Recall from §1 in (Harvey et al. 2001) that
Hk(X,X; S1)
H
(X -
X; S1)
Choose f
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
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d
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/e230578674de8bd8d7e7c75059c704fde4c54e88.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/22bb0dc9fba7bc39e8b4f0cee3f352e86073bbef.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/22bb0dc9fba7bc39e8b4f0cee3f352e86073bbef.gif)
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and so it descends to the quotient Hk(X,X; S1).
To see that j1 is injective, let f = a + db as above and suppose a = dc + S
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb20670a483ebe1df01eef0c838bac9e6cfd4c0.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb20670a483ebe1df01eef0c838bac9e6cfd4c0.gif)
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We now prove the exactness of (A) in the middle. Suppose a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
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We now prove the surjectivity of . Fix u
Hk + 1(X,
X;
) and choose a cycle R
u. Then there is a smooth form
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/544e6707b7dcd0bf697eef2db9d46b1edc5dd2ee.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/e5bab2648db79514ff54dae2e9fc32de49a449a1.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/ab2193432e73434fa41de610b02f2e771c6092ad.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
Now consider an element a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/ab2193432e73434fa41de610b02f2e771c6092ad.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/33db7d2d52734296681d5a7d2871c84068402456.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb20670a483ebe1df01eef0c838bac9e6cfd4c0.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/33db7d2d52734296681d5a7d2871c84068402456.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/33db7d2d52734296681d5a7d2871c84068402456.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/e5bab2648db79514ff54dae2e9fc32de49a449a1.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/33db7d2d52734296681d5a7d2871c84068402456.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f04b680a527c9e65f2ab6cc1b56e87eb60047941.gif)
Note that
)
(2.5)
3. THE STAR PRODUCT
In this section we prove the following.
THEOREM 3.1. There are functorial bilinear mappings
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d388c1fe40e6200bcf833fc51bf3a27cfc375743.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f4fe722c839b2db6a06a378db30b2cdb6bbe53b6.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/854e92f9c30b9d74a8d4169629eac94fafdf395c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/989459efcf3c6335631246210ed79b110e4d87e3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d388c1fe40e6200bcf833fc51bf3a27cfc375743.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f4fe722c839b2db6a06a378db30b2cdb6bbe53b6.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/854e92f9c30b9d74a8d4169629eac94fafdf395c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/989459efcf3c6335631246210ed79b110e4d87e3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
which make (X,
X) a graded commutative ring and
(X) a graded
(X,
X)-module. With this structure the maps
,
are ring and module homomorphisms.
PROOF. Fix a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b65d9bbcd873629b31dd2600ace9d5495e2948b2.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
da = - R and db =
- S
with
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/544e6707b7dcd0bf697eef2db9d46b1edc5dd2ee.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/8470b43a0d998f4717d0f561c8ec96cf8ea536ca.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6c9aa531e1baf9df15070dcf53604ab8c3e80765.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/543d6798934256084b6b0e63fc93faf9b8aa6c0e.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1a438138eae3dac4e89e01fcb0686425c68dfce0.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1a438138eae3dac4e89e01fcb0686425c68dfce0.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c53abbe089de429c25d403f21808f6235621916c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1a438138eae3dac4e89e01fcb0686425c68dfce0.gif)
(3.2)
and if S Z
(X,
X) or if a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bfa5ac085f57ab0b1ace19d2e0e1213980b80cda.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
(3.3)
Since a is smooth near X and a
= 0, a
(3.4)
The arguments from (Harvey et al. 2001) easily adapt to show that
a b![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/5ebde2ff6935e65fe5411cc868badffce556f64d.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f04b680a527c9e65f2ab6cc1b56e87eb60047941.gif)
4. SMOOTH PONTRJAGIN DUALS
The exact sequences of Proposition 2.2 show that (X,
X) has a natural topology making it a topological group (in fact a topological ring) for which
and
are continuous homomorphisms. Essentially it is a product of the standard C
-topology on forms with the standard topology on the torus Hk(X,
X;
)/H
(X,
X;
). It can also be defined as the quotient of the topology induced on sparks by the embedding
k(X,
X)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/8512bcd2246bd8fdb74d39ea811c1c9322004660.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bd4ca6ca9599ecf7fedc4df176110b8b40735c10.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0abc5bfbb162ad0d686932dbe200168d9ebee529.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1483667d0ebcbc939dda7e82c5d4f4401428fc5d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3461fecb682d679ffc3b2024b601502c91783cf8.gif)
It is natural to consider the dual to (X,
X) in the sense of Pontrjagin. For an abelian topological group A we denote by A
(4.1)
where is the restriction mapping.
