INVOLUTIONS WHOSE FIXED SET HAS THREE OR FOUR COMPONENTS: A SMALL
CODIMENSION PHENOMENON
E. M. BARBARESCO, P. E. DESIDERI and P. L. Q. PERGHER∗
Abstract
LetT :M →Mbe a smooth involution on a closed smooth manifold andF =n j=0Fj the fixed point set ofT, whereFjdenotes the union of those components ofFhaving dimensionj and thusnis the dimension of the component ofFof largest dimension. In this paper we prove the following result, which characterizes a small codimension phenomenon: suppose thatn≥4 is even andFhas one of the following forms: 1)F =Fn∪F3∪F2∪{point}; 2)F=Fn∪F3∪F2; 3)F =Fn∪F3∪ {point}; or 4)F=Fn∪F3. Also, suppose that the normal bundles ofFn,F3 andF2inMdo not bound. Ifkdenote the codimension ofFn, thenk≤4. Further, we construct involutions showing that this bound is best possible in the cases 2) and 4), and in the cases 1) and 3) whennis of the formn=4t, witht≥1.
1. Introduction
Throughout this paper, the involved cobordism notions will be understood in the unoriented sense. LetF be a disjoint (finite) union of smooth and closed manifolds andM be a smooth and closed manifold equipped with a smooth involutionT : M → M whose fixed point set isF. Suppose thatF is not a boundary. Ifnis the dimension of a component ofF of maximal dimension andk is the codimension of this component, thenk ≤ 32n; this follows from the famous Five Halves Theorem of J. Boardman, announced in [1], and its strengthened version of [11]. In fact, the Five Halves Theorem asserts that this is valid whenMis not a boundary, and in [11] R. E. Stong and C. Kosniowski established the same conclusion under the weaker hypothesis that(M, T )is a nonbounding involution. The assertion then follows from the fact that the equivariant cobordism class of(M, T )is determined by the cobordism class of the normal bundle ofF inM (see [4]).
The generality of this result, which is valid for everyn ≥ 1, allows the possibility that fixed components of all dimensionsj, 0 ≤ j ≤ n, occur; in this way, it is natural to ask whether there exist better bounds forkwhen we
∗The authors were partially supported by CNPq, FAPESP and CAPES.
Received 18 November 2010, in final form 10 May 2011.
omit some components ofFand restrict the set of involved dimensionsn. This question is inspired by the following results of the literature:
1) (R. E. Stong and C. Kosniowski, [11], 1978): ifF = Fn has constant dimensionn(and soFnis not a boundary), thenk≤n. For each fixedn, with the exception of the dimensionsn=1 andn=3, the maximal valuek=nis achieved by taking the involution(Fn×Fn, T ), whereFnis any nonbounding n-dimensional manifold andT is thetwist involution,T (x, y) =(y, x); that is, one has in this case an (best possible) improvement for the Boardman bound by omitting thej-dimensional components ofF withj < nand excluding n=1 and 3.
2) (D. C. Royster, [15], 1980): in this case, the result in question is referring to an intriguing improvement for the Boardman bound, which characterizes a small codimension phenomenon, given bynodd and the omission of the j-dimensional components of F with 0 < j < n. Specifically, ifnis odd andF has the formF = Fn∪ {point}, thenk ≤ 1. Evidently, this bound is best possible, and is realized by the involution(RPn+1, T ), whereRPn+1 is the(n+1)-dimensional real projective space and T[x0, x1, . . . , xn+1] = [−x0, x1, . . . , xn+1], withnodd.
This class of problems was introduced by P. Pergher in [12], where the above Royster result was enlarged in the following way: ifF has the form F = Fn ∪ {point}, where n = 2p withp odd, thenk ≤ p+3. This case (F =Fn∪ {point}) was completed by R. Stong and P. Pergher in [14], in the following (best possible) way: writingn= 2pq, wherep ≥0 andq is odd, thenk ≤n+p−q+1 ifp≤qandk ≤n+2p−qifp > q.
