Lefschetz-Pontrjagin Duality for Differential Characters

REESE HARVEY1 and BLAINE LAWSON2

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 byBlaine 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

Hk(X, ∂X)×Hn−k−1(X) −→ S1

given by(α, β) → (α∗β)[X]induces isomorphisms

D_:Hk_{(X, ∂X)→Hom∞(H}n−k−1_{(X), S}1₎ D′_:Hn−k−1_{(X)→Hom∞(H}k_{(X, ∂X), S}1₎

onto the smooth Pontrjagin duals. In particular,D_andD′ _{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

∗_{Invited paper}

∗∗_{Foreign Member of Academia Brasileira de Ciências} Correspondence to: Blaine Lawson

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 classesck are defined and the total class gives a natural transformation

c=1+c1+c2+ · · · :_K(X)−→_H∗(X)

from theK-theory of bundleswith 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), (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 groupsHk(X)of differential characters carry a natural topology. The connected component of 0 in this group consists of thesmooth 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 au-thors showed that differential characters satisfyPoincaré-Pontrjagin duality: On an orientedn dimensional manifoldXthe pairing

Hk(X)×Hncpt−k−1(X) −→ S 1

given by

(α, β) → (α∗β)[X]

(whereH∗cptdenotes characters with compact support) induces injective maps

Hk(X) → HomHncpt−k−1(X), S

1 _and

Hncpt−k−1(X) → HomH

k_{(X), S}1

with dense range in the groups of continuous homomorphisms into the circle. Moreover this range consists exactly of the smoothhomomorphisms. These are defined precisely in §4 but can be thought of roughly as follows. The connected component of 0 inHk(X)consists essentially (i.e., up to a finite-dimensional torus factor) of the exact(k+1)-formsdEk+1_{(X)with theC}∞_-topology.

Now Hom(dEk+1_{(X), S}1_{) = Hom(dE}k+1_{(X),R)is just the vector space dual. This is simply a} quotient of the space ofcurrents_{, the(n−k−1)-forms with distribution coefficients. The smooth}

dual corresponds to those forms which have smooth coefficients.

To do this we introduce the relative groups_H∗(X, ∂X)and develop the theory from (Harvey et al. 2001) for this case. The main theorem asserts the existence of a pairing

Hk(X)×_Hn−k−1_{(X, ∂X) −→ S}1

given by(α, β) → (α∗β)[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∂ :_Hk_(X)→Hk+1_{(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

LetX be a compact oriented differentiablen-manifold with boundary∂X. LetE∗_{(X)denote the} de Rham complex of differential forms which are smooth up to the boundary, and set

E∗_{(X, ∂X)} = {φ ∈E∗_(X) : φ_∂X=0}.

The cohomology of this complex is naturally isomorphic toH∗(X, ∂X;_R). LetC∗(X)denote the

complex ofC∞-singular chains onX andC∗(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.

Definition1.1. The group ofdifferential characters of degreekonX is the set of homomor-phisms

ˆ

Hk(X;_R/Z) ≡ {α∈Hom(Zk(X), S1) : δ(α)∈Ek+1_(X)}

whereδdenotes the coboundary. Similarly the group ofrelative differential charactersof degree kon(X, ∂X)is defined to be

ˆ

Hk(X, ∂X;_R/Z) ≡ {α ∈Hom(Zk(X, ∂X), S1) : δ(α)∈Ek+1_{(X, ∂X)}}

There is an alternative de Rham-Federer approach to these groups. Set

Ek

L1_loc(X) ≡ k-forms onXwithL 1

loc-coefficients

Rk_{(X) ≡ the rectifiable currents of degreek(dimensionn−k) onX} Ek

L1 loc

(X, ∂X) ≡ {a ∈Ek L1

loc

(X) : ain smooth in a neighborhood of∂Xanda_∂X=0}

Rk

cpt(X−∂X) ≡ {R∈R

k_{(X) : supp(R)⊂X−∂X}}

Definition1.2.An elementa ∈Ek

L1_loc(X)is called asparkof degreekonXif

da = φ−R where φ∈Ek+1_{(X) and R}∈Rk+1_{(X). (1.3)}

Denote bySk_{(X)the group of all such sparks and byT}k_{(X)the subgroup of alla∈ S}k_(X)such thata =db+Swhereb∈Ek−1

L1_loc(X) S∈R k

cpt(X). Then the group of deRham-Federer characters of degreekonXis defined to be the quotient

Hk(X) ≡ Sk_(X)/Tk_(X).