DEFINITION 4.2. An element f
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/568de9f3cbb8c295e0ee5d2e5c908721807f6aa8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/15345b8410c447f63d510daf8c4272352a03e464.gif)
f (a)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b0aa16e904ad9334cb8c3af7ae1c5171f6992c9f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1e19b85dca957f311e8563ec2372eecec32276ca.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
for a a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f56cca751de6d6711d2f013dfb5cde8cc9ae2040.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/500f304e0de1dafc967a6c45ae8a48c19bdf3bba.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
PROPOSITION 4.3. The smooth Pontrjagin dual (X,
X)
¥ is dense in
(X,
X)
PROOF. Applying to
(X,
X) gives an exact sequence
0
T![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
where T = Hk(X, X;
)/H
(X,
X;
), with dual sequence
(4.4)
Observe that T = H
(X,
X;
)
H
(X;
), and that d
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
with exact rows. Since
n - k - 1(X) is dense in![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0814f54fb4c590ee4651b79be3152581dbf1eb8c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f04b680a527c9e65f2ab6cc1b56e87eb60047941.gif)
There is a parallel story for (X). The analogue of 2.2(B) gives an exact sequence
(4.5)
DEFINITION 4.6. An element f
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/568de9f3cbb8c295e0ee5d2e5c908721807f6aa8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/15345b8410c447f63d510daf8c4272352a03e464.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
f (a)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b0aa16e904ad9334cb8c3af7ae1c5171f6992c9f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1e19b85dca957f311e8563ec2372eecec32276ca.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
for a a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03f6696dfa1cf9ea8eb6992c3900836a214be09b.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f56cca751de6d6711d2f013dfb5cde8cc9ae2040.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/500f304e0de1dafc967a6c45ae8a48c19bdf3bba.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
PROPOSITION 4.7. The smooth Pontrjagin dual (X)
¥ is dense in
(X)
.
PROOF. Applying to
(X) gives an exact sequence
0
T![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/a156408f6657fefd8cb676ef1090097ab35ed67d.gif)
where T = Hk(X; )/Hk(X;
), with dual sequence
(4.8)
Observe now that T = Hk(X;
)
Hn - k(X,
X;
), and d
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/382fa7bcaf69d3480c478f1854aa08a574b8d90a.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/073070a32ef4c85d4fda61a3ca9a33a9edfe9a12.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/867597bb6e6e47dbc66bfd89b21261cec942b780.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/15345b8410c447f63d510daf8c4272352a03e464.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f32201fcd71598c82af90bab518e4cd9033c3c8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f32201fcd71598c82af90bab518e4cd9033c3c8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/caee456914ee3ab25d27b148dfbcad26db1d5853.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1577ac97abf30d5721697a5c1d616ebd99719d2f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/ce95788df539119c09d0958a637e5bb216145703.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1fd8cc33d086526154daec46d8fd912d6b743b7f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/073070a32ef4c85d4fda61a3ca9a33a9edfe9a12.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f000f6ae097d8b64755663043e76d915f8a5dee2.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
with exact rows. Since
n - k - 1(X,![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0814f54fb4c590ee4651b79be3152581dbf1eb8c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f04b680a527c9e65f2ab6cc1b56e87eb60047941.gif)
5. LEFSCHETZ-PONTRJAGIN DUALITY
This brings us to the main result of the paper.
