Lefschetz-Pontrjagin duality for differential characters

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: Ch^k(X,dX) x Ch^{n-k-1}(X) -->S^1, given by (a,b) l-->(a*b)[X], induces isomorphisms: D : Ch^k(X,dX) -->Hom^{smooth}(Ch^{n-k-1}(X), S^1) D': Ch^{n-k-1}(X) -->Hom^{smooth}(Ch^k(X, dX), S^1) onto the smooth Pontrjagin duals. In particular, D and D' 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,dX). The relation of the sequence to the duality mappings is analyzed.


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 c k are defined and the total class gives a natural transformation c = 1 + c 1 + c 2 + · · · : K(X) −→ H * (X) from the K-theory of bundles with connection to differential characters which satisfies the Whitney sum formula: c(E ⊕ F ) = c(E) * c(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), ), analogous to those given for cohomology, which are useful for example in the theory of singular connections Lawson 1993, 1995). Furthermore, the groups H k (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 , 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 H k (X) × H n−k−1 cpt (X) −→ S 1 given by (α, β) → (α * β) [X] (where H * cpt denotes characters with compact support) induces injective maps H k (X) → Hom H n−k−1 cpt (X), S 1 and H n−k−1 cpt (X) → Hom H k (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 H k (X) consists essentially (i.e., up to a finite-dimensional torus factor) of the exact (k + 1)-forms dE k+1 (X) with the C ∞ -topology. Now Hom(dE k+1 (X), S 1 ) = Hom(dE k+1 (X), R) 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 H * (X, ∂X) and develop the theory from ) for this case. The main theorem asserts the existence of a pairing [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 ∂ : H k (X) → H k+1 (X, ∂X), yielding long sequences which intertwine the duality mappings and reduce to the standard picture under the natural transformation to integral cohomology.

DIFFERENTIAL CHARACTERS ON MANIFOLDS WITH BOUNDARY
Let X be a compact oriented differentiable n-manifold with boundary ∂X. Let E * (X) denote the de Rham complex of differential forms which are smooth up to the boundary, and set The cohomology of this complex is naturally isomorphic to H * (X, ∂X; R). 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 homomor-phismsĤ where δ denotes the coboundary. Similarly the group of relative differential characters of degree k on (X, ∂X) is defined to bê H k (X, ∂X; R/Z) ≡ {α ∈ Hom(Z k (X, ∂X), S 1 ) : δ(α) ∈ E k+1 (X, ∂X)} Inclusion and restriction give maps H k (X, ∂X) There is an alternative de Rham-Federer approach to these groups. Set Denote by S k (X) the group of all such sparks and by T k (X) the subgroup of all a ∈ S k (X) such Then the group of deRham-Federer characters of degree k on X is defined to be the quotient Given a spark a ∈ S k (X) we denote its associated character by a ∈ H k (X).
We define relative sparks and relative deRham-Federer characters on (X, ∂X) by 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 ) one concludes: and φ has integral periods on cycles in X. In the case that φ ∈ E k+1 (X, ∂X) and supp(R) ⊂ X−∂X, one has that dR = 0 and φ has integral periods on all relative cycles in (X, ∂X).
Corollary 1.6. Taking d 1 a = φ and d 2 a = R from the decomposition (1.3) gives well-defined mappings Proposition 1.7. There are natural isomorphisms Proof. The first is proved in ). The argument for the second is exactly the same.
Remark 1.8. In  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 H * (X) and H * (X, ∂X). The proof of the equivalence of these definitions closely follows the arguments in §2 of ) 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

