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Anais da Academia Brasileira de Ciências
versão impressa ISSN 0001-3765versão On-line ISSN 1678-2690
An. Acad. Bras. Ciênc. v.73 n.2 Rio de Janeiro jun. 2001
http://dx.doi.org/10.1590/S0001-37652001000200001
Lefschetz-Pontrjagin Duality for Differential Characters^{*}
REESE HARVEY^{1} and BLAINE LAWSON^{2}
^{1}The Department of Mathematics, P.O. Box 1892, Rice University, Houston, TX 77251-1892
^{2}The 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 ^{**}
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) S^{1}
given by
(a, b) (a b)[X]
(where denotes characters with compact support) induces injective maps
(X) Hom(X), S^{1} and (X) Hom(X), S^{1}
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^{k + 1}(X) with the C^{}-topology. Now Hom(d^{k + 1}(X), S^{1}) = Hom(d^{k + 1}(X),) is just the vector space dual. This is simply a quotient of the space of currents, the (n - k - 1)-forms with distribution coefficients. The smooth dual corresponds to those forms which have smooth coefficients.
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) S^{1}
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
^{*}(X, X) = { ^{*}(X) : = 0}.
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(Z_{k}(X), S^{1}) : (a) ^{k + 1}(X)}
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(Z_{k}(X,X), S^{1}) : (a) ^{k + 1}(X,X)}
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
(X) k-forms on X with L-coefficients |
^{k}(X) the rectifiable currents of degree k (dimension n - k) on X |
(X,X) {a (X) : a in smooth in a neighborhood of X and a = 0} |
(X - X) {R ^{k}(X) : supp(R) X - X} |
DEFINITION 1.2. An element a (X) is called a spark of degree k on X if
da = - R where ^{k + 1}(X) and R ^{k + 1}(X). | (1.3) |
Denote by ^{k}(X) the group of all such sparks and by ^{k}(X) the subgroup of all a ^{k}(X) such that a = db + S where b (X) S (X). Then the group of deRham-Federer characters of degree k on X is defined to be the quotient
(X) ^{k}(X)/^{k}(X).
Given a spark a ^{k}(X) we denote its associated character by a (X).
We define relative sparks and relative deRham-Federer characters on (X,X) by
^{k} (X, X) {a (X, X) : da = - R, ^{k + 1}(X,X) and R (X - X)} |
^{k} (X, X) {a ^{k}(X, X) : a = db + S, b (X,
X) and S (X - X)} |
(X, X) ^{k}(X, X)/^{k}(X, X). |
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 ^{k + 1}(X) and R is integrally flat. If da = - R' is a similar decomposition, then = and R = R'. Furthermore,
d = 0 and dR = 0
and has integral periods on cycles in X. In the case that ^{k + 1}(X,X) and supp (R) X - X, one has that dR = 0 and has integral periods on all relative cycles in (X, X).
Set
(X) = { (X) : d = 0 and has integral periods}
(X, X) = { (X, X) : d = 0 and has
integral periods on relative cycles in (X,X) } | (1.5) |
Z(X) = {R (X) : dR = 0}
Z(X,X) = {R ¸(X - X) : dR = 0} |
COROLLARY 1.6. Taking d_{1}a = and d_{2}a = R from the decomposition (1.3) gives well-defined mappings
d_{1} : S^{k}(X) (X), d_{2} : S^{k}(X) Z(X),
and
d_{1} : S^{k}(X, X) (X, X),d_{2} : S^{k}(X, X) Z(X - X)
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 ^{k}(X) such that a is smooth near
X, a = 0, and da = - R where ^{k + 1}(X, X)
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) (X, X) induces an isomorphism
(X, X) (X, X)/(X, X)
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 (X, X) is said to be smooth if a = a for a smooth form a ^{k} (X,X). The set of smooth characters is denoted (X,X). There is a natural isomorphism
(X, X) ^{k}(X, X)/(X, X)
PROPOSITION 2.2. The mappings d_{1} and d_{2} induce functorial short exact sequences:
0 H^{k} (X, X; S^{1})(X, X)(X, X) 0, | (A) |
0 (X, X)(X,X)H^{k + 1}(X,X; ) 0. | (B) |
PROOF. Note that X has a cofinal system of tubular neighborhoods each of which is diffeomorphic to X x [0, 1). We shall use the following elementary result.
LEMMA 2.3. For any a ^{k}(X x [0, 1)) such that da = 0 and a = 0, there exists b ^{k - 1}(X x [0, 1)) such that db = a and b = 0.