With the casesF =FnandF =Fn∪ {point}completed, the next natural step is the caseF = Fn∪Fj, 0 < j < n. Concerning this more general case, recently some advances have been obtained; specifically, we find best possible bounds forF = Fn∪F1 in [9] and [10],F = Fn∪F2in [5], [6]
and [7],F =Fn∪Fn−1in [8] andF =Fn∪Fj withFj indecomposable in [13]. Among these results, one particularly finds other cases involving small codimension phenomena: ifF = Fn ∪F1 andn is even, then k ≤ 2 [9], and ifF =Fn∪F2andnis odd, thenk ≤3 [6]. Further, these bounds are best possible. For F = Fn, F = Fn ∪Fn−1, F = Fn ∪F1 (nodd) and F =Fn∪F2(neven), small codimensions do not occur, which means thatk is not limited as a function ofn.
If the fixed point set F of an involution (M, T ) is expressed as F = n
j=0Fj, whereFjdenotes the union of those components ofFhaving dimen- sionj, then in all the discussion above eachFj can be supposed connected.
This follows from the fact that, given twok-dimensional vector bundles over closedj-dimensional manifolds,ξ →V andν→N, the connected sum ofξ andν,ξ ν(which is ak-dimensional vector bundle overV N), is cobordant as
a bundle to the disjoint unionξ∪ν. From [4], we then conclude that(M, T )is equivariantly cobordant to an involution whosej-dimensional part of the fixed point set is connected. So the casesF =FnandF = Fn∪Fj, 0≤j < n, can be referred to as theone component caseandtwo components case, re- spectively. Further, if the normal bundle of someFjinMis a boundary, it can be equivariantly removed to give a new involution, equivariantly cobordant to (M, T )and with fixed point setF −Fj (see [4]). Therefore suchFj have no influence in the context of looking for bounds for the possible codimensions.
This leads us to assume throughout this paper, without mention, that the normal bundle over each mentionedFj does not bound.
The goal of this paper is to start the study of small codimension phenomena when the number of components ofF is greater than 2. While in the one and two components cases one has standard sources of examples, the difficulty with more components lies in constructing examples to detect the sharpness of the obtained bounds. Also, the characteristic number computations to find bounds require more sophistication.
We will prove the following
Theorem1. Let(M, T )be an involution having fixed point set of the form F =Fn∪P, wheren≥4is even andP has the possible forms:
1) P =F3∪F2∪ {point}; 2) P =F3∪F2;
3) P =F3∪ {point}; 4) P =F3.
As previously mentioned, suppose that the normal bundle to each Fj does not bound. Then, if k is the codimension of Fn, k ≤ 4. Further, there are involutions showing that this bound is best possible in the cases2)and4), and in the cases1)and3)withnof the formn=4t,t ≥1.
Section 2 will be devoted to the construction of the above mentioned max- imal involutions. The main tool will be a combination of a construction of P. Conner and E. Floyd of [4] with a result of R. E. Stong and P. Pergher of [14]. In Section 3 we prove the part “k≤4”, using the Conner and Floyd the- ory and some special polynomials in the characteristic classes of total spaces of projective space bundles, introduced by Stong and Pergher in [14].
2. Maximal involutions
In order to construct the examples, we need to give some preliminaries and to establish some notations.
If(M, T )is an involution pair with fixed point setF andη → F is the normal bundle ofF inM, we callη→F thefixed dataof(M, T ).
For a vector bundleη→Fand a natural numberp≥1, writepη→Ffor the Whitney sum ofpcopies ofη; R →F will denote the one dimensional trivial vector bundle. Ifηisk-dimensional, writeW (η)=1+w1(η)+w2(η)+
· · · +wk(η)∈ H∗(F, Z2)for the Stiefel Whitney class ofη; ifF is a closed smoothn-dimensional manifold,W (F )=1+w1(F )+w2(F )+· · ·+wn(F ) will denote the Stiefel Whitney class of the tangent bundle ofF.