Given a sparka ∈Sk_{(X)we denote its associated character bya ∈H}k_(X).

We definerelative sparksandrelative deRham-Federer characters on(X, ∂X)by

Sk_{(X, ∂X)}≡ {a ∈Ek

L1_loc(X, ∂X) : da=φ−R, φ ∈E

k+1_{(X, ∂X)andR ∈R}k+1

cpt (X−∂X)} Tk_{(X, ∂X)}≡ {a ∈Sk_{(X, ∂X)} : a =db+S, b∈Ek−1

L1_loc(X, ∂X)andS∈R k

cpt(X−∂X)}

Hk(X, ∂X) ≡ Sk_{(X, ∂X)/}Tk_{(X, ∂X).}

The decomposition (1.3) is unique. In fact we have the following. Recall that a currentT is said to beintegrally flatif it can be written asT =R+dS whereR andSare rectifiable. Then from §1.5 in (Harvey et al. 2001) one concludes:

Proposition 1.4. Letabe any current of degreekonXsuch thatda=φ−Rwhereφ ∈Ek+1_(X) andR is integrally flat. Ifda = φ′ −R′is a similar decomposition, then φ = φ′ andR = R′. Furthermore,

dφ = 0 and dR_X−∂X= 0

andφhas integral periods on cycles inX. In the case thatφ ∈Ek+1_{(X, ∂X)and supp(R)⊂X−∂X,} one has thatdR=0andφ has integral periods on all relative cycles in(X, ∂X).

Set

Zℓ

0(X)= {φ ∈E ℓ

(X):dφ =0 andφhas integral periods} Zℓ

0(X, ∂X)= {φ ∈E

ℓ_{(X, ∂X):dφ=0 andφ has}

integral periods on relative cycles in(X, ∂X)} Zrectℓ (X)= {R∈R

ℓ

(X):dR_X−∂X=0} Zℓ_rect(X, ∂X)= {R∈Rℓ

cpt(X−∂X):dR=0}

Corollary 1.6. Takingd1a =φ andd2a =Rfrom the decomposition (1.3) gives well-defined mappings

d1:Sk(X) −→ Zk+1

0 (X), d2:S

k_{(X) −→ Z}k+1

rect(X),

and

d1:Sk(X, ∂X) −→ Zk+1

0 (X, ∂X), d2:S k

(X, ∂X) −→ Zkrect+1(X−∂X)

Proposition 1.7. There are natural isomorphisms

:Hk(X)−→ ˆ=∼ Hk(X;R/Z) and :Hk(X, ∂X)−→ ˆ=∼ Hk(X, ∂X; R/Z)

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 forH∗(X)andH∗(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 currentRonXis calledintegrally flatifR =S+dT whereSandT are rectifiable. Denote byD′k_{(X) ≡ {}En−k_(X)}′

the space of currents of degreekonX. LetSk

max(X, ∂X)denote the set ofa ∈ D

′k

(X)such thata is smooth near

∂X, a_∂X=0, and da=φ−R where φ ∈Ek+1_{(X, ∂X)}

andRis integrally flat. LetTk

max(X, ∂X)denote the subgroup of elements of the formdb+Swhereb is smooth near∂X,b_∂X=0, andSis integrally flat. Then the inclusionSk_{(X, ∂X)⊂S}k

max(X, ∂X) induces an isomorphism

Hk(X, ∂X) ∼= Sk

max(X, ∂X)/T k

max(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 α ∈ _Hk_{(X, ∂X) is said to be smoothif α = a for a smooth} forma ∈ Ek_{(X, ∂X). The set of smooth characters is denoted}

Hk∞(X, ∂X). There is a natural

isomorphism

Hk∞(X, ∂X) ∼= E

k

(X, ∂X)/Zk

Proposition 2.2. The mappingsd1andd2induce functorial short exact sequences:

0→Hk(X, ∂X;S1)−→j1 _Hk(X, ∂X)−→δ1 Zk+1

0 (X, ∂X)→0, (A)

0→_Hk

∞(X, ∂X)

j2

−→_Hk_{(X, ∂X)} δ2

−→Hk+1(X, ∂X; _Z)→0. (B)

Proof. Note that∂Xhas a cofinal system of tubular neighborhoods each of which is diffeomorphic to∂X× [0,1). We shall use the following elementary result.

