Betti Numbers of Locally Standard 2-Torus Manifolds∗

Junda CHEN Zhi L

1 Introduction

Locally standard(Z2)n-actions on smooth closedn-manifolds belong to a class of particularly nicely behaving actions,introduced by Davis and Januszkiewicz[4].Here a smooth connected closedn-manifold with a locally standard-action is called a locally standard 2-torus manifold.Generally,the orbit spaceQnof a locally standard 2-torus manifoldMnis anndimensional nice manifold with corners.IfQnis a simple convex polytopePn,thenMnis called a small cover by Davis and Januszkiewicz.There are strong links of small covers with combinatorics of polytopes with the following two key points(see[4]):

(1)Each small coverπ:Mn→Pncan be recovered fromPnwith an associated characteristic function;

(2)The algebraic topology of a small coverπ:Mn→Pn,such as(equivariant)cohomology and mod 2 Betti numbers etc.,can be explicitly expressed in terms of the combinatorics ofPn.For example,the mod 2 Betti numbers···,ofMnagree with theh-vector,···,

Given a locally standard 2-torus manifoldπ:generallyQnis not a simple convex polytope,and it may be not contractible.We know from[6]thatif and only if the action onMnis free,soMncan actually be regarded as a principal-bundle overQnin this case.However,ifthen it admits a simplicial poset structure,so we can define anh-vector h(Qn)onQn.In addition,like the case of small covers,Qnalso admits a characteristic functionλ.As shown in[7],the pair(Qn,λ)only provides information on the set of non-free orbits inMn,so generally it is not enough to recoverMn(of course,it is enough ifQnis contractible).Indeed,we also need another data(which provides information on the set of free orbits inM),i.e.,a principalbundleξoverQndetermined uniquely byMnup to isomorphism.Then it was shown in[7]that the orbit spaceQnwith two dataλandξcan reproduceMnup to equivariant homeomorphism,denoted byM(Q;λ,ξ).Generally the algebraic topology of a locally standard 2-torus manifoldπ:withis far from known except for the formula of the Euler characteristic ofMnin terms ofQn(see[7]).The purpose of this paper is to consider the following question.

Question 1.1Letπ:be a locally standard 2-torus manifold.How do we read the mod 2 Betti numbers b(Mn)from

We give an answer of Question 1.1 in the case where∂Qnis the boundary of a simple convex polytope.Our result is stated as follows.

Theorem 1.1Suppose that π:is a locally standard2-torus manifold such that∂Qnis the boundary of a simple convex polytope Pn.Let ξ=(E,p,Q)be the associated principal(Z2)n-bundle over Q.Then for n＞2,

and for n=2,

The arrangement of this paper is as follows.In Section 2,we introduce the concept of locally standard 2-torus action and manifold with corners,and show the reproducing process of locally standard 2-torus manifolds.In Section 3,when the boundary of the orbit spaceQis the boundary of a simple convex polytope,we obtain the relationship among the Betti numbers of locally standard 2-torus manifold,the Betti numbers of the orbit space and theh-vector of the orbit space,which gives the proof of Theorem 1.1.

2 Locally Standard 2-Torus Manifolds

2.1 Manifold with corners

Let|xi≥ 0,i=1,···,n}.For anyx=its codimensionc(x)is the number ofxiwhich are equal to 0.

Definition 2.1(see[3])An n-manifold Qnwith corners is a Hausdorffspace together with a maximal atlas of local charts onto open subsets ofRn+so that the overlap maps are homeomorphisms(diffeomorphisms)which preserve codimension.

Given a manifoldQnwith corners.For anyq∈Qn,let(U,ϕ)be a local chart ofq.Then,the codimensionc(q)ofqis defined asc(ϕ(q)).This is well defined because the overlap maps preserve codimension.An open face ofQnof codimensionlis a connected component of the inverse imagec−l(l).A face is the closure of an open face.Specially,an open face of codimension one is called an open facet,and its closure is called a facet.For anyq∈Qn,let Σ(q)be theset of facets which containq.A manifoldQnwith corners is nice if Card(Σ(q))=c(q)for anyq∈Qn.

Example 2.1Anyn-dimensional simple convex polytope is a nice manifold with corners.

Remark 2.1It should be pointed out that the notion of a manifold with corners is also a natural generalization of the notions of ordinary manifolds with or without boundary.Actually,given a manifoldQnwith corners,it is easy to see that the codimension of each point inQnis zero if and only if the boundary ofQnis empty.Furthermore,Qnis a manifold with boundary if and only if there is at least one pointqinQnsuch thatc(q)＞0.

2.2 Locally standard 2-torus manifolds

The standard action of(Z2)non Rnis defined by

Its orbit space is a positive cone∈Rn|xi≥ 0,i=1,···,n}.The action ofon ann-dimensional smooth closed manifoldMnis locally standard if the action locally looks like the standard action ofMore precisely,for each pointxinMn,there is a-invariant neighborhoodVxofxsuch thatVxis weakly equivariantly homeomorphic to a-invariant open setWin the standard representation Rn(i.e.,there is a homeomorphismf:Vx→Wand an automorphismθ:→such thatf(gv)=θ(g)f(v)forv∈Vx).

