Suqian ZHAO Yanying WANG Jingyan LI

Abstract Let Z2 denote a cyclic group of 2 order and Z22 =Z2×Z2 the direct product of groups. Suppose that (M,Φ) is a closed and smooth manifold M with a smooth Z22-action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M,Φ) bounding equivariantly.

Keywords Z22-action, Fixed data, Characteristic number, Simultaneous bordism,Stiefel-Whitney class

1 Introduction

LetMbe a smooth, closed manifold andT:M→Ma smooth involution defined onM(i.e.,Z2-action, whereZ2denotes a cyclic group of 2 order). It is well known that the fixed point setF= {x∈M|T(x) =x} of the involutionTis a finite and disjoint union of closed submanifolds ofM. In this setting, for a givenF, one naturally considers to classify the pairs (M,T) for which the fixed point set ofTisFup to equivariant bordism. Letνdenote the normal bundle ofFinM. It is known that the equivariant bordism class of (M,T)is determined by the bordism class of the bundle (F,ν), and the bordism class of the bundle(F,ν) is determined by its characteristic numbers (see [2]). For the vector bundleν→F, there are the associated sphere bundleS(ν) →Fand a fibre preserving fixed point free involution(S(ν),T)which on each fibre agree with the antipodal map of sphere. The bundleS(ν)/T→Fis denoted byRP(ν)→F; that is the real projective space bundle associated to vector bundleν. Further, the real projective space bundleRP(ν) bounds in the bordism of the classifying spaceRP(∞) forZ2, where the map intoRP(∞) classifies the double cover ofRP(ν) by the sphere bundleS(ν) (see [2, p.88]). Conversely, being given a vector bundleξoverFfor whichRP(ξ) bounds in the sense just described, there is an involution fixingFwith normal bundleν=ξ. Using the above results, in [4], Kosniowski and Stong gave a formula to express relationship between Stiefel-Whitney numbers ofMand that of the bundle (F,ν), and proved the following result: IfMmis a closed and smoothm-dimensional manifold with a smooth involutionT:Mm→Mmsuch that the fixed point setFofThas constant dimensionnwithm>2n, then (Mm,T) bounds equivariantly. For the fixed point setFbeing the disjoint union of some spaces and product spaces such as the disjoint union of projective spaces and the product spaces of projective spaces, by computing characteristic numbers and using the formula in [4], one has given equivariant bordism classification of (M,T) with a givenF(see[3, 6–10, 17–18, 20–21]).

Fork >1, letZk2denote the direct product ofkgroupsZ2.Zk2is often considered as the group generated byksmooth commuting involutionsT1,T2,···,TkonMm. Thekcommuting involutions determine a smoothZk2-action Φ:Zk2×Mm→Mm. The fixed data ofZk2-action Φ consists ofη=whereFΦ={x∈Mm|Ti(x)=x,i=1,2,···,k}is the fixed point set ofZk2-action andη=is the normal bundle ofFΦinMdecomposed into eigenbundlesερwithρrunning through the 2k−1 nontrivial irreducible representations ofZk2(see [16]). An equivariant bordism classification ofMmwithZk2-actions is closely related to the fixed data ofZk2-action (see [19]). In a series of papers, the equivariant bordism classification of (Mm,Φ)with a given condition on the fixed data of Φ has been studied (see [5, 11–16, 19]). Fork >1 andFΦ=RP(l)∪RP(n), whereRP(·) denotes a real projective space, the classifications in cases (l,n)=(0,odd),(0,even) were completely solved in [11–13]. In [15], Pergher,Ramos and Oliveira solved the case (l,n) = (2,n) (nis even), wheren≥4. Later, in [16], Pergher and Ramos solved the case (l,n) = (2s,n) (nis even), wheres≥1 andn≥2s+1, which extended the previous case (s=1).

The purpose of this paper is to extend above results forZ22-actions. Let Φ:Z22×M→Mbe a smooth action of the groupZ22= {T1,T2|T2i= 1,i= 1,2, T1T2=T2T1} on a smooth closed manifoldM. LetT3=T1T2. The fixed data of Φ is (FΦ;ε1,ε2,ε3), whereFΦ= {x∈M|Ti(x)=x,i=1,2,3}is the fixed point set ofZ22-action onM,εi(i=1,2,3)is the normal bundle ofFΦinFTi={x∈M|Ti(x)=x}. We have the following result.