THE EXACT SEQUENCES
The fundamental exact sequences established by Cheeger and Simons in (Cheeger and Simons 1985) carry over to the relative case.
Proposition 2.2. The mappings d 1 and d 2 induce functorial short exact sequences: Proof. Note that ∂X has a cofinal system of tubular neighborhoods each of which is diffeomorphic to ∂X × [0, 1). We shall use the following elementary result.
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, We shall also need the following result. On any manifold Y let arises naturally in sheaf theory. However, the following equivalent definition will also be useful here.
, and note that in N ≡ Y − K we have that a = −db. By standard de Rham theory there exists an L 1 loc -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 η ∈ C ∞ 0 (Y ) with η ≡ 1 in a neighbothood of K, let χ = 1 − η and set a = a + d(χb 0 + χb ∞ ) and b = b − χb 0 − χb ∞ with χ as above. Then f = a + d b and a has compact support in Y .
Observe now that f − a is d-closed and has compact support in Y . Since H * (E * cpt (Y )) ∼ = H * (F * cpt (Y )) we conclude that there exist a smooth form ω and a flat form g, both having compact support on Y such that f − a = ω + dg. Now by the paragraph above we can write We first prove the surjectivity of δ 1 . Fix φ ∈ Z k+1 0 (X, ∂X). Then by Lemma 2.3 there is a neighborhood N ∼ = ∂X × [0, 1) of ∂X and a form A ∈ E k (N ) with dA = φ and A ∂X = 0. Choose χ ∈ C ∞ 0 (N ) with χ ≡ 1 in a neighborhood of ∂X, and set φ 0 = φ − d(χA). Now supp(φ 0 ) ⊂⊂ X − ∂X and φ 0 has integral periods, so there exists a cycle R ∈ Z rect (X, ∂X) with Then d 1 (χ A + a) = φ and surjectivity is proved.
We now construct the map j 1 . Recall from §1 in ) that , and write f = a + db where a and b are L 1 loc -forms with compact support in X − ∂X (cf. Lemma 2.4). Then a ∈ S k (X, ∂X) and we set j 1 (f ) ≡ a ∈ H k (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 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 ∈ T k (X, ∂X) where c is smooth and zero on ∂X. By Lemma 2.3 there exists an L 1 loc -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 ∈ S k (X, ∂X) and δ 1 ( a ) = 0. Then da = −R ∈ R k+1 cpt (X − ∂X). Thus, in a neighborhood N of ∂X we have that a is smooth, da = 0 and a ∂X = 0. By Lemma 2.3 there exists b ∈ E k−1 (N ) with db = a and b ∂X = 0. Then a = a − d(χb), with χ as above, is equivalent to a in H k (X, ∂X). Since a has compact support in X − ∂X and d a = −R, we see that a lies in the image of j 1 .
We now prove the surjectivity of δ 2 . Fix u ∈ H k+1 (X, ∂X; Z) and choose a cycle R ∈ u.
Then a ∈ S k (X, ∂X) and δ 2 ( a ) = u. Now consider an element a ∈ S k (X, ∂X) with δ 2 ( a ) = 0. Then da = φ − R where φ is smooth and R = dS for some S ∈ R k cpt (X − ∂X). Then a = a − S ≡ a in H k (X, ∂X) and d a = 0 on X. Since a is smooth near ∂X, standard de Rham theory shows that there is an L 1 loc -form b with compact support in X − ∂X such that a − db is smooth. Hence, a = a ∈ H k ∞ (X, ∂X).

THE STAR PRODUCT
In this section we prove the following. Proof. Fix α ∈ H k (X, ∂X) and β ∈ H (X). Then from  we know that there exist sparks a ∈ α and b ∈ β with rect (X, ∂X) and S ∈ Z +1 rect (X), so that the wedgeintersection products R ∧ b and R ∧ S are well defined. Furthermore, if supp S ⊂⊂ X − ∂X we can also assume that a ∧ S is well defined. We then define and if S ∈ Z +1 rect (X, ∂X) or if a ∈ E k cpt (X − ∂X), we can also define Since a is smooth near ∂X and a ∂X = 0, a * b also has these properties (as well as a * b when it is defined). Note that The arguments from ) easily adapt to show that a * b depends only on a and b , and that a * b = a * b (when it is defined). Associativity, commutativity, etc. are straightforward. Equation (3.4) establishes the homomorphism propertes of δ 1 and δ 2 .