From [4], one has an algebraic scheme to determine the cobordism class of η, given by the set ofWhitney numbers(orcharacteristic numbers) ofη; such modulo 2 numbers are obtained by evaluatingn-dimensionalZ2-cohomology classes of the form wi1(F )wi2(F ) . . . wir(F )wj1(η)wj2(η) . . . wjs(η) ∈ Hn(F, Z2) (that is, with i1+i2+ · · · +ir +j1+j2+ · · · +js = n) on the fundamental homology class [F]∈Hn(F, Z2).
The following construction of P. Conner and E. Floyd will be useful (see [4]): let(Mn, T )be an involution defined on a closedn-dimensional manifold Mnwith fixed dataη→F. OnS1×Mn, consider the involutions−Id×T and c×Id, whereS1is the unit circle in the complex numbers, Id is the identity map andcis the complex conjugation. Note that−Id×Tis free and commutes with c×Id, hencec×Id induces an involution on the orbit spaceS1×Mn/−Id×T, which is a closed(n+1)-dimensional manifold. This involution, denoted by (Mn, T ), has (R⊕η → F )∪(R → Mn) as fixed data. If Mn bounds, R→Mnbounds as a line bundle, so(Mn, T )is equivariantly cobordant to an involution with fixed dataR⊕η→F. IfS1×Mn/−Id×T is a boundary, we can repeat the process taking2(Mn, T ), and so on.
We are now ready to give the required examples. Forn ≥ 1, denote by λn →RPnthe canonical line bundle over then-dimensional real projective space, and forn ≥ 0 andp ≥ 0, consider the involution(RPn+p+1, Tn,p) defined in homogeneous coordinates by
Tn,p[x0, x1, . . . , xn+p+1]=[−x0,−x1, . . . ,−xn, xn+1, . . . , xn+p+1].
The fixed data ofTn,pis
((p+1)λn→RPn)∪((n+1)λp →RPp).
Now take n ≥ 4 even. Since n+ 3 is odd, RPn+3 bounds, and thus (RPn+3, Tn,2) is equivariantly cobordant to an involution (Bn+4, f ) with fixed data
(3λn⊕R→RPn)∪((n+1)λ2⊕R→RP2); also, the fixed data of(RPn+4, Tn,3)is
(4λn →RPn)∪((n+1)λ3→RP3).
Setα ∈ H1(RPn, Z2)andβ ∈ H1(RP3, Z2)for the generators of the one- dimensionalZ2-cohomology groups. Note that 4λn, 3λn⊕Rand(n+1)λ2⊕R do not bound because RPn and RP2 do not bound; also, (n+ 1)λ3 and 4λn∪(3λn⊕R)do not bound because
(w1((n+1)λ3))3[RP3]=((n+1)β)3[RP3]=β3[RP3]=1 and (w1(4λn∪(3λn⊕R)))n[RPn∪RPn]
=(w1(4λn))n[RPn]+(w1(3λn⊕R))n[RPn]
=(4α)n[RPn]+(3α)n[RPn]=0+αn[RPn]=1.
Thus(RPn+4, Tn,3)and(RPn+4, Tn,3)∪(Bn+4, f )are involutions showing that the boundk ≤4 is best possible in the cases 4) and 2).
The examples for the cases 1) and 3) are a little more subtle. Writen=2dq, whered ≥ 1 andq is odd. In [14], R. Stong and P. Pergher showed that the underlying manifold of j(RPn+1, Tn,0) bounds forj = 1 ifd = 1, and forj ≤ 2d−2 if d > 1. Ifn is of the formn = 4t, t ≥ 1, then d ≥ 2, and thus the underlying manifold of2(RPn+1, Tn,0)bounds. It follows that 3(RPn+1, Tn,0) is equivariantly cobordant to an involution(Wn+4, g)with fixed data
(λn⊕3R →RPn)∪((n+4)R→ {point}).
Nowλn⊕3Rand(n+4)R → {point}do not bound, and 4λn∪(λn⊕3R) does not bound because
(w1(4λn∪(λn⊕3R)))n[RPn∪RPn]=αn[RPn]=1.
Thus the involution(RPn+4, Tn,3)∪(Wn+4, g)realizes the bound k ≤ 4 in the case 3).
