Singularities and Higher Torsion inSymplectic Cobordism I

Boris I. Botvinnik and Stanley O. Kochman

Abstract

In this paper we construct higher two–torsion elements of all orders in thesymplectic cobordism ring. We begin by constructing higher torsion ele-ments in the symplectic cobordism ring with singularities using a geometricapproach to the Adams-Novikov spectral sequence in terms of cobordismwith singularities. Then we show how these elements determine particularelements of higher torsion in the symplectic cobordism ring.

1 Introduction

The symplectic cobordism ring MSp∗ is the homotopy of the Thom spectrumMSp and classifies up to cobordism the ring of smooth manifolds with an Sp-structure on their stable normal bundles. Although MSp∗ only has two–torsion,its ring structure is far more complicated than any of the other cobordism ringsMG∗ for the classical Lie groups G = O, SO, Spin, U , SU which have beencompletely computed. Over the past thirty years, these cobordism rings MG∗have had a major impact on differential topology and homotopy theory. On theother hand, if the complexity of the ring MSp∗ were understood, then symplecticcobordism theory MSp∗(·) would have the potential to become a powerful toolin algebraic topology.

The symplectic cobordism ring MSp∗ is still far from being computed andunderstood despite much research on the subject over the past twenty years.It seems beyond present methods to completely compute MSp∗ in the nearfuture. Nevertheless, we can try to determine some general structural propertiesof this ring. The most striking example of such a result is the application of theNilpotence Theorem [5] to MSp∗ which says that all of its torsion elements arenilpotent. Another basic structural question is:

Do there exist elements of order 2k in the ring MSp∗ for all k ≥ 1? (1)

Note that the corresponding structural property is well-known for all other classi-cal cobordism rings as well as for framed cobordism, the stable homotopy groupsof spheres. This paper gives an affirmative answer to (1).

1991 Mathematics Subject Classification. Primary 55N22, 55T15, 57R90.This research was partially supported by a grant from the Natural Sciences and EngineeringResearch Council of Canada.

1

2

We begin by describing the background of our research. In the torsion ofMSp∗ there are the fundamental Ray elements [12]: φ0 = η ∈ MSp1 whichcomes from framed cobordism, and φi ∈ MSp8i−3 for i ≥ 1. These are nonze-ro indecomposable elements of order two, and all torsion elements of MSp∗can be constructed from these Ray elements by using Toda brackets. Theseφi determine basic patterns in all approaches to understanding the structure ofsymplectic cobordism. In particular, projections of these elements to the Adamsand Adams–Novikov spectral sequences for MSp have had a major impact onthe description of their structure.

The approach based on the Adams spectral sequence (ASS)

E∗,∗2 = Cotor∗A (H∗ (MSp;Z/2),Z/2)∗ =⇒MSp∗

was developed in North America. Computations through the 29 stem were madeby D. Segal [13] in 1970. Subsequently the second author in [6], [7], [8] comput-ed the E2 and E3 –terms, showed that the spectral sequence does not collapse,computed the image of MSp∗ in N∗, found elements of order four beginning indegree 111 and computed the first 100 stems.

The other approach based on the Adams-Novikov spectral sequence (ANSS)

E∗,∗2 = Ext∗,∗ABP

(BP ∗ (MSp) , BP ∗) =⇒MSp∗.

was developed in the former Soviet Union. In particular, V. Vershinin [15] com-puted the ANSS through the 52 stem and showed that the first element of orderfour in MSp∗ occurs in degree 103 (unpublished).

It became apparent from both approaches that if there was torsion of ordergreater than four in MSp∗ then it would occur in such a high degree that it wouldnot be reasonable to try to discover it through stem by stem computations. Inaddition, there were no candidates for elements in E2 of either the ASS or ANSSwhich might represent elements of higher torsion. (The only such family ofcandidates in the ASS was shown in [7] to be the image of higher differentials.)The determination of elements of higher torsion required new geometric ideas.