Theorem 1.1Let(M,Φ)be a closed and smooth manifold M with a smooth Z22-action whose fixed point set is the disjoint union of two real projective spaces with dimension2m+1,that is, FΦ=RP1(2m+1)∪RP2(2m+1), where RPi(2m+1)denotes the i-th copy. Let

be the fixed data ofΦ, where µiand νidenote the normal bundle of components of FΦrespectively. If at least twoµ′is have dimension greater than2m+1,and at least one νihas dimension greater than2m+1, then(M,Φ)bounds equivariantly.

Example 1.1LetM=RP(4m+ 3) ×RP(4m+ 3). Considering the involutionT1:RP(4m+3) →RP(4m+3) on the (4m+3)-dimensional real projective spaceRP(4m+3)given by

T1fixes the disjoint unionRP1(2m+1)∪RP2(2m+1).

AZ22-action Φ is defined by (T1×T1,S), whereS(x,y) = (y,x). The fixed data of Φ is(RP1(2m+1);µ2m+2,µ2m+2,τ(RP1(2m+1)))∪(RP2(2m+1);ν2m+2,ν2m+2,τ(RP2(2m+1))),whereτ(RP1(2m+1)means the tangent bundle. (M,Φ)satisfies the hypothesis of the theorem and (M,Φ) bounds equivariantly.

2 Preliminaries

Let Φ be a smoothZ22-action onMwith fixed data (FΦ;ε1,ε2,ε3). Eachs-dimensional component of(FΦ;ε1,ε2,ε3)can be considered as an element of Ns(BO(s1)×BO(s2)×BO(s3)),the bordism ofs-dimensional manifolds with a map into BO(s1)×BO(s2)×BO(s3), wheresiis the dimension ofεiover the component and BO(si) is the classifying space forsi-dimensional vector bundles (this is the simultaneous bordism between lists of vector bundles: Two lists of vector bundles over closedn-dimensional manifolds, (Fn;ε1,ε2,ε3) and (Vn;µ1,µ2,µ3), are simultaneously bordant if there exists an (n+1)-dimensional manifoldWn+1with boundary∂(Wn+1)=Fn∪Vn(disjoint union)and a list of vector bundles overWn+1,(Wn+1;η1,η2,η3),so thatηρ(ρ=1,2,3) restricted toFn∪Vnis equivalent toερ∪µρ) (see [14]).

According to[19],the equivariant bordism class of(M,Φ)is determined by the simultaneous bordism class of(F;ε1,ε2,ε3). Also,if(M,Φ)has the fixed data(FΦ;ε1,ε2,ε3), which is simultaneously bordant to (V;µ1,µ2,µ3), then there exists (N,Ψ) with fixed data (V;µ1,µ2,µ3),hence (N,Ψ) is equivariantly bordant to (M,Φ). On the other hand, as in the casek= 1,the simultaneous bordism class of (F;ε1,ε2,ε3) is determined by its characteristic numbers:writeW(F) = 1+ω1+ω2+··· for the Stiefel-Whitney classes of the tangent bundle ofF,andW(ερ) = 1+υρ1+υρ2+··· for the Stiefel-Whitney classes of the bundlesερ(ρ= 1,2,3).Then a characteristic number of (F;ε1,ε2,ε3) is an evaluation of the formK[F], whereKis a product ofωi,sand,ρ∈{1,2,3}, and [F] is the fundamentalZ2-homology class ofF;again, as in the casek= 1,K[F] must be understood as a sum ΣsKs[Fs], whereFsis the union of thes-dimensional components ofF, andKsis the part ofKwith degrees. If every characteristic number of (F;ε1,ε2,ε3) is zero, then (F;ε1,ε2,ε3) bounds simultaneously, hence(M,Φ) bounds equivariantly.

Now letF0⊂Mbe any component ofFT1. Writel= dim(F0), and denote byFi0⊂F0,0 ≤i < l, the union of thei-dimensional components ofFΦthat are contained inF0. Then,for each 0 ≤i < l, one has that dim(ε2) + dim(ε3) is equal to dim(M) −loverFi0. Letr= dim(M)−l. ConsiderRP(ε1) →Fi0,which is the real projective space bundle associated toε1→Fi0, and denote byξ→RP(ε1)line bundle of the double coverS(ε1)→RP(ε1), whereS(ε1) is the sphere bundle ofε1. Then, for each 0 ≤i

whereε1,ε2andε3are considered as bundles overFi0or the pull back overRP(ε1). This object represents an element in the bordism group Nl−1(BO(1)×BO(r)). For our purpose, let us recall the following lemmas.