THEOREM 5.1. Let X be a compact, oriented n-manifold with boundary X. Then the biadditive mapping
(X,
X) x
(X)
S1
given by
(a,b) (a
b) [X]
induces isomorphisms
:
(X,
X)
(X)
¥
and
:
(X)
(X,
X)
¥
PROOF. Fix a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0cde8fa16bebc4a179bcb93840aded391a7b4e6f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9e948caf827bcfa82bf15a8c5940f1bec042b2cf.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
since d2b = 0. It follows that = 0.
Hence, da = - R
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0abc5bfbb162ad0d686932dbe200168d9ebee529.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0abc5bfbb162ad0d686932dbe200168d9ebee529.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f511bcc6ab77b0468ec118e3af5e36652e72cfaa.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f511bcc6ab77b0468ec118e3af5e36652e72cfaa.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f511bcc6ab77b0468ec118e3af5e36652e72cfaa.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f511bcc6ab77b0468ec118e3af5e36652e72cfaa.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/26fd4656601423d13ba0d3787f0adcb35635ddce.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b1e5300cd5cbebd2986a04030bd305d52796eec1.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/edeffe205fc6e60d0e2b6ba2a4767e7c49ea692c.gif)
where Lk denotes the de Rham-Seifert linking between the groups Hn - k - 1(X - X;
)tor and Hk(X,
X;
)tor. By the non-degeneracy of this pairing we conclude that
a = 0.
Therefore a ker(
)
ker(
) can be represented by a smooth d-closed form a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/7bd0c10d2be48861d7c6eb113a36937ac9ef9c28.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/fc15c698545767056345d608266e02968fcbac5c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
Hence, a represents the zero class in
and by (2.2) and (2.5) we conclude that a = 0. Thus the map is injective.
To see that is surjective consider the commutative diagram with exact rows:
where the top row is 2.2(A) and the bottom row is the dual of 2.2(B). By definition
0 is onto the smooth elements in![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9e948caf827bcfa82bf15a8c5940f1bec042b2cf.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9bb668828bcac453ed2719fd6356fcb2b8f5d5be.gif)
The proof that is an isomorphism is parallel. Fix b
(X) and suppose (a
b)[X] = 0 for all a
(X,
X). We shall show that b = 0. Choose a spark b
b and write db =
- S as in 1.4. Then for all smooth forms a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/0cde8fa16bebc4a179bcb93840aded391a7b4e6f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3607c2542aeef8e6bf1a86c28c0c8b088e68d3b3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b0aa16e904ad9334cb8c3af7ae1c5171f6992c9f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
since d2a = 0. It follows that = 0.
Hence, db = - S
[S] Hn - k(X;
)tor
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
Choose u Hk + 1(X,
X;
)tor
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f511bcc6ab77b0468ec118e3af5e36652e72cfaa.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f9c634c46710c9baacd3168a2d6c39a822847e54.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b0aa16e904ad9334cb8c3af7ae1c5171f6992c9f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/02ea4d3db597d4deef205c9abc75331c8cd98403.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
where Lk denotes the de Rham-Seifert linking as before. We conclude that a = 0.
Therefore b ker(
)
ker(
) can be represented by a smooth d-closed form b
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6c9aa531e1baf9df15070dcf53604ab8c3e80765.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/32f92e5d840889d510333eca1110cba7dbad7498.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2c46d33e989f13143e43ef3e8a01d38f2122e206.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/947c51f9398a64ac020ea1dd20dcbafa6ab649ac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b0aa16e904ad9334cb8c3af7ae1c5171f6992c9f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f29a2f0cc9c68fbc8b3629d69ae0c3e8172ea41.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/ad6ae48dc2c3f486601c636a9763490042221fb3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
Hence, b represents the zero class in
Hom(Hn - k - 1(X;
),
)/
Hom(Hn - k - 1(X;
),
)
Hn - k - 1(X;
)/Hn - k - 1(X;
)free,
and by (2.2) and (2.5) we conclude that b = 0. Thus the map is injective.
