Wei WANG

1 Introduction

Let W be a compact smooth manifold.One knows that the boundary of W is a closed smooth manifold.Conversely,given a closed smooth manifold M,how to know M is the boundary of some compact smooth manifold?

Thom[1]solved this question completely in 1950’s.In fact,he proved the following theorem.

Theorem 1.1LetMbe a closed smooth manifold.Mis the boundary of a compact smooth manifold if and only if all Stiefel-Whitney characteristic numbers ofMare zero.

This theorem leads to the connection between characteristic numbers and the question whether a manifold is the boundary of some compact smooth manifold with boundary.

It is known that there are various kinds of characteristic classes and characteristic numbers defined in ordinary cohomology ring,for example,Chern classes and Chern numbers.It is natural to consider the relation between vanishing of Chern numbers and being the boundary of some manifold with some stably complex structure.

Recall that a unitary manifold(or a stably complex manifold)is a smooth manifold whose stable tangent bundle admits a complex structure.Let M be a unitary manifold with stable tangent bundle TM.The corresponding Chern classes(resp.Chern numbers)of TM can be defined in integral cohomology ring H∗(M;Z)(resp.H∗(pt;Z)).In his paper[10],Milnor proved the following theorem.

Theorem 1.2LetMbe a unitary manifold.Mis the boundary of a compact unitary manifoldWwith the induced unitary structure ofWif and only if all Chern numbers ofMare zero.

We can further consider this question in other generalized cohomology.In complex K-theory,if M is a unitary manifold,then one can define K-theoretic Chern classes(resp.K-theoretic Chern numbers)in K∗(M)(resp.K∗(pt)).In Section 2,we will use the Riemann-Roch relation of Atiyah-Hirzebruch type to show the following proposition.

Proposition 1.1LetMbe a closed unitary manifold.All integral cohomology Chern numbers ofMvanish if and only if allK-theoretic Chern numbers ofMvanish.

Therefore,it follows that a unitary closed M is the boundary of some compact unitary manifold if and only if all K-theoretic Chern numbers of M vanish.

In equivariant case,suppose that the transformation group is G.We can still ask this question:Is it true that the vanishing of some equivariant characteristic numbers implies being the boundary of some G-manifold?

When the transformation group G=Tkis a torus,recall that M is a unitary Tk-manifold provided that M is a Tk-manifold whose stable tangent bundle TM admits a complex structure comparable with the Tk-action.In his paper[12],tom Dieck proved the following theorem.

Theorem 1.3(see[12])LetMbe a unitaryTk-manifold.Mis the boundary of a compact unitaryTk-manifold with the induced unitaryTk-structure if and only if all the equivariantK-theoretic characteristic numbers vanish.

In tom Dieck’s theorem,equivariant K-theoretic characteristic numbers play an important role.We want to know the question whether it is possible the vanishing of equivariant cohomology characteristic numbers can still determine that M is the boundary of a compact unitary Tk-manifold with the induced unitary Tk-structure.

Inspired by Proposition 1.1,if we could prove the equivalence of the vanishing of these two kinds of equivariant characteristic numbers,then the answer of the question above would be yes.Hence,our main theorem in this paper is the following result.

Theorem 1.4LetMbe a unitaryTk-manifold with stable complex tangent bundleTM∈KTk(M).Then all integral equivariant cohomology Chern numbers ofMvanish if and only if all equivariant K-theoretic Chern numbers ofMvanish.

Therefore,one obtains the answer of the question above.

Theorem 1.5LetMbe a unitaryTk-manifold.Mis the boundary of a compact unitaryTk-manifold with the induced unitaryTk-structure if and only if all the equivariant cohomology characteristic numbers vanish.

Remark 1.1 In[9],Zhi L¨u and the author gave the first proof of Theorem 1.5 directly and also consider more cases.The proof in this paper is another aspect of the understanding of the relation between various kinds of equivariant Chern numbers.