Finally, note that 4λn∪(3λn⊕R)∪(λn⊕3R)does not bound because the base spaceRPn∪RPn∪RPnis cobordant toRPn. Then this vector bundle is cobordant to a nonbounding 4-dimensional vector bundleμ→Gn, where Gn is a connected closedn-dimensional manifold. From [4], there exists an involution(Kn+4, h)equivariantly cobordant to(RPn+4, Tn,3)∪(Bn+4, f )∪ (Wn+4, g), and with fixed data
(μ→Gn)∪((n+1)λ3→RP3)
∪((n+1)λ2⊕R→RP2)∪((n+4)R→ {point}).
Then(Kn+4, h)realizes the bound in question in the case 1).
3. A small codimension phenomenom
Let(M, T )be an involution having fixed point set of the formF =Fn∪P, wheren≥4 is even andPis as described in the statement of the main theorem.
We will prove that, ifkis the codimension ofFn, thenk≤4.
Ifη → F is the fixed data of an involution, denote byEthe total space of the real projective space bundle associated to η, RP (η) → F, and by ξ → E the line bundle of the double cover S(η) → E, S(η) the sphere bundle of η. It is known, from the Conner and Floyd exact sequence of [4], thatξ → E bounds as a line bundle. Then, if dim(E) = r, any class P = P (w1(E), . . . , wr(E), w1(ξ )) ∈ Hr(E, Z2), given by a polynomial of dimensionrin the classeswi(E)andw1(ξ ), gives the zero characteristic num- berP[E]; in this case,P[E] splits into a modulo 2 sum of factors corresponding to the connected components ofF.
Returning to our particular case, denote by Ei, i = 0,2,3 and n, the total space of the projective space bundle corresponding to thei-dimensional component, and byξi →Ei the corresponding line bundle. Our strategy will be first to select four special polynomials,P1,P2,P3andP4, with dimension n+k−1, leading to a modulo 2 system of equations
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩ 4
i=1
P1[Ei]=0 4
i=1
P2[Ei]=0 4
i=1
P3[Ei]=0 4
i=1
P4[Ei]=0
P1, P2, P3 andP4 will be built under the hypothesis, by contradiction, that k >4. After computation of the termsPi[Ej], this system will be reduced to a system of equations in four variables, these variables being four special charac- teristic numbers of the normal bundle over the 3-dimensional component. The zero solution will be the unique solution of this system. Then we will prove that these four characteristic numbers determine the cobordism class of any bundle over a 3-dimensional closed manifold. This will imply that the normal bundle over the 3-dimensional component bounds, giving the contradiction.
Denote by ηi → Fi, i = 0,2,3 and n, the normal bundle over the i- dimensional component. To avoid excessive notation, write W (Fi) = 1+
w1+w2+ · · · +wi,W (ηi)=1+v1+ · · · +vn+k−i andW (ξi)=1+cfor the Stiefel-Whitney classes.
From [2, p. 517], the Stiefel-Whitney class ofEi is
W (Ei)=(1+w1+· · ·+wi)((1+c)n+k−i+(1+c)n+k−i−1v1+· · ·+vn+k−i), where here we are suppressing bundle maps.
To construct the polynomialsPi,i =1,2,3 and 4, we consider the class W (E i)= W (Ei)
(1+c)k = 1+w1(Ei)+w2(Ei)+ · · ·
(1+c)k =1+W1+W2+ · · · This kind of class was introduced by Stong and Pergher in [14]. EachWjis a polynomial in the classesws(Ei)andc, which means that it can be used to yield the required polynomials. Thus, sincek >4, it makes sense to setP1= W1n+4ck−5,P2=W1nW22ck−5andP3=W1n−1W2W3ck−5. We recall that, if Sq is the Steenrod operation, then the Wu formula implies that Sqjevaluated on a characteristic class of a bundle gives a polynomial in the characteristic classes of this bundle. Therefore, by the Cartan formula, Sq1(W3)is a polynomial in the classesws(Ei)andc; we defineP4=W1nSq1(W3)ck−5. Next we compute Pi[Ej] fori =1,2,3,4 andj =0,2,3, n.