The surjectivity of follows as before from the commutative diagram with exact rows:
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/dcfd34ac7545e23a335eb345d6436b9787b9d767.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/867597bb6e6e47dbc66bfd89b21261cec942b780.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/15345b8410c447f63d510daf8c4272352a03e464.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1577ac97abf30d5721697a5c1d616ebd99719d2f.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f32201fcd71598c82af90bab518e4cd9033c3c8.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6f32201fcd71598c82af90bab518e4cd9033c3c8.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/ce95788df539119c09d0958a637e5bb216145703.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/921eca07cfb781134e3297e7fe9c8e4933c04ead.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2ebc28f4927db4cba514679b16e62c7fbd3f4566.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d388c1fe40e6200bcf833fc51bf3a27cfc375743.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
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![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f000f6ae097d8b64755663043e76d915f8a5dee2.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/216740ff2ac532daf2a71ad7c3ba5fb9bf7df971.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d4a0b00f91e60b4f32b46755fc42d7d72178e387.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/295fbfd4c5e8394941299279df2a69af2a2fb2ae.gif)
This completes the proof.
6. COBOUNDARY MAPS
It is natural to ask if there is a coboundary mapping with the property that the sequence
is exact. The differential-form-component of characters makes this impossible. However, there do exist natural coboundary maps with the following properties:
(1) Under the sequence (6.1) becomes the standard long exact sequence in integral cohomology.
(2) Under the sequence (6.1) becomes a sequence of smooth d-closed forms which induces the standard long exact sequence in real cohomology.
Recall that the definitions of Thom maps and Gysin maps for differential characters depend essentially on a choice of "normal geometry''. This will also be true for our coboundary maps. Fix a tubular neighborhood N0 of X in X and an identification N0
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f42f4982b500b7102e7e223409ad43405198b1e3.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/c53abbe089de429c25d403f21808f6235621916c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2e5e11c220e5e07b1644dfb4b336c121a5521fae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2e5e11c220e5e07b1644dfb4b336c121a5521fae.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb20670a483ebe1df01eef0c838bac9e6cfd4c0.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/5408234479575474a66d1b837cad6585ad4661c4.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/d2a1696678b4af58c51aa40394389f547a535006.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/58890ef192064f61ab143345b3bf57fc50bb7571.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/2f50ecf5558ecc2662acbcb8410e6fc0df7bf2e1.gif)
Note that d = d
- [
N] has compact support in X -
X.
DEFINITION 6.2. We define the coboundary map =
:
(
X)
(X,
X) by
(a) = (
a)
.
Verification of (1) and (2) above is straightforward, and the details are omitted.
7. SEQUENCES AND DUALITY
At the level of cohomology the long exact sequences for the pair (X,X) are related by the duality mappings. There is an analogous diagram for differential characters:
and it is natural to ask whether this diagram commutes (up to sign). The square on the left is evidently commutative. The other two squares commute up to an error term which we now analyse.
We begin with the square on the right. Fix a
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/567bb32d04f113fc7df7c1d1f4733aad12fa6fac.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/241f4f7d8e0069491e21127762022aa1e385ef9d.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/e5bab2648db79514ff54dae2e9fc32de49a449a1.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9334182737ad05f205e2d29970ccdaeb0198f559.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03dab58c781e0e9048d73279349cbba02e01f87a.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9b8a441c58e5984e647c5bb031805f051f9a6a04.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9b8a441c58e5984e647c5bb031805f051f9a6a04.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/03dab58c781e0e9048d73279349cbba02e01f87a.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9b8a441c58e5984e647c5bb031805f051f9a6a04.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/f42f4982b500b7102e7e223409ad43405198b1e3.gif)
Now we may assume that S =
S0 for some S0
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/bc0da3c07fe399a4062b7f8d5868416fb1bce846.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/671dec7efb2b857ca4beca7c4d2f7e6c68ad8774.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/b1e5300cd5cbebd2986a04030bd305d52796eec1.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/edeffe205fc6e60d0e2b6ba2a4767e7c49ea692c.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9b8a441c58e5984e647c5bb031805f051f9a6a04.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/9b8a441c58e5984e647c5bb031805f051f9a6a04.gif)
Now d (a*b) =
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/1a438138eae3dac4e89e01fcb0686425c68dfce0.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/6eb0a3c39e363ab5cc64e08f00d1aaf588722f70.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/11eed58f60acf48fcb72466db6e6a3d74bfae699.gif)
![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/e6c2a2a3d54cdb627c4a32f681769e5e147993c3.gif)
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Thus for example we see that (
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A similar analysis applies to the middle square in the diagram and we have the following.