This paper is organized as follows.In Section 2,we will recall ordinary Chern numbers and K-theoretic Chern numbers in nonequivariant case and will prove Proposition 1.1.In Section 3,we turn to equivariant ordinary Chern numbers and K-theoretic Chern numbers and prove the Tk-version Riemann-Roch relation of Atiyah-Hirzebruch type in Section 3.3.Combining the results in Section 4,we will finish the proof of our main theorem in Section 4.

2 K-Theoretic Chern Classes and Integral Cohomology Chern Numbers:Nonequivariant Case

2.1 K-theoretic Chern classes and Chern numbers

First let us recall the definitions of K-theoretic Chern classes and Chern numbers and some standard facts(see[2]).Let ξ be a complex vector bundle over a space X and λt(ξ) ∈ K∗(X)[[t]]be the power series

where λi(ξ)denotes the i-th exterior power of the vector bundle ξ.Then λtcan be extended to a homomorphism

Furthermore,one can define the operations γi(x)by putting

Proposition 2.1(see[2])Letξandηbe two complex bundles over a finiteCWcomplexX.One has

Now we can use these operations γito define K-theoretic Chern classes.Let M be a closed unitary manifold with stable complex tangent bundle TM∈K(M).Then the total K-theoretic Chern class of M is defined to be

with the i-th K-theoretic Chern class

Remark 2.1 In[5],the i-th K-theoretic Chern class of M is also defined by(−1)iγi(TM).In this paper,it is more convenient to use the definition of γi(TM)as above.

Using the Gysin map,we can define the K-theoretic Chern numbers of M.Namely,let p:M →pt be the constant map and:K∗(M)→ K∗(pt)be the Gysin map induced by p in complex K-theory.Then the K-theoretic Chern numbers of M are defined to be

where each ω =(i1,i2,···,is)is a partition of|ω|=i1+i2+ ···+is,and(M)means

2.2 Riemann-Roch relation of Atiyah-Hirzebruch type

In this part,we will review the Riemann-Roch relation of Atiyah-Hirzebruch type(see[3,8]).

definition 2.1For any topological spaceX,we denote byH∗∗(X;R)the direct product ofHi(X;R)with coefficient ringR.More precisely,

For any two elementsa={xi},b={yi}inH∗∗(X;R),the producta·b={zi}is defined to be

Remark 2.2 When we discuss the Borel construction EG×GX(which is not compact)of X,it is better to use the direct productinstead of the cohomology ring⊕(X;Q).

We know that the Chern character ch is a natural transformation from complex K-theory to ordinary cohomology theory with rational coefficients

Let X be a compact space and ξ be a complex vector bundle over X.Then one has Thom isomorphisms in complex K-theory and ordinary cohomology theory:

With respect to these Thom isomorphisms ψ!and ψ∗,one has the following diagram:

This diagram is not commutative.However,we have the following nonequivariant Riemann-Roch relation.

Proposition 2.2LetXbe a compact space andξbe a complex vector bundle overX.Then for eachα ∈ K∗(X),

whereTd(ξ) ∈ H∗∗(X;Q)is the total Todd class ofξ.

Proof We refer to[8,p.182]or[4,Proposition 3.5].

With respect to the Gysin maps,Proposition 2.2 leads to the Riemann-Roch relation of Atiyah-Hirzebruch type(see[3]or[8,Theorem 26.5.2]).Indeed,this kind of Riemann-Roch relation can be stated as follows.

Proposition 2.3LetMandNbe two closed unitary manifolds andf:M→Nbe a smooth map.Then for eachα ∈ K∗(M),

whereare the Gysinmaps induced byfin complexK-theory and ordinary cohomology theory respectively withm=dimMandn=dimN.

Remark 2.3 As a special case of Proposition 2.3,let p:M →pt be the constant map.Then for each α ∈ K∗(M),

2.3 K-theoretic Chern numbers and ordinary cohomology Chern numbers

In this part,we will use the Riemann-Roch relation of Atiyah-Hirzebruch type to prove the following proposition.