PROOF. Write a = a_{1} + dt a_{2} where a_{1} and a_{2} are forms on X whose coefficients depend smoothly on t [0, 1), or in other words, a_{1}(t), a_{2}(t) are smooth curves in ^{k}(X) and ^{k - 1}(X) respectively with a_{1}(0) = 0. Now da = d_{x}a_{1} + dt - dt d_{x}a_{2} = 0. We conclude that d_{x}a_{1} = 0 and d_{x}a_{2} = . Since a_{1}(0) = 0 we have
a_{1}(t) = (s) ds = d_{x}a_{2}(s) ds = d_{x}a_{2}(s) ds
Set b a_{2}(s) ds, and note that: b = 0, d_{x}b = a_{1} and = a_{2}. Hence, a = db. |
We shall also need the following result. On any manifold Y let
^{k}(Y) (Y) + d(Y)
denote flat currents and (Y) those with compact support. Note that d^{k}(Y) = d(Y). This definition of (Y) arises naturally in sheaf theory. However, the following equivalent definition will also be useful here.
LEMMA 2.4. (Y) = (Y) + d(Y) and so d(Y) = d(Y).
PROOF. Fix f (Y) and write f = a + db where a (Y) and b (Y). Let K = supp(f ), and note that in N Y - K we have that a = - db. By standard de Rham theory there exists an L-form b_{0} on N such that a_{} a + db_{0} is smooth on N. Furthermore since a_{} is weakly exact on N there exists a smooth form b_{} with a_{} = - db_{} on N. Choose C(Y) with 1 in a neighbothood of K, let = 1 - and set = a + d (b_{0} + b_{}) and = b - b_{0} - b_{} with as above. Then f = + d and has compact support in Y.
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 (X,X). Then by Lemma 2.3 there is a neighborhood N X x [0, 1) of X and a form A ^{k}(N) with dA = and A = 0. Choose C(N) with 1 in a neighborhood of X, and set = - d (A). Now supp () X - X and has integral periods, so there exists a cycle R Z(X, X) with [ - R] = 0 in H(X - X;). By Lemma 2.4 there are L-forms a, b with compact support in X - X such that d (a + db) = da = - R. Then d_{1}(A + a) = and surjectivity is proved.
We now construct the map j_{1}. Recall from §1 in (Harvey et al. 2001) that
H^{k}(X,X; S^{1}) H(X - X; S^{1})
Choose f (X - X) with df = R (X - X), and write f = a + db where a and b are L-forms with compact support in X - X (cf. Lemma 2.4). Then a ^{k}(X, X) and we set j_{1}(f ) a (X, X). Note that if f = a' + db' is another such decomposition, then a - a' = d (c' - c) and a = a'. Clearly j_{1} = 0 on
d(X - X) + (X - X) = d(X - X) + (X - X),
and so it descends to the quotient H^{k}(X,X; S^{1}).
To see that j_{1} is injective, let f = a + db as above and suppose a = dc + S ^{k}(X, X) where c is smooth and zero on X. By Lemma 2.3 there exists an L-form e, smooth near X, such that c_{0} = c - de 0 near X. Then a = dc_{0} + S 0 in H^{k}(X,X; S^{1}).
We now prove the exactness of (A) in the middle. Suppose a ^{k}(X, X) and (a) = 0. Then da = - R (X - X). Thus, in a neighborhood N of X we have that a is smooth, da = 0 and a = 0. By Lemma 2.3 there exists b ^{k - 1}(N) with db = a and b = 0. Then = a - d (b), with as above, is equivalent to a in (X,X). Since has compact support in X - X and d = - R, we see that lies in the image of j_{1}.
We now prove the surjectivity of . Fix u H^{k + 1}(X, X; ) and choose a cycle R u. Then there is a smooth form (X, X) such that - R = df for f (X - X). By Lemma 2.4 f = a + db where a is L with compact support in X - X. Then a ^{k}(X,X) and (a) = u.
Now consider an element a ^{k}(X,X) with (a) = 0. Then da = - R where is smooth and R = dS for some S (X - X). Then = a - S a in (X,X) and d = 0 on X. Since is smooth near X, standard de Rham theory shows that there is an L-form b with compact support in X - X such that - db is smooth. Hence, a = (X, X).
Note that
ker() ker() | (2.5) |
3. THE STAR PRODUCT
In this section we prove the following.