1) SinceF0= {point},E0=RPn+k−1and W (E 0)= (1+c)n+k
(1+c)k =(1+c)n.
Becausenis even,W1=nc=0. ThusPi =0 fori=1,2,3,4.
2) OnF2, one has
W (E 2)=(1+w1+w2)[(1+c)n−2+(1+c)n−3v1+(1+c)n−4v2].
Becausenis even,
W1=(n−2)c+v1+w1=v1+w1.
Since dim(F2) = 2 and n ≥ 4, by dimensional reasons Pi = 0 for i = 1,2,3,4.
3) OnFn, one has
W (E n)=(1+w1+ · · · +wn) 1+ v1
1+c + v2
1+c2 + v3
(1+c)3+ · · · + vk
(1+c)k
.
Then,
W1=w1+v1,
W2=v1c+v2+w1v1+w2,
and W3=v1c2+w1v1c+v3+w1v2+w2v1+w3. Since Sq1(c2)=0, the Cartan formula gives
Sq1(v1c2+w1v1c)=v12c2+w1v1c2+w1v12c+w12v1c and soW1nSq1(W3)=0 by dimensional reasons. Also
W1n+4=(w1+v1)n+4,
W1nW22=(w1+v1)n(v12c2+v22+w21v21+w22) and W1n−1W2W3=(w1+v1)n−1(v1c+v2+w1v1+w2)
·(v1c2+w1v1c+v3+w1v2+w2v1+w3).
Note that each term of the above polynomials has a factor of dimension at leastn+1 coming from the cohomology of Fn, which givesPi = 0 for i=1,2,3,4.
4) OnF3, one has
W (E 3)=(1+w1+w2+w3)[(1+c)n−3
+(1+c)n−4v1+(1+c)n−5v2+(1+c)n−6v3].
Becausenis even and by dimensional reasons,W1= (n−3)c+v1+w1 = c+v1+w1and
W1n+4=(v1+w1+c)n+4=
n+4
j=0
n+4 j
(v1+w1)jcn+4−j
=
cn+4, ifn≡0 mod 4,
cn+4+(v12+w12)cn+2, ifn≡2 mod 4.
Thus
P1[E3]=
cn+k−1[E3], ifn≡0 mod 4, (cn+k−1+(v21+w21)cn+k−3)[E3], ifn≡2 mod 4.
Denoting by
W (η3)= 1
W (η3) =1+v1+v2+v3
the dual Stiefel-Whitney class ofη3, the Conner formula of [3] gives that xcn+k−i[E3]=xv4−i[F3],
for every x ∈ Hi−1(F3) and 1 ≤ i ≤ 4 (this follows from the fact that H∗(E3, Z2) is a free H∗(F3, Z2)-module with basis {1, c, c2, . . . , cn+k−4}, with multiplication determined by the relationcn+k−3=v1cn+k−4+v2cn+k−5+ v3cn+k−6). Sincev1=v1andv3=v13+v3, this gives
P1[E3]=
v3[F3]=(v31+v3)[F3], ifn≡0 mod 4,
(v3+(v12+w21)v1)[F3]=(v3+w21v1)[F3], ifn≡2 mod 4.
To computePi[E3],i = 2,3 and 4, we use the same approach above and the following ingredients:
i) the relationsw3=w1w2=w13=0, which follow from the fact that every 3-dimensional manifold bounds; and the relationsw2v1 = w21v1,w1v12 = 0 and w1v2 = v1v2 +v3, which are valid for every vector bundle over a 3- dimensional manifold (see the proof of the fact stated at the end of the paper).