Lemma 2.3. For any a ∈ Ek_{(∂X × [0,1)) such thatda = 0 anda}

∂X= 0, there exists b ∈ Ek−1_{(∂X× [0,1))such thatdb=aandb}

∂X=0.

Proof. Write a = a1+dt ∧a2 wherea1anda2 are forms on X whose coefficients depend smoothly ont ∈ [0,1), or in other words,a1(t),a2(t)are smooth curves inEk_(X)andEk−1_(X) respectively witha1(0) = 0. Nowda = dxa1+dt ∧ ∂a1

∂t −dt ∧dxa2 = 0. We conclude that dxa1=0 anddxa2= ∂a1

∂t . Sincea1(0)=0 we have

a1(t) =

t

0 ∂a1

∂t (s) ds = t

0

dxa2(s) ds = dx t

0

a2(s) ds

Setb≡₀ta2(s) ds, and note that: b_∂X=0,dxb=a1and ∂b_∂t =a2. Hence,a =db.

We shall also need the following result. On any manifoldY let

Fk_{(Y )≡E}k

L1_loc(Y )+dE k−1 L1

loc (Y )

denoteflat currentsandFk

cpt(Y )those with compact support. Note thatdFk(Y )=dE_Lk1 loc

(Y ). This definition ofFk

cpt(Y )arises naturally in sheaf theory. However, the following equivalent definition will also be useful here.

Lemma 2.4. Fk

cpt(Y ) = E_Lk1 loc,cpt

(Y )+dEk−1

L1_loc,cpt(Y ) and so dF k

cpt(Y ) = dE_Lk1 loc,cpt

(Y ).

Proof. Fixf ∈ Fk

cpt(Y )and writef = a+dbwherea ∈ E_Lk1 loc(Y )

andb ∈ Ek−1

L1_loc(Y ). Let K = supp(f ), and note that in N ≡ Y −K we have that a = −db. By standard de Rham theory there exists anL1_loc-formb0onN such thata∞ ≡ a+db0is smooth onN. Furthermore

sincea∞is weakly exact onN there exists a smooth formb∞witha∞ = −db∞onN. Choose

η∈ C₀∞(Y )withη ≡1 in a neighbothood ofK, letχ =1−ηand seta =a+d(χ b0+χ b∞)

andb=b−χ b0−χ b∞withχas above. Thenf =a+dbandahas compact support inY.

Observe now thatf −a isd-closed and has compact support in Y. SinceH∗(E∗

cpt(Y )) ∼= H∗(F∗

We first prove the surjectivity of δ1. Fix φ ∈ Zk+1

0 (X, ∂X). Then by Lemma 2.3 there is a neighborhoodN ∼= ∂X× [0,1)of∂X and a formA ∈ Ek_{(N )with dA = φ andA}

∂X= 0. Choose χ ∈ C₀∞(N ) with χ ≡ 1 in a neighborhood of ∂X, and set φ0 = φ −d(χ A). Now supp(φ0)⊂⊂X−∂X andφ0has integral periods, so there exists a cycleR ∈ Z_rectℓ (X, ∂X)with [φ0−R] =0 inHcpt∗ (X−∂X;R). By Lemma 2.4 there areL1loc-formsa, bwith compact support inX−∂Xsuch thatd(a+db)=da=φ0−R. Thend1(χ A+a)=φand surjectivity is proved.

We now construct the mapj1. Recall from §1 in (Harvey et al. 2001) that

Hk(X, ∂X;S1)∼=H_cptk (X−∂X;S1)∼= {f ∈ Fk

cpt(X−∂X):df ∈R k+1

cpt (X−∂X)} dF_cptk−1_(X−∂X)+Rk_cpt(X−∂X)

Choosef ∈ Fk

cpt(X−∂X)withdf =R ∈ R k+1

cpt (X−∂X), and writef =a+dbwhereaand b areL1_loc-forms with compact support inX −∂X (cf. Lemma 2.4). Thena ∈ Sk_{(X, ∂X)and} we setj1(f )≡ a ∈_Hk_{(X, ∂X). Note that iff =a}′₊

db′is another such decomposition, then a−a′=d(c′−c)anda = a′. Clearlyj1=0 on

dFk−1

cpt (X−∂X)+R k

cpt(X−∂X)=dE k−1

cpt (X−∂X)+R k

cpt(X−∂X),

and so it descends to the quotientHk_{(X, ∂X;S}1_).