PROPOSITION 7.3. The duality diagram above commutes in the limit as
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This is the best one can expect. The "commutators'' in this diagram do not lie in the smooth dual. Of course by Propositions 4.3 and 4.7 they do lie in its closure.
Here is an explicit example of this non-commutativity. Let X = S2x D3 be the product of the 2-sphere and the 3-disk. Choose sparks a
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(a
b)[![](https://minio.scielo.br/documentstore/1678-2690/9NqB878VfXgSLr55ccNVghy/3000b78a762b7d0a6d10e04baa2c0adc14b2ffd7.gif)
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ACKNOWLEDGMENTS
Research of both authors was partially supported by the NSF. Research of the second author was also partially supported by IHES and CMI.
RESUMO
Uma teoria de caracteres diferenciais é aqui desenvolvida para variedades com bordo. Isto é feito tanto do ponto de vista de Cheeger-Simons como do deRham-Federer. O resultado central deste artigo é a formulação e a prova de um teorema da dualidade de Lefschetz-Pontrjagin, que afirma que o pareamento
dado por (a, b) (a * b) [X] induz isomorfismos
sobre os duais diferenciáveis de Pontrjagin. Em particular, e
são injetivos com domínios densos no grupo de todos os homeomorfismos contínuos no círculo. Uma aplicação de cobordo é introduzida, a qual fornece uma sequência longa para os grupos de caracteres associados ao par ( X,
X). A relação desta sequência com as aplicações de dualidade é analisada.
Palavras-chave: caracteres diferenciais, dualidade de Lefschetz, teoria de deRham.
HARVEY FR, LAWSON HB AND ZWECK J. 2001. The deRham-Federer theory of differential characters and character duality, Stony Brook preprint.
HARVEY FR AND ZWECK J. 2001. Divisors and Euler sparks of atomic sections, Indiana Math J, to appear.
SIMONS J. 1974. Characteristic forms and transgression: characters associated to a connection, Stony Brook preprint.
- CHEEGER J. 1973. Multiplication of Differential Characters, Instituto Nazionale di Alta Mathematica, Symposia Mathematica XI: 441-445.
- CHEEGER J AND SIMONS J. 1985. Differential Characters and Geometric Invariants, Lect Notes in Math 1167: 50-80. New York, Springer-Verlag.
- DE RHAM G. 1955. Variétés Différentiables, formes, courants, formes harmoniques, Hermann, Paris.
- FEDERER H. 1969. Geometric Measure Theory, Springer-Verlag, New York.
- GILLET H AND SOULÉ C. 1989. Arithmetic chow groups and differential characters, Algebraic K-theory; Connections with Geometry and Topology, JARDINE AND SNAITH (eds.), Kluwer Academic Publishers, 30-68.
- HARRIS B. 1989. Differential characters and the Abel-Jacobi map, in "Algebraic K-theory; Connections with Geometry and Topology," JARDINE AND SNAITH (eds.), Kluwer Academic Publishers, pp. 69-86.
- HARVEY FR AND LAWSON HB. 1993. A theory of characteristic currents associated with a singular connection, Astérisque 213: 1-268.
- HARVEY FR AND LAWSON HB. 1995. Geometric residue theorems, Amer J Math 117: 829-873.
Publication Dates
-
Publication in this collection
08 June 2001 -
Date of issue
June 2001
History
-
Accepted
12 Feb 2000 -
Received
05 Feb 2001