ii) W2=
n−3 2
c2+(n−4)v1c+(n−3)w1c+v2+w1v1+w2
=
w1c+v2+w1v1+w2, ifn≡0 mod 4, c2+w1c+v2+w1v1+w2, ifn≡2 mod 4;
iii) W3=
n−3 3
c3+
n−4 2
v1c2+
n−3 2
w1c2+(n−5)v2c +(n−4)w1v1c+(n−3)w2c+w2v1+w1v2+v3
=
⎧⎨
⎩
(v2+w2)c+w2v1+w1v2+v3, ifn≡0 mod 4, c3+(v1+w1)c2+(v2+w2)c
+w2v1+w1v2+v3, ifn≡2 mod 4;
iv)
Sq1(W3)†=
⎧⎪
⎨
⎪⎩
(v2+w2)c2+(v1v2+v3)c, ifn≡0 mod 4, c4+(v21+w12+v2+w2)c2
+(v1v2+v3)c, ifn≡2 mod 4;
v)
W1n= n j=0
n j
(v1+w1)jcn−j =cn+ n
2
(v21+w12)cn−2
=
cn, ifn≡0 mod 4,
cn+(v12+w21)cn−2, ifn≡2 mod 4, and
W1n−1=cn−1+(v1+w1)cn−2 +
n−1 2
(v12+w21)cn−3+
n−1 3
(v1+w1)3cn−4
=
⎧⎨
⎩
cn−1+(v1+w1)cn−2
+(v12+w12)cn−3+(v1+w1)3cn−4, ifn≡0 mod 4,
cn−1, ifn≡2 mod 4.
With these data in hand we obtain, by doing a routine calculation, that P2[E3]=
w21v1[F3], ifn≡0 mod 4, v3[F3], ifn≡2 mod 4, P3[E3]=
w1v2[F3], ifn≡0 mod 4, (v3+w12v1+w1v2)[F3], ifn≡2 mod 4, and
P4[E3]=
(w21v1+v3)[F3], ifn≡0 mod 4, (v31+w21v1)[F3], ifn≡2 mod 4.
Taking into account that a cohomology classv ∈ H3(F3, Z2) is zero if and only ifv[F3] = 0, we conclude that our initial system is reduced to the
†In fact, by the Wu formula one has Sq1(v2+w2)=v1v2+v3+w1w2+w3=v1v2+v3; also, Sq1(w2v1+w1v2+v3)=0. The remaining calculation follows from the Cartan Formula.
following systems of equations in the variablesv13, v3, w1v2andw12v1:
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
v31+v3=0 w12v1=0 w1v2=0 w21v1+v3=0
if n≡0 mod 4,
and ⎧
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎩
v3+w21v1=0 v3=0 v3+w21v1+w1v2=0 v13+w21v1=0
if n≡2 mod 4.
In both cases,v31 =v3=v2w1 =w21v1 =0 is the unique solution. Thus, the following fact will end our task:
Fact. The cobordism class of any vector bundle over a 3-dimensional manifoldF3is determined by the Whitney numbersv31[F3],v3[F3],v2w1[F3] andv1w21[F3].
Proof. The complete list of Whitney numbers in this case is w3[F3], w1w2[F3],w13[F3],v3[F3],v1v2[F3],v31[F3],w1v12[F3],w1v2[F3],w21v1[F3] andw2v1[F3].
The first reduction is given by the fact that F3 bounds: w3[F3] = w1w2[F3]=w31[F3]=0.
Now letU =1+u1be the Wu class ofF3. Then
W (F3)=1+w1+w2+w3=Sq(U )=1+u1+u21, which givesw2=u21=w21and the next reduction,w2v1=w21v1.
Ifx ∈ H2(F3, Z2), it is known that Sq1(x) = u1x; also, Sq1(v21) = 0.
Then one has the reductionw1v21=u1v12=Sq1(v12)=0.
Finally, by the Wu formula,
w1v2=u1v2=Sq1(v2)=v1v2+v3, which makesv1v2[F3] unnecessary.
Acknowledgement. We would like to express our gratitude to the referee for suggestions that helped to clarify considerably the original version; we also thank Bjørn Ian Dundas for the guidance concerning the referee suggestions.
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DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE SÃO CARLOS CAIXA POSTAL 676
SÃO CARLOS, SP 13565-905 BRAZIL
E-mail:evelinmbarbaresco@dm.ufscar.br patricia.desideri@gmail.com pergher@dm.ufscar.br