To see thatj1is injective, letf =a+dbas above and supposea = dc+S ∈ Tk_{(X, ∂X)} wherecis smooth and zero on∂X. By Lemma 2.3 there exists anL1loc-forme, smooth near∂X, such thatc0=c−de≡0 near∂X. Thena=dc0+S≡0 inHk_{(X, ∂X;S}1_).

We now prove the exactness of (A) in the middle. Supposea ∈ Sk_{(X, ∂X)andδ1(a)=0.} Thenda = −R ∈ Rk+1

cpt (X−∂X). Thus, in a neighborhoodN of∂Xwe have thatais smooth, da =0 anda_∂X=0. By Lemma 2.3 there existsb ∈Ek−1_{(N )withdb=a andb}

∂X=0. Then a =a−d(χ b), withχas above, is equivalent toainHk_{(X, ∂X). Sinceahas compact support in} X−∂Xandda= −R, we see thatalies in the image ofj1.

We now prove the surjectivity of δ2. Fix u ∈ Hk+1_{(X, ∂X;}

Z) and choose a cycleR ∈ u. Then there is a smooth formφ ∈Zk+1

0 (X, ∂X)such thatφ−R =df forf ∈F k

cpt(X−∂X). By Lemma 2.4f =a+dbwhereaisL1_locwith compact support inX−∂X. Thena∈ Sk_{(X, ∂X)} andδ2(a)=u.

Now consider an elementa ∈ Sk_{(X, ∂X) withδ2(a) = 0. Thenda = φ−R whereφ is} smooth andR =dSfor someS∈Rk

cpt(X−∂X). Thena=a−S ≡ainH

k_{(X, ∂X)andda=0} onX. Sinceais smooth near∂X, standard de Rham theory shows that there is anL1_loc-formbwith compact support inX−∂Xsuch thata−dbis smooth. Hence,a = a ∈Hk

∞(X, ∂X).

Note that

ker(δ1)∩ker(δ2) ∼= H

k_{(X, ∂X;R)} Hk_{(X, ∂X;Z)free} ∼=

Hk

cpt(X−∂X;R) Hcptk (X−∂X;Z)free

3. THE STAR PRODUCT

In this section we prove the following.

Theorem 3.1. There are functorial bilinear mappings

Hk(X, ∂X)×_Hℓ_{(X, ∂X) →}∗

Hk+ℓ+1(X, ∂X) and

Hk(X, ∂X)×_Hℓ(X) →∗ _Hk+ℓ+1(X, ∂X)

which makeHk_{(X, ∂X)a graded commutative ring andH}∗_{(X)a graded}

Hk_{(X, ∂X)-module. With} this structure the mapsδ1,δ2are ring and module homomorphisms.

Proof. Fixα ∈_Hk_{(X, ∂X)andβ ∈}

Hℓ(X). Then from (Harvey et al. 2001) we know that there exist sparksa∈αandb∈βwith

da = φ−R and db = ψ−S

withφ ∈ Zk+1

0 (X, ∂X), ψ ∈ Z ℓ+1

0 (X), R ∈ Z k+1

rect(X, ∂X)andS ∈ Z ℓ+1

rect(X), so that the wedge-intersection productsR∧bandR∧Sare well defined. Furthermore, if suppS ⊂⊂X−∂Xwe can also assume thata∧Sis well defined. We then define

a∗b = a∧ψ+(−1)k+1R∧b, (3.2)

and ifS∈Zrectℓ+1(X, ∂X)or ifa∈Ecptk (X−∂X), we can also define

a∗b = a∧S+(−1)k+1φ∧b. (3.3)

Sinceais smooth near∂Xanda_∂X=0,a∗balso has these properties (as well asa∗bwhen it is defined). Note that

d(a∗b) = d(a∗b) = φ∧ψ−R∧S (3.4)

The arguments from (Harvey et al. 2001) easily adapt to show thata∗bdepends only ona andb, and that a ∗b = a∗b (when it is defined). Associativity, commutativity, etc. are straightforward. Equation (3.4) establishes the homomorphism propertes ofδ1andδ2.