SMOOTH PONTRJAGIN DUALS
The exact sequences of Proposition 2.2 show that H * (X, ∂X) has a natural topology making it a topological group (in fact a topological ring) for which δ 1 and δ 2 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; R)/H k free (X, ∂X; Z). It can also be defined as the quotient of the topology induced on sparks by the embedding S k (X, ∂X) → (a, d 1 a, d 2 a). (Similar remarks apply to H * (X).) It is natural to consider the dual to H * (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 : A → S 1 . Then 2.2(B) yields a dual sequence where ρ is the restriction mapping.
The set of these is called the smooth Pontrjagin dual of H k (X, ∂X) and is denoted by H k (X, ∂X) ∞ = Hom ∞ ( H k (X, ∂X), S 1 ).
Observe that T = H k free (X, ∂X; Z) ∼ = H n−k free (X; Z), and that dE k (X, ∂X) = {dE 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 dE k (X, ∂X). This gives a commutative diagram There is a parallel story for H * (X). The analogue of 2.2(B) gives an exact sequence Definition 4.6. An element f ∈ H k ∞ (X) is called smooth if there exists a form ω ∈ Z n−k 0 (X, ∂X) such that The set of these is called the smooth Pontrjagin dual of H k (X) and is denoted H k (X) ∞ = Hom ∞ ( H k (X), S 1 ).
Proposition 4.7. The smooth Pontrjagin dual H k (X) ∞ is dense in H k (X) .
Proof. Applying δ 1 to H k ∞ (X) gives an exact sequence Observe now that T = H k (X; Z) ∼ = H n−k (X, ∂X; Z), and dE k (X) = {dE k (X)} is the space of currents of degree n − k − 1 on X restricted to the closed subspace dE k (X). This gives a commutative diagram: with exact rows. Since E n−k−1 (X, ∂X) is dense in D n−k−1 (X), the result follows.

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 that α = 0. Choose a spark a ∈ α and write da = φ − R as in 1.4. Then for all smooth forms b ∈ E n−k−1 (X) we have by (3.3) that cpt (X−∂X; Z) tor ∼ = H n−k−1 (X− ∂X; Z) tor . Choose any u ∈ H n−k (X; Z) tor ∼ = H k (X, ∂X; Z) 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 = − 1 m 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; Z) tor and H k (X, ∂X; Z) tor . By the non-degeneracy of this pairing we conclude that δ 2 α = 0.
Therefore α ∈ ker(δ 1 )∩ker(δ 2 ) can be represented by a smooth d-closed form a ∈ E 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 n−k rect (X), i.e., any k-dimensional rectifiable current S ∈ R k (X) with dS ∈ R k−1 (∂X), we can find ψ ∈ E n−k (X) and b ∈ E n−k−1 L 1 loc (X) with db = ψ − S. Then by (3.3) we have that Hence, a represents the zero class in Hom(H k (X, ∂X; Z), R)/ Hom(H k (X, ∂X; Z), Z) ∼ = H k (X, ∂X; R)/H k (X, ∂X; Z) free , and by (2.2) and (2.5) we conclude that α = 0. Thus the map D is injective.
To see that D 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 D 0 is onto the smooth elements in H n−k−1 (X) and therefore the map D is surjective.
The proof that D is an isomorphism is parallel. Fix β ∈ H n−k−1 (X) and suppose (α * β)[X] = 0 for all α ∈ H k (X, ∂X). We shall show that β = 0. Choose a spark b ∈ β and write db = ψ − S as in 1.4. Then for all smooth forms a ∈ E k (X, ∂X) we have by (3.3) that Hence, db = −S ∈ R n−k (X) is a relative cycle with torsion homology class Choose u ∈ H k+1 (X, ∂X; Z) tor ∼ = H n−k−1 (X; Z) 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 = − 1 m 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 where Lk denotes the de Rham-Seifert linking as before. We conclude that δ 2 α = 0. Therefore β ∈ ker(δ 1 )∩ker(δ 2 ) can be represented by a smooth d-closed form b ∈ E n−k−1 (X). Now for any cycle R ∈ Z k+1 rect (X, ∂X), i.e., any (n − k − 1)-dimensional rectifiable current R ∈ R n−k−1 (X − ∂X) with dR = 0, we can find φ ∈ E k+1 (X, ∂X) and a ∈ E k L 1 loc (X, ∂X) with da = φ − R. Then by (3.2) we have that Hence, b represents the zero class in Hom(H n−k−1 (X; Z), R)/ Hom(H n−k−1 (X; Z), Z) ∼ = H n−k−1 (X; R)/H n−k−1 (X; Z) free , and by (2.2) and (2.5) we conclude that β = 0. Thus the map D is injective.
The surjectivity of D follows as before from the commutative diagram with exact rows: This completes the proof.
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 × D 3 be the product of the 2-sphere and the 3-disk. Choose sparks α ∈ S 1 (S 2 ) and b ∈ S 2 (D 3 ) with da = ω − [x 0 ] and db = − [0] for some x 0 ∈ S 2 , where ω and are unit volume forms on S 2 and D 3 respectively. Direct calculation shows that (1 − χ) < 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. sobre os duais diferenciáveis de Pontrjagin. Em particular, D e D 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.