Lemma 2.1(see [14])The object(RP(ε1);ξ,ε2⊕(ε3⊗ξ))bounds as an element ofNl−1(BO(1)×BO(r)).

Remark 2.1If (FΦ;ε1,ε2,ε3) is the fixed data of aZ22-action, then the same is true for (FΦ;εi,εj,εk), where (i,j,k) is any permutation of (1,2,3). Then, in the above lemma,(RP(ε1);ξ,ε2⊕(ε3⊗ξ))can be replaced by(RP(εi);ξ,εj⊕(εk⊗ξ))for any permutation(i,j,k) of (1,2,3).

Lemma 2.2(see [14])Let(M,Φ)be a Z22-action with fixed data(FΦ;ε1,ε2,ε3). Suppose that V⊂M is an h-dimensional component of FΦ. Let P be the component of FT1that contains V. Suppose that P satisfies the following conditions:

(1) dim(P)>2h;

(2)V is the unique component of FΦcontained in P.Then(V;ε1,ε2,ε3)bounds simultaneously.

3 Proof of Theorem

We review some general background information.

Let

be the Stiefel-Whitney class ofRP1(2m+1) andλ1→RP1(2m+1) the canonical real line bundle overRP1(2m+1). From the structure of Grothendieck ringKO(RP1(2m+1)),one has that any bundleµi→RP1(2m+1) (i=1,2,3)is stably equivalent toliλ1→RP1(2m+1) for someli≥0, which implies that

is the Stiefel-Whitney class ofµi→RP1(2m+ 1),i= 1,2,3, whereαis the generator ofH1(RP1(2m+1);Z2),mi=dim(µi). If 2ais the greatest power of 2 of the 2-adic expansion of 2m+1, andli≡p(mod 2a+1), then (1+α)li= (1+α)p. So we could assumeli≤2a+1−1.Throughout the paper, we use the fact that a binomial coefficientabis nonzero modulo 2 if and only if the 2-adic expansion ofbis a subset of the 2-adic expansion ofa.

Letc∈H1(RP1(µ1);Z2) be the first Stiefel-Whitney class of the line bundleξ→RP1(µ1)for the double coverS(µ1) →RP(µ1). From [2, p.75] , one knows that the Stiefel-Whitney class ofRP1(µ1) is

with a relation

The Stiefel-Whitney class ofξis

and the Stiefel-Whitney class of the bundleµ2⊕(µ3⊗ξ) is

On the componentRP2(2m+1), we write

whereβ∈H1(RP2(2m+1);Z2)is the generator,ni=dim(νi)fori=1,2,3. If 2ais the greatest power of 2 of the 2-adic expansion of 2m+1, andti≡q(mod 2a+1), then (1+β)ti=(1+β)q.We can assumeti≤2a+1−1.

Also, we denote byλ→RP2(ν1) the line bundle for the double coverS(ν1)→RP(ν1), and by

its Stiefel-Whitney class. One has

and

For any integerr, we introduce the following characteristic classes which were initially introduced in [17],

and

The classesW[r]tandU[r]lare polynomials inWi(RP1(µ1)),c,Wj(µ2⊕(µ3⊗ξ)), hence they can be used to give characteristic numbers. Also,these classes satisfy the following special properties:

and in the same way,

Now we consider aZ22-action (M,Φ) with fixed data

where at least twoµ′ishave dimension greater than 2m+1, and at least oneνihas dimension greater than 2m+1. In order to obtain our result,we only need to prove(RP1(2m+1);µ1,µ2,µ3)∪(RP2(2m+1);ν1,ν2,ν3)bounds,that is,to show that every number of(RP1(2m+1);µ1,µ2,µ3)is equal to every characteristic number of (RP2(2m+1);ν1,ν2,ν3).For our purpose, we need the following notations: For a sequenceω= (j1,j2,···,js) of natural numbers, one lets |ω| =j1+j2+···+js, and forµ= 1+µ1+···+µp, letµω=µj1µj2···µjsbe the product of the classesµj.

Taking sequencesω= (j1,j2,···,js) andωi= (ji1,ji2,···,jis) fori= 1,2,3, with |ω|+=2m+1, we consider the characteristic numbers

and

We will prove

Fori= 1,2,3, denote byPithe component ofFTicontainingRP1(2m+1) andQithe component ofFTicontainingRP2(2m+1). Then eitherPi=QiorPi∩Qi=∅.