4. SMOOTH PONTRJAGIN DUALS

The exact sequences of Proposition 2.2 show thatH∗(X, ∂X)has a natural topology making it a topological group (in fact a topological ring) for whichδ1andδ2are continuous homomorphisms. Essentially it is a product of the standard C∞-topology on forms with the standard topology on the torusHk(X, ∂X;_R)/H_freek (X, ∂X; _Z). It can also be defined as the quotient of the topology induced on sparks by the embedding Sk_{(X, ∂X) ⊂ F}k_(X)×Ek+1_{(X, ∂X)×R}k+1

It is natural to consider the dual to_H∗(X, ∂X) in the sense of Pontrjagin. For an abelian topological groupAwe denote byA⋆ ≡Homcont(A, S1_{)the group of continuous homomorphisms} h:A→S1_{. Then 2.2(B) yields a dual sequence}

0→Hk+1(X, ∂X;_Z)⋆ →_Hk_{(X, ∂X)}⋆ _→ρ

Hk∞(X, ∂X) ⋆_→

0. (4.1)

whereρis the restriction mapping.

Definition 4.2. An element f ∈ _Hk

∞(X, ∂X)⋆ is calledsmooth if there exists a formω ∈

Zn−k

0 (X)such that

f (α) ≡

X

a∧ω (mod Z)

fora ∈α ∈_Hk

∞(X, ∂X). An elementf ∈Hk(X, ∂X)⋆is calledsmoothifρ(f )is smooth. The

set of these is called thesmooth Pontrjagin dualofHk_{(X, ∂X)and is denoted byH}k_{(X, ∂X)}⋆∞ ₌

Hom∞(Hk(X, ∂X), S1).

Proposition 4.3. The smooth Pontrjagin dualHk_{(X, ∂X)}⋆∞_{is dense inH}k_{(X, ∂X)}⋆_.

Proof. Applyingδ1toHk

∞(X, ∂X)gives an exact sequence

0→T →_Hk

∞(X, ∂X)→dEk(X, ∂X)→0

whereT =Hk_{(X, ∂X;R)/H}k

free(X, ∂X;Z), with dual sequence

0→dEk_{(X, ∂X)}⋆ _→

Hk∞(X, ∂X) ⋆ _→

T⋆ →0 (4.4)

Observe thatT⋆ = H_freek (X, ∂X;_Z) ∼= H_freen−k(X; _Z), and thatdEk_{(X, ∂X)}⋆ _{= {dE}k_{(X, ∂X)}}′

(the topological vector space dual) which is exactly the space of currents of degreen−k−1 on Xrestricted to the closed subspacedEk_{(X, ∂X). This gives a commutative diagram}

0 −−−→ dEn−k−1_{(X) −−−→ Z}n−k

0 (X) −−−→ H n−k

free(X; Z) −−−→ 0 



 

  ∼₌

0 −−−→ dD′n−k−1_(X) −−−→ _Hk

∞(X, ∂X)⋆ −−−→ T⋆ −−−→ 0

with exact rows. SinceEn−k−1_{(X)is dense inD}′n−k−1

(X), the result follows.

There is a parallel story forH∗(X). The analogue of 2.2(B) gives an exact sequence

0→Hk+1(X; _Z)⋆→_Hk_(X)⋆ _→ρ Hk∞(X)

⋆

.→0. (4.5)

Definition 4.6. An element f ∈ Hk

∞(X)

⋆ _{is called smooth if there exists a form ω ∈}

Zn−k

0 (X, ∂X)such that

f (α) ≡

X

fora∈α∈_Hk

∞(X)An elementf ∈Hk(X)

⋆_{is calledsmoothif}

ρ(f )is smooth. The set of these is called thesmooth Pontrjagin dualofHk_{(X)and is denotedH}k_(X)⋆∞ _=Hom

∞(Hk(X), S1). Proposition 4.7. The smooth Pontrjagin dualHk(X)⋆∞ _{is dense inH}k_(X)⋆_.