THEOREM 3.1. There are functorial bilinear mappings
(X, X) x (X, X) | (X, X) and |
(X, X) x (X) | (X, X) |
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 (X, X) and b (X). Then from (Harvey et al. 2001) we know that there exist sparks a a and b b with
da = - R and db = - S
with (X, X), (X), R Z(X, X) and S Z(X), so that the wedge-intersection products R b and R S are well defined. Furthermore, if suppS X - X we can also assume that a S is well defined. We then define
a b = a + (- 1)^{k + 1}R b, | (3.2) |
and if S Z(X,X) or if a (X - X), we can also define
ab = a S + (- 1)^{k + 1} b. | (3.3) |
Since a is smooth near X and a = 0, a b also has these properties (as well as ab when it is defined). Note that
d (a b) = d (ab) = - R S | (3.4) |
The arguments from (Harvey et al. 2001) easily adapt to show that a b depends only on a and b, and that a b = ab (when it is defined). Associativity, commutativity, etc. are straightforward. Equation (3.4) establishes the homomorphism propertes of and .
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 H^{k}(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) ^{k}(X) x ^{k + 1}(X, X) x (X - X) sending a (a, d_{1}a, d_{2}a). (Similar remarks apply to (X).)
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 Hom_{cont}(A, S^{1}) the group of continuous homomorphisms h : AS^{1}. Then 2.2(B) yields a dual sequence
0 H^{k + 1}(X, X; ) (X, X)(X, X) 0. | (4.1) |
where is the restriction mapping.
DEFINITION 4.2. An element f (X, X) is called smooth if there exists a form (X) such that
f (a) a (mod )
for a a (X, X). An element f (X, X) is called smooth if (f ) is smooth. The set of these is called the smooth Pontrjagin dual of (X,X) and is denoted by (X, X)^{¥} = Hom_{}((X, X), S^{1}).
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 (X, X) d^{k}(X, X) 0
where T = H^{k}(X, X; )/H(X, X; ), with dual sequence
0 d^{k}(X, X) (X, X) T 0 | (4.4) |
Observe that T = H(X, X; ) H(X; ), and that d^{k}(X, X) = {d^{k}(X, X)}' (the topological vector space dual) which is exactly the space of currents of degree n - k - 1 on X restricted to the closed subspace d^{k}(X, X). This gives a commutative diagram
with exact rows. Since ^{n - k - 1}(X) is dense in ^{ n - k - 1}(X), the result follows.
There is a parallel story for (X). The analogue of 2.2(B) gives an exact sequence
0 H^{k + 1}(X; ) (X)(X). 0. | (4.5) |
DEFINITION 4.6. An element f (X) is called smooth if there exists a form (X, X) such that
f (a) a (mod )
for a a (X) An element f (X) is called smooth if (f ) is smooth. The set of these is called the smooth Pontrjagin dual of (X) and is denoted (X)^{¥} = Hom_{}((X), S^{1}).
PROPOSITION 4.7. The smooth Pontrjagin dual (X)^{¥} is dense in (X).
PROOF. Applying to (X) gives an exact sequence
0 T (X) d^{k}(X) 0,
where T = H^{k}(X; )/H^{k}(X; ), with dual sequence
0 d^{k}(X) (X) T 0. | (4.8) |
Observe now that T = H^{k}(X; ) H^{n - k}(X, X; ), and d^{k}(X) = {d^{k}(X)}' is the space of currents of degree n - k - 1 on X restricted to the closed subspace d^{k}(X). This gives a commutative diagram:
^{n - k - 1}(X, X) | (X, X) | H^{n - k}(X, X; ) | 0 | |||
^{n - k - 1}(X) | (X) | T | 0 |
with exact rows. Since ^{n - k - 1}(X, X) is dense in ^{n - k - 1}(X), the result follows.
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) S^{1}
given by
(a,b) (a b) [X]
induces isomorphisms
: (X, X)(X)^{¥}
and
: (X)(X, X)^{¥}
PROOF. Fix a (X,X) and suppose (a b)[X] = 0 for all b (X). We shall show that a = 0. Choose a spark a a and write da = - R as in 1.4. Then for all smooth forms b ^{n - k - 1}(X) we have by (3.3) that
since d_{2}b = 0. It follows that = 0.
