On Z3-Actions on Spin 4-Manifolds∗

Ximin LIU Changtao XUE

1 Introduction

Let G be a finite group,and X be an n-dimensional manifold.If Top(G,X)denotes the set of equivalent classes of topological G-actions on X.Recall that two topological G-actions on X are equivalent if there exists a homeomorphism f of X such that one action is conjugate to the other by f.A topological finite group G-action on an n-dimensional manifold X is called locally linear if for any point x∈X,there exists a Gx-invariant neighborhood Vxof x such that Vxis homeomorphic to Rn,and Gxacts on Vxin a linear orthogonal way,where Gxis the isotropy group of x.Similarly,LL(G,X)denotes the set of equivalent classes of locally linear G-actions on X.If a smooth structure on X is specified,C∞(X,G)denotes the set of equivalent classes of smooth G-actions on X with respect to differmorphisms preserving the smooth structure.It is well known that the three classes of group actions have the relation

where ϕ is the map forgetting the smooth structure.For a 4-dimensional manifold X the map ϕ is not surjective.An action is called pseudofree if it is free outside of a finite set of points.

It is proved by Kwasik and Vogel[11]that the existence of nontrivial locally linear involutions on simply-connected closed topological 4-manifolds implies that the vanishing of the Kirby-Siebenmann obstruction.In the present paper,for a topological Z3-action on 4-dimensional manifold X,we obtain the following necessarily condition for it to be locally linear.

Theorem 1.1LetXbe a closed,simply-connected,smooth,spin4-manifold whose intersection form is isomorphic to2k(−E8)⊕lH,whereHis the hyperbolic form.If a pseudofree,topologicalZ3-action onXis locally linear,thenSign(g,X)≡ −k mod 3.

The proof of the above theorem is based on the properties of Kirby-Sibenmann invariant and Rochlin invariant in[7].

The smoothability of a locally linear orientation preserving pseudofree action on a smooth 4-manifold has been an open question.Kwasik and Lawson[10]provided the first example answering this question in the negative mainly by gauge theory,and in some cases of involutions,Rohlin’sµ-invariant is used.In recent years,many nonsmoothable group actions on 4-manifolds are constructed by many authors(see[3–4,9–14,16]).For example,the authors proved the existence of nonsmoothable involutions on a large class of spin 4-manifolds in[16],where we use the Rochlin’s theorem.In[13–14],Liu and Nakamura constructed groups actions on elliptic surfaces which are not smooth with respect to in finitely many smooth structures including the standard smooth structure.They used the mod p vanishing theorem of Seiberg-Witten invariants in[6]to get nonsmoothable group actions on elliptic surfaces.

In this paper,we restrict our attention to Z3-actions on spin 4-manifolds and provide an example of nonsmoothable locally linear Z3-actions on certain elliptic surfaces(see Theorem 4.2).

2 Preliminaries

In this section,a constraint on smooth Z3-actions and some method of constructing locally linear Z3-actions are given.We also collect some fomulae which will be used in calculation.

Let G be the cyclic group of order 3(G=Z3),and suppose that G acts locally linearly and pseudofreely on a spin 4-manifold X.Now let bibe the i-th Betti number of X,and b+(resp.b−)be the rank of a maximal positive(resp.negative)de finite subspace H+(X;R)(resp.H−(X;R))of H2(X;R).For any G-space V,let VGbe the fixed point set of the G-action.Let=dimH•(X;R)G,where • =2,+,−.The Euler number of X is denoted by χ(X)and the signature of X by Sign(X).

When we fix a generator g of G,the representation at a fixed point can be described by a pair of nonzero integers(a,b)modulo 3 which is well-defined up to order and changing the sign of both together.Hence,there are two types of fixed points:

(1)The type(+):(1,2)=(2,1).

(2)The type(−):(1,1)=(2,2).

Let m+be the number of fixed points of the type(+),and m−be the number of fixed points of the type(−).

2.1 The realization theorem of locally linear Z3-actions

To construct locally linear Z3-actions,we use the following special case of the realization theorem by Edmonds and Ewing[5].

Theorem 2.1(see[5])Suppose that we are given a fixed point data

whereai,bi∈Z3{0},and aZ3-invariant bilinear unimodular even formΨ:V×V →Z,whereVis a finitely generatedZ-freeZ[Z3]-module.Then the dataDand the form(V,Ψ)are realizable by a locally linear,pseudofree,G-action on a closed,simply-connected,topological4-manifold if and only if they satisfy the following two conditions:

(1)The conditionREP:As aZ[Z3]-module,VT⊕F,whereTis a trivialZ[Z3]-module withrankZT=n,andFis a freeZ[Z3]-module.

(2)The conditionGSF:TheG-signature formula is satisfied,i.e.,

where

Note that the realization theorem for all cyclic groups of prime order provided by Edmonds and Ewing[5]also has a condition TOR.However,the TOR condition is redundant for prime numbers p less than 23.Since the form Ψ is assumed even,the homeomorphism type of X is unique by Freedman’s theorem(see[7]).

Setis invariant under the action of G.