Proof.Applyingδ1toHk∞(X)gives an exact sequence

0→T →_Hk_∞(X)→dEk_(X)→0,

whereT =Hk(X; _R)/Hk(X; _Z), with dual sequence

0→dEk_(X)⋆ _→ Hk∞(X)

⋆ _→

T⋆ →0. (4.8)

Observe now thatT⋆ =Hk(X;_Z)∼=Hn−k(X, ∂X;_Z), anddEk_(X)⋆_{= {dE}k_(X)}′_{is the space}

of currents of degree n−k−1 onX restricted to the closed subspace dEk_{(X). This gives a} commutative diagram:

En−k−1_{(X, ∂X) −−−→}d _Zn−k

0 (X, ∂X) −−−→ Hn

−k_{(X, ∂X; Z) −−−→ 0} 



 

  ∼₌

D′n−k−1

(X) −−−→d _Hk

∞(X)⋆ −−−→ T⋆ −−−→ 0

with exact rows. SinceEn−k−1_{(X, ∂X)is dense inD}′n−k−1

(X), the result follows.

5. LEFSCHETZ-PONTRJAGIN DUALITY

This brings us to the main result of the paper.

Theorem 5.1. LetXbe a compact, orientedn-manifold with boundary∂X. Then the biadditive mapping

Hk(X, ∂X)×_Hn−k−1_{(X) −→ S}1

given by

(α, β) → (α∗β)[X]

induces isomorphisms

D:_Hk_{(X, ∂X)−→}∼=

Hn−k−1(X)⋆∞

and

D′ _:Hk_(X)−→∼= _Hn−k−1_{(X, ∂X)}⋆∞

thatα = 0. Choose a sparka ∈ α and writeda =φ −R as in 1.4. Then for all smooth forms b∈En−k−1_{(X)we have by (3.3) that}

α∗ b [X] = (−1)k+1

X

φ∧b ≡ 0 mod Z

sinced2b =0. It follows thatφ=0. Hence, da= −R∈Rk+1

cpt (X−∂X)is a cycle with[R] ∈H k+1

cpt (X−∂X;Z)tor∼=Hn−k−1(X−

∂X;_Z)tor. Choose anyu∈Hn−k(X;_Z)tor∼=Hk(X, ∂X; _Z)tor, and choose a relative cycleS∈u. Letmbe the order ofu. Then there is a(k+1)-chainT onXwithdT =mSrel∂X. Setb= −_m1T and considerbas a spark of degreen−k−1 onXwithdb= −S. Now we may assumeSandT to have been chosen so that supp(S)∩supp(R)= ∅andT meetsRproperly. Then

0 = α∗ b [X] ≡ (−1)k+1R∧b[X] mod Z

≡ (−1)k+11

mR∧T [X] mod Z ≡ (−1)k+1Lk([R],[S]) mod Z

≡ (−1)k+1Lk(δ2α, u) mod_Z

where Lk denotes the de Rham-Seifert linking between the groups Hn−k−1(X−∂X;Z)tor and

Hk(X, ∂X;_Z)tor. By the non-degeneracy of this pairing we conclude thatδ2α =0.

Thereforeα ∈ker(δ1)∩ker(δ2)can be represented by a smoothd-closed forma∈Ek_{(X, ∂X).} In fact by Lemma 2.3 we may chooseato have compact support inX−∂X. Now for any cycle S∈Znrect−k(X), i.e., anyk-dimensional rectifiable currentS∈Rk(X)withdS ∈Rk−1(∂X), we can

findψ∈En−k_(X)andb∈En−k−1 L1

loc

(X)withdb=ψ−S. Then by (3.3) we have that

0 = α∗ b [X] ≡ a∧S[X] modZ

≡

S

a mod Z.