Ann-dimensional 2-torus manifold is a connected closed smooth manifold of dimensionnwith an effective smooth action ofGenerally,a 2-torus manifold may not be locally standard.We say thatMis a locally standard 2-torus manifold if it is a 2-torus manifold and admits a locally standard action.

Example 2.2The canonical(Z2)n-action onRPn,defined by

is a locally standard 2-torus manifold.

Lemma 2.1(see[3])Let π:M→Q be a locally standard2-torus manifold.Then the orbit space Q is a nice manifold with corners,and for any x∈M,the rank of isotropy group Gx equals c(π(x)).

Now letQbe a nicen-manifold with corners withThen it is easy to see that all faces ofQform a simplicial poset with∂Qas the minimum element with respect to inverse inclusion.Following[4],letfidenote the numbers of faces of codimensioni+1 inQ.Define a polynomial ΨQ(t)of degreenby

and lethidenote the coefficients of the termin(t),where 0 ≤i≤n.Then(h0,h1,···,hn)is called theh-vector ofQ,denoted by h(Q).

A locally standard 2-torus manifold is called a small cover if its orbit space is a simple convex polytope.Davis and Januszkiewicz showed in[4]the following theorem.

Theorem 2.1(Davis–Januszkiewicz)Let π:M→P be a small cover.Then

whereb(M)=(b0,b1,···,bn)is the vector formed by all mod2Betti numbers of M.

2.3 Reconstruction of locally standard 2-torus manifolds

Suppose thatπ:M→Qis a locally standard 2-torus manifold.

IfQis a simple convex polytopeP,thenMis a small cover.In this case,Davis and Januszkiewicz showed in[4]that there exists a characteristic functionλfrom all facets ofPtosatisfying that whenever some facetshave a nonempty intersection,···,are linearly independent.Furthermore,Davis and Januszkiewicz usedλto blow down the product bundle×Pto recoverM.

In the setting of locally standard 2-torus manifolds,if the boundary ofQis nonempty,as shown in[7],an analogous reconstruction as above can still be carried out well,but two data forQwill be needed.One data is a characteristic function onQ,and the other data is a principalbundle overQ.Actually,if the boundary ofQis nonempty,sinceQis nice,then∂Qis the union of its all facets,and each facet ofQcorresponds to a nonzero elementv∈(Z2)nsuch that the inverse image of this facet is fixed by the rank-one subgroup determined byv.Thus there is a characteristic function

satisfying the condition that whenever the intersectionis nonempty,all elements of{λ(Fij)|j=1,···,l}are linearly independent,whereF(Q)denotes the set of all facets ofQ.Note that for anyk-faceFofQ,sinceQis nice,there existn−kfacetssuch thatFis a component ofFurthermore,determine a rankn−ksubgroup ofdenoted byGF.Besides,a principal-bundleξ=(E,p,Q)overQcan be produced fromπ:M→Qin the following way:We take a small invariant open tubular neighborhood ofπ−1(F)inMfor everyF∈F(Q)and remove all such neighborhoods fromM,wherepis the projection.This gives a principal-bundle overQ,which is unique up to isomorphism.Now we can reproduce a locally standard 2-torus manifold from these two data.First,define an equivalence relation∼onEas follows:Foru1,u2∈E,

whereFis the face ofQcontainingp(u1)=p(u2)in its relative interior andGFis the subgroup ofdetermined byF.Then the quotient spaceE/∼,denoted byM(Q,ξ,λ),naturally inherits the-action fromE.It was shown in[7]the following proposition.

Proposition 2.1Let π:M→Q withbe an n-dimensional locally standard2-torus manifold with ξ as the associated principal(Z2)n-bundle and λas the characteristic function.Then there is an equivariant homeomorphism from M(Q,ξ,λ)to M which covers the identity on Q.

Remark 2.2Given a nice manifold with cornersQof dimensionn,it is well-known that the isomorphism classes of all principal-bundles overQbijectively correspond to all elements ofHowever,generallyQmay not admit any characteristic function(see[4]).IfQadmits a characteristic function,then in the above way we can construct all possible locally standard 2-torus manifolds overQas

whereP(Q)denotes the set of all principal-bundles overQand Λ(Q)consists of all characteristic functions onQ.In particular,ifQis a simple convex polytopeP,then the set

consists of all possible small covers overP.