2.2 The Lefschetz fixed points theorem and the G-signature theorem

Here we collect some classical formulae.We refer the reader to see[1–2]and the excellent exposition in[4]for more details.Let X be a closed,oriented smooth 4-manifold,and let cyclic group G ≡ Zpof prime order act on X effectively via orientation-preserving diffeomorphisms.Then the fixed-point set F,if nonempty,will consist of isolated points and surfaces.If a generator g of G is fixed,each fixed point m∈F is associated with a nonzero integers pair(am,bm),where−p＜am,bm＜p,and they are uniquely determined up to a change of order or a change of sign simultaneously,such that the induced g-action on the tangent space at m is given by the complex linear transformation(z1,z2)(ξamz1,ξbmz2),where ξ=expFor each connected surface Y⊂F,the action of g on the normal bundle of Y in X is given by z→ ξcYz for an integer cYwith 0＜cY＜p,which is uniquely determined up to a sign modulo p.

Theorem 2.2(Lefschetz Fixed Point Theorem)LetT:X→Xgenerate an action ofZpon X,a closed,oriented smooth4-manifold.ThenL(T,X)= χ(F),whereχ(F)is the Euler characteristic of the fixed-point setFandL(T,X)is the Lefschetz number of the mapT,which is defined by

For a simply-connected4-manifoldX,the formula isχ(F)=2+tr(g)|H2(X;R).

Theorem 2.3(G-Signature Theorem)Set

Then

whereY·Ydenotes the self-intersection number of Y.

The weaker version of the G-signature theorem is used more often since the convenient for calculation.

Theorem 2.4(G-Signature Theorem–The Weaker Version)

where the termsdefmanddefYare called signature defects.They are given by the following formulae:

if the local representation of G atmis given by(z1,z2) → (ξkz1,ξkqz2),andis invariant under the action of G.

2.3 Kirby-Siebenmann invariant and Rochlin invariant

Let us review some properties of Kirby-Siebenmann invariant and Rochlin invariant in this part(see[4,7]for more details).

Let X be a compact topological 4-manifold whose boundary has a unique smooth structure.There is an obstruction in ks(X)∈ H4(X,∂X;Z2)to extend the smooth structure to a smooth structure on X×R.Each component of X has H4(−,∂X;Z2)Z2.define ks(X)to be the sum of these invariants in Z2over all components.The ks(X)is the stable smoothing obstruction if X is connected.Suppose(M,τ)is a closed spin 3-manifold,where τ is a spin structure.There is a smooth spin 4-manifold(W,τ′)bounded by(M,τ)since the 3-dimensional smooth bordism group is trivial.Then Rochlin invariant roc(M,τ)is defined to be the signature of W,mod 16.This invariant may depend on the spin structure.The invariant is well defined if there is a unique spin structure.

Suppose that a smooth,simply-connected,spin 4-manifold X admits a locally linear,topological action of a finite group G.Then the quotient space X/G is a spin 4-orbifold with only isolated singular points.By removing a regular neighborhood of the singular set,we get a spin 4-manifold with boundary which denoted by N,and the boundary of N inherits a spin structure from that of N which denoted by ∂η.Then,the Kirby-Siebenmann invariant of N and the Rochlin invariant of(∂N,∂η)are constrained as follows.

Theorem 2.5(see[7])8·ks(N)≡ Sign(N)+roc(∂N,∂η)mod 16.

By the results about the Kirby-Siebenmann invariant in[3],it is very easy to get the following theorem.

Note that the G-signature theorem is also valid for locally linear,topological actions of prime orders in dimension 4.The actions in this paper are pseudofree,i.e.,the fixed point set only contains isolated fixed points,so the above formulae will be more concise.Set

Theorem 2.6The Kirby-Siebenmann invariantks(N)=0is a necessary condition for theG-action to be smoothable.

3 The Proof of Theorem 1.1

Let X be a closed,simply-connected,smooth,spin 4-manifold which has the intersection form isomorphic to 2k(−E8)⊕lH,where k and l are positive integers.Therefore l≥ 2k+1 by Furuta’s inequality(see[8]).Suppose there is a locally linear pseudofree Z3-acttion on X.The fixed points of a pseudofree Z3-action on X can be divided into two types by considering their local representation:the type(+)and the type(−),and let k+,k−be the numbers of the fixed points of the type(+),type(−)in the fixed point set separately.We can see that the corresponding 4-manifold N has k++k−boundary components:There are k+L(3,2)and k−L(3,1).

Lemma 3.1The Rochlin invariants ofL(3,1)andL(3,2)are2and−2,mod 16,respectively.

Proof Since lens space L(3,1)bounds a spin 4-manifold Y obtained by plumbing on a chain which has two vertices both weighted by 2,the signature of Y is 2.Note that L(3,2)=−L(3,1).Then,the Rochlin invariants of L(3,1)and L(3,2)can be obtained by definition.

Lemma 3.2For any locally linearZ3-acttion onX,ks(N)≡0 mod 2.

Proof Since the obstruction in question is natural for coverings and in this case can be thought of as a multiple of the “top class” in H4(N,∂N;Z2).