Proposition 2.4LetMbe a closed unitary manifold with stable complex tangent bundleTM.All integral cohomology Chern numbers ofMvanish if and only if allK-theoretic Chern numbers ofMvanish.

Proof First,assume that all integral cohomology Chern numbers of M vanish.Then for any K-theoretic Chern numberwith partition ω =(i1,i2,···,is),one has

where p!:H∗∗(M;Q) → H∗∗(pt;Q)is the Gysin map induced by p in ordinary cohomology.Since p!(ch((M))·Td(TM))is represented by a combinations of ordinary cohomology Chern numbers,it follows that

Moreover,since the Chern character ch:K∗(pt)→ H∗∗(pt;Q)is injective,one has

Conversely,assume that all K-theoretic Chern numbers of M vanish.Then for any cohomology Chern number p!(cω(M)),consider the corresponding K-theoretic Chern class cKω(M).One has

By the Riemann-Roch relation,one has

which implies that p!(cω(M))=0 in H∗∗(pt;Q).This means that p!(cω(M))=0 in H∗∗(pt)Z.The proof is completed.

3 K-Theoretic Chern Classes and Integral Cohomology Chern Numbers:Equivariant Case

3.1 Equivariant Chern classes and Chern numbers

In this subsection,we recall the definitions of three kinds of equivariant Chern classes.Let M be a unitary G-manifold and E be a unitary G-vector bundle over M.Then applying the Borel construction to E → M gives a unitary G-vector bundle EG×GE over EG×GM.First,recall that the total equivariant cohomology Chern class of E is defined to be the total Chern class of EG×GE over EG×GM:

Similarly,in equivariant K-theory K(EG×G−),the total equivariant Chern class of E over M is defined to be

In equivariant K-theory KG(−)(see[11]),the total equivariant Chern class of E is defined to be

where the operations γi(E)are defined in the same way as was done in the nonequivariant case(see[7,12]).

In particular,let M be a unitary G-manifold with stable complex G-vector bundle TM.The corresponding total equivariant Chern class of M is defined by the corresponding total equivariant Chern class of TM as follows:

We know that the equivariant Gysin maps are well-defined in these equivariant cohomology theories.Let p:M →pt be the constant map.Then we have three corresponding equivariant Gysin maps

Hence,the corresponding equivariant Chern numbers of M are defined to be

respectively,where ω =(i1,i2,···,is)is a partition of|ω|=i1+i2+ ···+is,andmeans the product

3.2 Inverse limits in equivariant cohomology theory of Borel type

Assume that G is a compact Lie group.Let M be a compact unitary G-manifold and EG×GM be the Borel construction of M.EG×GM admits a fi ltration:

where EG(n)can be some compact smooth manifold.The following proposition is well-known.

Proposition 3.1In ordinary cohomology theory,one has

In complexK-theory,one also has

For a unitary G-vector bundle EG×GE over EG×GM,let EG×GD(E)be the disk bundle of EG×GE and EG×GS(E)be the sphere bundle of EG×GE.There is a diagram

where ψ!and ψ∗are Thom isomorphisms in K-theory and cohomology theory,respectively.

Proposition 3.2For anyx∈K(EG×GM),one has

whereTdG(E):=Td(EG×GE)∈ H∗∗(EG×GM;Q)is the total Todd class of the unitary vector bundleEG×GE.

Proof By Proposition 3.1,we will use the finite approximation method to finish the proof.Let in:EG(n)×GM → EG×GM be the inclusion induced by EG(n)→ EG.For the Todd genus TdG(E),since i∗n(TdG(E))=i∗n(Td(EG ×GE))=Td(EG(n) ×GE),one has

Taking the inverse,one has

Let x ∈ K∗(EG×GM)andOne has the following commutative diagram:

where jnis the bundle map induced by in.The vector bundle EG(n)×GE is the pull-back of EG×GE,so the Thom class t(EG(n)×GE)of EG(n)×GE satisfies(t(EG×GE))=t(EG(n)×GE).It follows that

By the definition of Thom isomorphism,we have that ψn!(xn)=(xn)·t(EG(n)×GE)and ψ!(x)= π!(x)·t(EG ×GE),where πn(resp. π)denotes the projective map EG(n)×G(D(E)/S(E))→EG(n)×GM(resp.EG×G(D(E)/S(E))→EG×GM).Sinceit follows that

Similarly,one can show that

On the other hand,since EG(n)×GM is compact,by the nonequivariant Riemann-Roch relation(see Proposition 2.2),we have

Taking the inverse limit,one has

as desired.