Suppose thatPi∩Qi=∅holds for somei∈{1,2,3}. Let us suppose first that this number is 2 or 3. Because of the hypothesis concerning the number of bundles with dimension greater than 2m+1, there existsi∈{1,2,3} such thatPi∩Qi= ∅and dim(µi)>2m+1. By applying Lemma 2.2 on the componentRP1(2m+1) ⊂Pi, one concludes that (RP1(2m+1);µ1,µ2,µ3) bounds simultaneously, thus it can be equivariantly removed to give aZ22-action(N,Ψ), equivariantly bordant to (M,Φ), and with fixed data (RP2(2m+1);ν1,ν2,ν3). Since at least oneνjhas dim(νj)>2m+1, using Lemma 2.2 on the componentRP2(2m+1)⊂Qj,one concludes that(RP2(2m+1);ν1,ν2,ν3)bounds simultaneously. It follows that(N,Ψ)(and thus (M,Φ)) bounds equivariantly, and the theorem is proved.

In this way, we could suppose that there exists a uniquei∈{1,2,3} such thatPi∩Qi=∅orPi=Qi.

By making permutation oni∈{1,2,3} if necessary, we can suppose with out loss that

SinceP1=Q1andP3=Q3,one has dim(ν1)=m1and dim(ν3)=m3. Now

thus dim(ν2)=m2.

From Lemma 2.1withF0=P1and=RP1(2m+1)∪RP2(2m+1),one has that

is bordant to

in the bordism group

Then any class of dimension 2m+m1given by a product of the class

evaluated on [RP(µ1)], gives the same characteristic number as the one obtained by the corresponding product of the classes

evaluated on [RP(ν1)]. To find the value of such numbers, we have a formula of Conner (see[1, Lemma 3.1]),

wherei+j=2m+m1andis the dual Stiefel-Whitney class ofµ1.

wherei+j= 2m+n1andis the dual Stiefel-Whitney class ofν1. We apply the above equations to prove that

bounds.

We knowW(µ1) = (1 +α)l1,W(µ2) = (1 +α)l2,W(µ3) = (1 +α)l3,W(ν1) = (1 +β)t1,W(ν2) = (1 +β)t2,W(ν3) = (1 +β)t3. Ift1,t2andt3are even (orl1,l2andl3are even), one concludes that (RP2(2m+1);ν1,ν2,ν3) (or (RP1(2m+1);µ1,µ2,µ3)) bounds simultaneously, thus it can be equivariantly removed to give aZ22-action (N,Ψ), equivariantly bordant to (M,Φ), and with fixed data (RP1(2m+1);µ1,µ2,µ3)(or (RP2(2m+1);ν1,ν2,ν3)).Since at least twoµ′is (or oneνi) have dim(µi)>2m+ 1 (has dim(νi)>2m+ 1), using Lemma 2.2 on the componentRP1(2m+1) ⊂Pi(orRP2(2m+1) ⊂Qi), one concludes that(RP1(2m+ 1);µ1,µ2,µ3) ((RP2(2m+ 1);ν1,ν2,ν3)) bounds simultaneously. It follows that(N,Ψ) (and thus (M,Φ)) bounds equivariantly, and the theorem is proved. So, we always suppose that not alll1,l2andl3are even (ort1,t2andt3are even). In this case, we only need to provel1=t1,l2=t2,l3=t3. The proof is divided into several cases.

Proposition 3.1l1=t1.

ProofIfl1/=t1, we will prove that there does not existZ22-action (M,Φ). We divided the arguments into the following cases.

(1)l1is odd andt1is even. Then onRP1(2m+1),

OnRP2(2m+1),

We form the class(W[0]1)2m+1cm1−1onRP1(2m+1),and the corresponding class onRP2(2m+1) is (W[0]1)2m+1dm1−1, then

In the same way, we can prove the case thatl1is even andt1is odd.

This is a contradiction.

(2)l1is odd andt1is odd andl1/=t1. Let

We form the class (W[0]1)2m+1−2sW[2s−1]2(2s−1)+2cm1−1−2sonRP1(2m+1), and the corresponding class onRP2(2m+1) is (W[0]1)2m+1−2sW[2s−1]2(2s−1)+2dm1−1−2s.Then

This is a contradiction.

(3)l1andt1are even andl1/=t1. Let

Since not alll1,l2andl3are even, one ofl2andl3must be odd. Ifl2is odd, then onRP1(2m+1),

OnRP2(2m+1),

We form the class (U[1]2)2m+1−2sW[2s−1]2(2s−1)+2cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (U[1]2)2m+1−2sW[2s−1]2(2s−1)+2dm1−2m−2.Then

This is a contradiction.