For a pseudofree Z3-action on X,we have the following constraint on Kirby-Siebenmann invariant and Rochlin invariant.

Proof By the G-Signature formula,we have

Here we use the facts that the signature defect of a type(+) fixed point is def+=,and that of a type(−) fixed point is def−= −,and Sign(g,X)=(k+−k−).

By the additivity of the Rochlin invariant,we have

Hence

Note that Sign(X/Z3)=Sign(N)for the standard choices of orientations.The proof is done.

Taking into account Lemma 3.2,and the formula in Theorem 3.1,then we have the next proposition.

Proposition 3.1For a generatorg∈Z3,Sign(g,X)≡−k mod 3.

4 Nonsmoothable Z3-Actions

In this section,an example of nonsmoothable locally linear Z3-actions on 4-dimensional manifolds X satisfying k≡1 mod 3 will be provided.

Let X be a 4-manifold as above,its intersection form isomorphic to 2k(−E8)⊕ lH,where k≡1 mod 3.Recall that if there is a locally linear pseudofree Z3-action on X,Sign(g,X)=13(m+−m−).By Proposition 3.1,m+−m−≡−3 mod 9.Note that#XG=m++m−and 2+tr(g|H2(X))≤ χ(X).By the Lefschetz fixed point theorem,

For the spin 4-manifold X,the G=Z3-action can lifts to a G-action on the Spincstructure naturally.Then,the G-index of the Dirac operator DXcan be written as indGR(G)Z[t]/(t3=1),where Cjis the complex 1-dimensional weight j representation of G and R(G)is the representation ring of G.The following mod p vanishing theorem can be used to prove the existence of certain nonsmoothable locally linear Z3-actions.

Theorem 4.1(see[6])Let Y be a smooth closed oriented4-dimensionalZp-manifold withb1=0andb+≥2,wherepis a prime.Suppose thatcis aSpinc-structure on whichZp-action lifts,and thatb+=bG+.If2kj≤ b+−1forj=0,···,p−1,then

As in[12],the coefficients kjare calculated by the G-spin theorem.For a generator g∈G,the Lefschetz number is calculated by the formula as

where ζ=We obtain

Then,we have

We can see that the condition 2kj≤b+−1 in Theorem 3.1 is equivalent to

At last,we give a result that there exist some nonsmoothable locally linear Z3-actions on certain elliptic surfaces.

An elliptic surface is a compact,complex surface E which comes with a holomorphic projection π :E →C onto a compact,connected complex curve,such that the generic fibers of π are elliptic curves.We will always assume that E is minimal elliptic,i.e.,not a blow-up of another elliptic surface.

The projection π(or elliptic fibration)of an elliptic surface has well-understood local behavior.It has only finitely many critical values,and away from these it is a bundle projection with torus fibers(called regular fibers).The singular fibers,or preimages of critical values,come in various types.For minimal elliptic surfaces,we may smoothly change π so that only two types of singular fibers occur:cusp fibers and(smooth)multiple fibers.A cusp fiber is a PL-embedded sphere with a unique non-locally fl at point,which is locally a cone on a(right-handed)trefoil knot.

Simply connected minimal elliptic surfaces without multiple fibers are completely classi fi ed up to diffeomorphism by a positive integer n.Each such manifold E(n)has a projection with exactly 6n cusp fibers and no multiple fibers.It follows that E(n)has Euler characteristic 12n.

A smooth multiple fiber is a smoothly embedded torus which is multiply covered by nearby regular fibers.In fact,it is essentially a Seifert multiple fiber crossed with S1.It follows that any elliptic surface can be obtained from one without multiple fibers by a process called logarithmic transform,which is essentially Dehn surgery along a fiber.

Now let E(n)be the relatively minimal simply-connected elliptic surface without multiple fibers,and with geometric genus pg=n − 1.Note that Sign(E(n))= −8n and χ(E(n))=12n.Thus E(2)=E(1)♯T2E(1)is the K3 surface.To see this just note that the Euler characteristic are additive under taking fiber connected sums over a torus.Hence Sign(E(2))=−16 and χ(E(2))=24 which characterizes K3 surface.Besides,the general surface E(n)can be constructed as a fiber connected sum E(n)=E(n − 1)♯T2E(1).So the intersection form of E(n)isomorphic to n(−E8)⊕(2n−1)H.Suppose n is even and n ≥ 2.The condition 2kj≤b+−1 in Theorem 3.1 is equivalent to

Recall that the Seiberg-Witten invariant of the E(n)isBy the mod p vanishing theorem of the Seiberg-Witten invariants,we have the following theorem.

Theorem 4.2Ifcn−2/0 mod 3andn ≡ 2 mod 6,then there exists a locally linearZ3-action onE(n)stisfying the following conditions,which is nonsmoothable with respect to in finitely many smooth structures onE(n):

The proof of above theorem is divided into two steps.In the first step,to construct a locally linear action,we use the realization theorem due to Edmonds and Ewing[5].In the second step,we use the mod p vanishing theorem of the Seiberg-Witten invariants to give a constraint on smooth Z3-action.In fact the proof can be done by imitating the method in[13],so we omit here.

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