3.3 An equivariant version of Riemann-Roch relation of Atiyah-Hirzebruch type

Let M and N be closed unitary G-manifolds and f:M →N be a G-map.Then f induces equivariant Gysin maps

in three equivariant cohomology theories H∗(EG×G−),K∗(EG×G−)and K∗G(−),respectively.

Since the Chern character is a natural transformation from K-theory to ordinary cohomology,one has the following diagram:

Still,this diagram is not commutative.In a way similar to the nonequivariant case,we have the following equivariant Riemann-Roch relation.

Theorem 3.1LetMandNbe closed unitaryG-manifolds with unitary stable tangent bundles,still denoted byTMandTN,respectively.Letf:M →Nbe aG-map.Then for anyx∈K∗(EG×GM),

Proof By the definition of equivariant Gysin map,choose an G-embedding f×e:M →N ×V with normal bundle η,where V is a G-representation.Then one can obtain the following diagram:

Note that the middle square in the above diagram is commutative.For any x∈K∗(EG×GM),by Proposition 3.2,we have the following relation:

Sincewe have

where TdG(V)=Td(EG×GV)is the equivariant total Todd class of the G-bundle V×N →N.Put these equalities together,

and one has

On the other hand,we see that TM ⊕ η =(f × e)∗T(N × V)f∗(TN ⊕ V),where f∗(TN ⊕V)is the pull-back of the bundle TN ⊕V→N via f:M →N.It follows that

Hence,

as desired.

There is a natural transformation from KG(−)to K(EG×G−)

for any G-space X and unitary G-vector bundle E over X.Then we define the equivariant Chern character as follows.

definition 3.1For anyG-spaceX,the equivariant Chern characterchGis defined by the composition ofαandch:

By choosing the Thom classes,for any G-map f:M→N between two closed unitary G-manifolds,one has the following commutative diagram:

Proposition 3.3LetMandNbe closed unitaryG-manifolds with unitary stable tangent bundles which are still denoted byTMandTN.Letf:M →Nbe aG-map.Then for anyx∈K∗G(M),

Proof By Theorem 3.1,for α(x)∈ K∗(EG×GM),one has

Sinceit follows that

as desired.

4 Tk-K-Theoretic Chern Numbers and Tk-integral Cohomology Chern Numbers

Now we are going to prove our main theorem.

Theorem 4.1LetMbe a unitaryTk-manifold with stable complex tangent bundleTM∈KTk(M).Then all integral equivariant cohomology Chern numbers ofMvanish if and only if all equivariant K-theoretic Chern numbers ofMvanish.

Proof First,assume that all integral equivariant cohomology Chern numbers of M vanish.Then for any equivariant K-theoretic Chern number

where ω is a partition.By Proposition 3.3,one has

Sinceis a combination of equivariant cohomology Chern classes,it follows thatcan be represented by a combination of equivariant cohomology Chern numbers.By assumption,one has

On the other hand,is injective and the equivariant K-theoretic Chern number

Second,assume that all the equivariant K-theoretic Chern numbers vanish.For any integral equivariant cohomology Chern numberconsider the equivariant K-theoretic Chern classOne has

and

It follows thatin H∗(BTk,Z)since H∗(BTk)is torsion-free(see[6]).Thus we have proved that all integral equivariant cohomology Chern numbers of M vanish.

AcknowledgementsThe author is grateful to Prof.Zhi L¨u for very helpful conversations and comments,and he would also like to thank Prof.Peter Landweber for very helpful suggestions and comments.which implies that

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