Ifl3is odd, onRP1(2m+1),

OnRP2(2m+1),

We form the class (U[0]2)2m+1−2sW[2s−1]2(2s−1)+2cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (U[0]2)2m+1−2sW[2s−1]2(2s−1)+2dm1−2m−2.Then

This is a contradiction.

Proposition 3.1 holds.

Proposition 3.2l2=t2.

ProofIfl2/=t2, we will prove that there does not existZ22-action (M,Φ). We divided the arguments into the following cases.

(1)l2is odd andt2is even. Then onRP1(2m+1),

OnRP2(2m+1),

We form the class (U[1]2)2m+1cm1−2m−2onRP1(2m+ 1), and the corresponding class onRP2(2m+1) is (U[1]2)2m+1dm1−2m−2. Then

In the same way, we can prove the case thatl2is even andt2is odd.

This is a contradiction.

(2)l2is odd andt2is odd andl2/=t2. Let

OnRP1(2m+1),

OnRP2(2m+1),

We form the classU[2s]2s+1(U[1]2)2m+1−2scm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) isU[2s]2s+1(U[1]2)2m+1−2sdm1−2m−2. Then

This is a contradiction.

(3)l2andt2are even andl2/=t2. Let

Since not alll1,l2andl3are even, one ofl1andl3must be odd. Ifl1is odd, then onRP1(2m+1),

OnRP2(2m+1),sincel1=t1,so

We form the class (W[0]2)2m+1−2sU[2s]2s+1cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (W[0]2)2m+1−2sU[2s]2s+1dm1−2m−2. Then

This is a contradiction.

Ifl3is odd, onRP1(2m+1),

OnRP2(2m+1),

We form the class (U[0]2)2m+1−2sU[2s]2s+1cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (U[0]2)2m+1−2sU[2s]2s+1dm1−2m−2. Then

This is a contradiction.

Proposition 3.2 holds.

Proposition 3.3l3=t3.

ProofIfl3/=t3, we will prove there does not existZ22-action (M,Φ). We divided the arguments into the following cases.

(1)l3is odd andt3is even. Then onRP1(2m+1),

OnRP2(2m+1),

We form the class (U[0]2)2m+1cm1−2m−2onRP1(2m+ 1), and the corresponding class onRP2(2m+1) is (U[0]2)2m+1dm1−2m−2. Then

In the same way, we can prove the case thatl3is even andt3is odd.

This is a contradiction.

(2)l3is odd andt3is odd andl3/=t3. Let

OnRP2(2m+1),

We form the class (U[0]2)2m+1−2sU[2s−1]2(2s−1)+2cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (U[0]2)2m+1−2sU[2s−1]2(2s−1)+2dm1−2m−2. Then

This is a contradiction.

(3)l3andt3are even andl3/=t3. Let

Since not alll1,l2andl3are even, one ofl1andl2must be odd. Ifl1is odd, then onRP1(2m+1),

OnRP2(2m+1),

We form the class (W[0]2)2m+1−2sU[2s−1]2(2s−1)+2cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (W[0]2)2m+1−2sU[2s−1]2(2s−1)+2cm1−2m−2.Then

This is a contradiction.

Ifl2is odd, onRP1(2m+1),

OnRP2(2m+1),sincel2=t2,so

We form the class (U[1]2)2m+1−2sU[2s−1]2(2s−1)+2cm1−2m−2onRP1(2m+1), and the corresponding class onRP2(2m+1) is (U[1]2)2m+1−2sU[2s−1]2(2s−1)+2dm1−2m−2. Then

This is a contradiction.

Proposition 3.3 holds.

From above discussions, we know thatl1=t1,l2=t2andl3=t3.If we take any sequencesω=(j1,j2,···,js) andωi=(ji1,ji2,···,jis) fori=1,2,3 with=2m+1, we have

That is,

We conclude that every characteristic number of (RP1(2m+1);µ1,µ2,µ3) is equal to the characteristic number of(RP2(2m+1),ν1,ν2,ν3). By[19](RP1(2m+1),µ1,µ2,µ3)∪(RP2(2m+1),ν1,ν2,ν3) bounds simultaneously, (M,Φ) bounds equivariantly, and the theorem is proved.

AcknowledgementThe authors would like to thank the anonymous reviewers for their careful reading and valuable suggestions.