3 Betti Numbers of Locally Standard 2-Torus Manifold

Throughout the following,assume thatQis a connected nicen-manifold with corners such that∂Qis the boundary of a simple convex polytopeP,andQ(i.e.,P)admits a characteristic functionλ.Choose a principal(Z2)n-bundleξ=(E,p,Q)overQ.Based upon the reconstruction of locally standard 2-torus manifolds,we can obtain a locally standard 2-torus manifoldπQ:M(Q,ξ,λ)→Qand a small coverπP:M(P,λ)→P.Ifn=1,thenQwill be a 1-simplex,so thatM(Q,ξ,λ)is exactly a circle with a refection.Thus,we assume thatn＞1 in the following discussion.

Lemma 3.1Let N=(∂P)(so N is an(n−1)-dimensional CW complex).Then themod2Betti numbers of N are

ProofThe simple convex polytopePhas a natural CW complex structure such that every open face ofPof dimensioniis ani-cell.This induces the cell decomposition of quotient spaceM(P,λ).Then we have the cellular chain complex with Z2coefficients ofM(P,λ)

such that(M(P,λ)))=(P),wheredenotes the number of faces of dimensioniinP.Obviously,Nis the(n−1)-skeleton ofM(P,λ).So it has the following cellular chain complex with Z2coefficients:

Then the required result follows from the facts that dimZ2(Cn(M(P,λ)))=2nand b(M(P,λ))=h(P).

Lemma 3.2The total space E of ξ=(E,p,Q)must still be a nice manifold with corners such that if n＞2,then its boundary∂E is a disjoint union of2ncopies of∂Q;and if n=2,then the∂E is a disjoint union of4copies of∂Q,or a disjoint union of2connected sums∂Q#∂Q,or the connected sum of4copies of∂Q.

ProofIt is obvious thatEis a nice manifold with corners.Consider the inverse imagep−1(∂Q),it is easy to see thatp−1(∂Q)=∂Eand it is still a principal-bundle.Since∂Qis the boundary ofP,we have that

This implies that ifn＞2,then(∂Q)→∂Qmust be a trivial principal-bundle.So∂E=×∂Q.Ifn=2,sinceis nontrivial,we have that(∂Q)→∂Qmay not be trivial.Since∂Q=∂Pis the boundary of a polygon,an easy argument can induce the required result in this case.

Theorem 3.1For n＞2,

and for n=2,

ProofWe know from[1]that∂Qhas a collar inQ,i.e.,there is an embedding

such thatϕ(0,q)=qfor anyq∈∂Q.SetandQ2=Obviously,Q′is homeomorphic toIt is easy to check thatis an excisive couple ofM(Q,ξ,λ)=Now let us look at the following Mayer-Vietoris sequence(with Z2coefficients)of the couple

Claim AIfn＞2,then the mod 2 Betti numbers

SinceQ′is in the interior ofQ,we see from the construction ofM(Q,ξ,λ)thatis a principalbundle overQ′.SinceQ′is homeomorphic toandn＞2,similarly to the argument of Lemma 3.2,we have thatis isomorphic to the product bundleand then Claim A follows from this sinceQ′is homeomorphic to

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Claim BIfn＞2,then

Since∂Qis the deformation retract ofQ1,this induces the deformation retraction ofonto,so=Then we have by Lemma 3.2 that==as desired.Furthermore,we obtain from Lemma 3.1 that

Claim CFor the mod 2 Betti numbers ofwe have=b(E).

This is because(Q2)is the deformation retract ofE.

Claim DM(Q,ξ,λ)is connected.

For every pointx∈M,there exists always a path connectxtoAlsois connected since=h0(P)=1,soM(Q,ξ,λ)is connected.

Together with Claims A–D,Lemma 3.2 and the Mayer-Vietoris sequence above,the required result in the casen＞2 follows immediately.

Finally,let us consider the case ofn=2.Ifn=2 then we have from[7]that

as desired.

Corollary 3.1If E is trivial,then

Corollary 3.2If Q is a simple convex polytope,then

Remark 3.1We see that Corollary 3.2 is just Theorem 2.1 of Davis-Januszkiewicz.So Theorem 3.1 is a generalization of Theorem 2.1.

Finally,we conclude this paper with the following remark on the construction ofM(Q,ξ,λ).

Remark 3.2We see from Lemma 3.2 that ifn＞2,∂Eis a disjoint union of 2ncopies of∂Q=∂P.Thus,we can obtain a closed manifoldfromEby gluing boundaries of 2ncopies,,···,ofPto thecomponents of∂Erespectively.On the other hand,we know from the reconstruction of locally standard 2-torus manifolds thatM(P,λ)is actually obtained from the 2ncopies,,···,ofPalong their boundaries viaλ,whileM(Q,ξ,λ)is obtained fromEby gluing the 2ncomponents(i.e.,∂,···,∂)of∂Eviaλ.Thus we have that ifn＞2,thenM(Q,ξ,λ)=obtained by doing connected sums of 2ntimes betweenM(P,λ)and^Ealong the interiors ofP1,···,.Actually,M(Q,ξ,λ)is exactly the equivariant connected sum ofM(P,λ)andalong a free orbit,denoted byM(Q,ξ,λ)=

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