Hence,arepresents the zero class in

Hom(Hk(X, ∂X;_Z),R)/Hom(Hk(X, ∂X; _Z),Z)∼=Hk(X, ∂X;_R)/Hk(X, ∂X;_Z)free,

and by (2.2) and (2.5) we conclude thatα=0. Thus the mapD_{is injective.} To see thatD_{is surjective consider the commutative diagram with exact rows:}

0 −−−→ Hk_{(X, ∂X;S}1₎ j1

−−−→ Hk_{(X, ∂X)} δ1

−−−→ Zk+1

0 (X, ∂X) −−−→ 0

∼ =

 

  D

  D0

0 −−−→ Hom(Hn−k(X;_Z), S1) −−−→ _Hn−k−1_(X)⋆ _−−−→ρ

Hn∞−k−1(X)⋆ −−−→ 0

The proof thatD′_{is an isomorphism is parallel. Fixβ ∈H}n−k−1_{(X)and suppose(α∗β)[X] =0} for allα ∈ _Hk_{(X, ∂X). We shall show thatβ =0. Choose a sparkb ∈β and writedb=ψ−S} as in 1.4. Then for all smooth formsa∈Ek_{(X, ∂X)we have by (3.3) that}

a ∗β[X] =

X

a∧ψ ≡ 0 mod Z

sinced2a =0. It follows thatψ=0.

Hence,db= −S ∈Rn−k_{(X)is a relative cycle with torsion homology class} [S] ∈Hn−k(X;_Z)tor∼=Hk(X, ∂X;_Z)tor.

Chooseu ∈ Hk+1(X, ∂X; _Z)tor =∼ Hn−k−1(X; Z)tor, and choose a cycleR ∈ uwith support in

X−∂X. Letmbe the order ofu. Then there is a(n−k−1)-chainT inX−∂XwithdT =mR. Seta = −1

mT and consideraas a spark of degreekonX withda = −R. Now we may assume RandT to have been chosen so that supp(R)∩supp(S)= ∅andT meetsSproperly. Then

0 = a ∗β[X] ≡ (−1)k+1a∧S[X] modZ

≡ (−1)k+11

mT ∧S[X] mod Z ≡ (−1)k+1Lk([R],[S]) mod Z

≡ (−1)k+1Lk(u, δ2β) mod Z

where Lk denotes the de Rham-Seifert linking as before. We conclude thatδ2α=0.

Thereforeβ ∈ker(δ1)∩ker(δ2)can be represented by a smoothd-closed formb∈En−k−1_(X). Now for any cycleR ∈ Zrectk+1(X, ∂X), i.e., any(n−k−1)-dimensional rectifiable currentR ∈ R_{n−k−1(X−∂X) with dR = 0, we can find φ ∈} Ek+1_{(X, ∂X) and a ∈ E}k

L1_loc(X, ∂X) with da=φ−R. Then by (3.2) we have that

0 = a ∗β[X] ≡ (−1)k+1R∧b[X] mod Z

≡ (−1)n(k+1)

R

b mod Z.

Hence,brepresents the zero class in

Hom(Hn−k−1(X;Z),R)/Hom(Hn−k−1(X; Z),Z)∼=Hn−k−1(X; R)/Hn−k−1(X;Z)free,

and by (2.2) and (2.5) we conclude thatβ =0. Thus the mapD′_{is injective.}

The surjectivity ofD′_{follows as before from the commutative diagram with exact rows:}

0 −−−→ Hn−k−1(X;S1) −−−→j1 _Hn−k−1_(X) δ1

−−−→ Zn−k

0 (X) −−−→ 0

∼ =

 

  D

  D0

0 −−−→ Hom(Hk+1_{(X, ∂X;Z), S}1_{) −−−→ H}k_{(X, ∂X)}⋆ _−−−→ρ Hk

∞(X, ∂X)

⋆ _{−−−→ 0.}

6. COBOUNDARY MAPS

It is natural to ask if there is a coboundary mapping∂ with the property that the sequence

· · · −→_Hk−1_(∂X)−→∂

Hk(X, ∂X)−→j _Hk_(X)−→ρ

Hk(∂X)−→∂ _Hk+1_{(X, ∂X)−→. . .} (6.1)

is exact. The differential-form-component of characters makes this impossible. However, there do exist natural coboundary maps∂ with the following properties:

(1) Underδ2the sequence (6.1) becomes the standard long exact sequence in integral cohomology.