Hence, da = - R (X - X) is a cycle with [R] H(X - X; )_{tor} H_{n - k - 1}(X - X; )_{tor}. Choose any u H^{n - k}(X; )_{tor} H_{k} (X, X; )_{tor}, and choose a relative cycle S u. Let m be the order of u. Then there is a (k + 1)-chain T on X with dT = mS rel X. Set b = - T and consider b as a spark of degree n - k - 1 on X with db = - S. Now we may assume S and T to have been chosen so that supp(S) supp(R) = and T meets R properly. Then
where Lk denotes the de Rham-Seifert linking between the groups H_{n - k - 1}(X - X; )_{tor} and H_{k}(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 ^{k} (X, X). In fact by Lemma 2.3 we may choose a to have compact support in X - X. Now for any cycle S Z(X), i.e., any k-dimensional rectifiable current S _{k}(X) with dS _{k - 1}(X), we can find ^{n - k}(X) and b (X) with db = - S. Then by (3.3) we have that
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 (X) and therefore the map is surjective.
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 ^{k}(X, X) we have by (3.3) that
a b [X] = a 0 mod
since d_{2}a = 0. It follows that = 0.
Hence, db = - S ^{n - k}(X) is a relative cycle with torsion homology class
[S] H^{n - k}(X; )_{tor} H_{k}(X,X; )_{tor}.
Choose u H^{k + 1}(X, X; )_{tor} H_{n - k - 1}(X; )_{tor}, and choose a cycle R u with support in X - X. Let m be the order of u. Then there is a (n - k - 1)-chain T in X - X with dT = mR. Set a = - T and consider a as a spark of degree k on X with da = - R. Now we may assume R and T to have been chosen so that supp(R) supp(S) = and T meets S properly. Then
0 = ab [X] | (- 1)^{k + 1}a S [X] mod |
(- 1)^{k + 1}T S [X] mod | |
(- 1)^{k + 1}Lk([R],[S]) mod | |
(- 1)^{k + 1}Lk(u,b) mod |
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 ^{n - k - 1}(X). Now for any cycle R Z(X, X), i.e., any (n - k - 1)-dimensional rectifiable current R _{n - k - 1}(X - X) with dR = 0, we can find ^{k + 1}(X, X) and a (X,X) with da = - R. Then by (3.2) we have that
0 = a*b [X] | (- 1)^{k + 1}R b [X] mod |
(- 1)^{n(k + 1)} b mod . |
Hence, b represents the zero class in
Hom(H_{n - k - 1}(X; ), )/Hom(H_{n - k - 1}(X; ), ) H^{n - k - 1}(X; )/H^{n - 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:
0 | H^{n - k - 1}(X; S^{1}) | (X) | (X) | 0 | ||||
_{0} | ||||||||
0 | Hom(H^{k + 1}(X,X; ), S^{1}) | (X, X) | (X,X) | 0. |
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 N_{0} of X in X and an identification N_{0} X x [0, 2), and let : N_{0}X be the projection. Set N = X x [0, 1) N_{0} and let _{N} be the characteristic function of this subset. Let be a smooth approximation to _{N}; specifically choose (x, t) = (t) where 1 near 0 and (t) = 0 for t 1. Then set
- _{N} (X)
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 (X) and b (X) and choose L-sparks a_{0} a and b b with da_{0} = - R_{0} and db = - S as usual. Let a = a_{0}, = and R = R_{0} denote the pull-backs to the collar neighborhood of X via the projection : N_{0}X defined in §6. Then
Now we may assume that S = S_{0} for some S_{0} ^{k + 1}(X), and we may further assume that supp(R_{0}) supp(S_{0}) = because dim(R_{0}) + dim(S_{0}) = n - 2. Hence, d_{2}(ab) = R_{0} S_{0} = (R_{0} S_{0}) = 0, and from (7.1) we see that
Now d (a*b) = - R S = and we can write = + dt as in the proof of Lemma 2.3. Since = we see that = 0 and we conclude that
Thus for example we see that (o)(a) = (- 1)^{n - 1}(o)(a) on all b which are -pull backs in N. Furthermore, we can consider the family of sparks r where r : X x [0,)X[0, 1) is given by r (x, t) = (x, t/). From (7.2) we see that
E() = 0.
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 0.
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 = S^{2} x D^{3} be the product of the 2-sphere and the 3-disk. Choose sparks a ^{1}(S^{2}) and b ^{2}(D^{3}) with da = - [x_{0}] and db = - [0] for some x_{0} S^{2}, where and are unit volume forms on S^{2} and D^{3} respectively. Direct calculation shows that
(ab)[X] = 1 but (ab)[X] = (1 - ) < 1.
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.
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^{*}Invited paper
^{**}Foreign Member of Academia Brasileira de Ciências
Correspondence to: Blaine Lawson
E-mail: blaine@math.sunysb.edu / harvey@math.rice.edu