(2) Underδ1the sequence (6.1) becomes a sequence of smoothd-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 neighborhoodN0of∂XinXand an identificationN0=∼∂X× [0,2), and letπ :N0→∂X be the projection. Set N = ∂X× [0,1) ⊂ N0 and letIN be the characteristic function of this subset. Letχbe a smooth approximation toIN; specifically chooseχ (x, t )=χ (t )whereχ ≡1 near 0 andχ (t )=0 fort ≥1. Then set

λ ≡ χ−_IN ∈H0(X)

Note thatdλ=dχ− [∂N]has compact support inX−∂X.

Definition 6.2. We define the coboundary map∂ =∂λ:Hk_{(∂X)−→H}k+1_{(X, ∂X)by}

∂(a) = (π∗a)∗λ.

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:

Hk(X, ∂X) −−−→j _Hk_{(X) −−−→}ρ

Hk(∂X) −−−→∂ _Hk+1_{(X, ∂X)}

D

 

D

 

D

 

D

 

Hn−k−1(X)⋆ j

∗

−−−→ _Hn−k−1_{(X, ∂X)}⋆ _−−−→∂∗

Hn−k−2(∂X)⋆ ρ

∗

−−−→ _Hn−k−2_(X)⋆

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 α ∈ _Hk_{(∂X) andβ ∈}

Hn−k−2(X)and choose L1loc-sparksa0 ∈ α andb ∈ β withda0 = φ0−R0 anddb = ψ−S as usual. Let a = π

φ =π∗φ0andR =π∗R0denote the pull-backs to the collar neighborhood of∂Xvia the projection π :N0→∂Xdefined in §6. Then

{(D_{◦∂)(α)}(β) = (π}∗_{a∗b∗λ)[X] = {(a∗b)∧dχ+(−1)}n_{d2(a∗b)λ}[X]. (7.1)}

Now we may assume thatS_N 0=π

∗_{S0for someS0∈R}k+1_{(∂X), and we may further assume that} supp(R0)∩supp(S0)= ∅because dim(R0)+dim(S0)=n−2. Hence,d2(a∗b)=π∗R0∧π∗S0= π∗(R0∧S0)=0, and from (7.1) we see that

(−1)n−1{(D_{◦∂)(α)}(β) = (−1)}n−1_{(a∗b)∧dχ[X]} = (a∗b)[∂X] −χ d(a∗b)[X] = {(ρ∗◦D_)(α)}(β)−χ d(a∗b)[X].

Nowd(a∗b)=φ∧ψ−R∧S=φ∧ψand we can writeψ =ψ1+dt∧ψ2as in the proof of Lemma 2.3. Sinceφ =π∗φ0we see thatφ∧ψ1=0 and we conclude that

{(ρ∗◦D)(α)}(β)+(−1)n{(D◦∂)(α)}(β) =

N

φ∧χ dt∧ψ2

=

∂X φ∧

1

0

χ (t ) dt∧ψ2

=

∂X

φ∧π∗{χ (t ) dt∧ψ2} ≡E(λ).

(7.2)

Thus for example we see that(ρ∗◦D_)(α)=(−1)n−1_{(D◦∂)(α)on allβwhich areπ}∗_{-pull backs in}

N. Furthermore, we can consider the family of sparksλǫ≡r_ǫ∗λwhererǫ :∂X×[0, ǫ)→∂X[0,1) is given byrǫ(x, t)=(x, t/ǫ). From (7.2) we see that

lim

ǫ→0E(λǫ)=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 asǫ →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. LetX =S2_×D3_{be the product of} the 2-sphere and the 3-disk. Choose sparksα ∈S1_(S2_{)andb∈ S}2_(D3_{)withda=ω− [x0]and} db=5− [0]for somex0∈S2, whereωand5are unit volume forms onS2andD3respectively. Direct calculation shows that

(a∗b)[∂X] = 1 but (a∗λ∗b)[X] =

D3

(1−χ )5 < 1.

ACKNOWLEDGMENTS

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

Hk(X, ∂X)×Hn−k−1(X) −→ S1

dado por(α, β) → (α∗β)[X]induz isomorfismos

D_:Hk(X, ∂X)→Hom∞(Hn−k−1(X), S1) D′_:Hn−k−1_{(X)→Hom∞(H}k_{(X, ∂X), S}1₎

sobre os duais diferenciáveis de Pontrjagin. Em particular,D_eD′_{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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