V. Vershinin’s paper [16] provided new perspectives for viewing the symplec-tic cobordism ring. He constructed a sequence

MSp∗ −→MSpΣ1∗ −→MSpΣ2

∗ −→ · · · −→MSpΣn∗ −→ · · · −→MSpΣ

∗

of cobordism ringsMSpΣn∗ of symplectic manifolds with singularities which startswith the mysterious ring MSp∗ and ends with MSpΣ

∗ , a polynomial ring over theintegers. In the two-local category, the spectrum MSpΣ splits as a wedge of sus-pensions of the spectrum BP . Here Σ = (P1, . . . , Pn, . . .) and Σn = (P1, ..., Pn)are sequences of closed Sp-manifolds which represent the Ray elements [P1] = ηand [Pi] = φ2i−2 for i ≥ 2. This led to the description of the Adams-Novikovspectral sequences for the spectra MSpΣn in terms of cobordism with singular-ities [2], and, in particular, to a precise formula for the Adams-Novikov differ-ential d1 that reduces the computation of the E2-term to elementary algebraicmanipulations.

3

The opportunity that we have had to work together at York University hasled to the understanding that the geometry of manifolds with singularities canbe used to uncover the deep interaction between the Adams and Adams-Novikovspectral sequences for MSp thereby constructing torsion elements of all orders2k in MSp∗.

First we construct higher torsion elements in MSpΣn∗ for n ≥ 3. The keys

to this construction are that the cobordism ring MSpΣn∗ has new elements

w1, . . . , wn that have the same degrees and behavior as the elements v1, . . . , vn ∈BP∗ and that the Toda brackets 〈φi, wk, φj〉 are defined for k ≤ n. In Section 5,we prove the following theorem.

Theorem A. For each n ≥ 3 and n < i1 < · · · < is, there exist indecomposableelements τn (i1, . . . , is) ∈MSpΣn

4∗+1 with the following properties:

(i) τn(i1) = φ2i1−2 ;

(ii) τn(i1, . . . , is) ∈ 〈φ2is−2 , wn−1, τn(i1, . . . , is−1)〉,

(iii) τn(i1, . . . , is) has order at least 2[(s+1)/2] for s ≥ 1.

Using Bockstein long exact sequences we deduce Theorem B which gives a pos-itive answer to question (1).

Theorem B. For each k ≥ 1 there exist elements of order 2k in the symplecticcobordism ring MSp∗.

However, Theorem B is just an existence theorem. It does not give a particularway to construct higher torsion elements in MSp∗. The remainder of this paper isdevoted to the construction of elements α(i1, . . . , is) ∈MSp∗ from the elementsτ3(i1, . . . , is) ∈MSpΣ3

∗ . Consider the diagram below

MSpΣ3∗ MSpΣ2

∗ MSpΣ1∗ MSp∗-β3 -β2 -β1

MSp∗

@@@I π

We define the elements

α′(i1, . . . , is) = β2 (β3 (τ3(i1, . . . , is)))

in the ring MSpΣ1∗ . Then we construct elements α(i1, . . . , is) ∈ MSp4∗+1 such

that π(α(i1, . . . , is)) and 2α′(i1, . . . , is) in MSpΣ1∗ project to the same elementof E3,4∗+1

2

(MSpΣ1

)in the Adams-Novikov spectral sequence. Finally we prove

that the α(i1, . . . , is) ∈MSp4∗+1 are elements of higher order in MSp∗.

Main Theorem. The element α (i1, . . . , is) ∈ MSp4∗+1 has order at least2[(s+1)/2]−3 for s ≥ 7 and 3 ≤ i1 < · · · < is.

4

We describe the contents of this paper in detail. In Section 2, we summarizethe basic facts about the spectra MSpΣn which we will be using. In Section3, we give the definition and basic properties of three-fold Toda brackets ofmanifolds with singularities. These Toda brackets will be used to inductivelydefine the elements we construct. In Section 3, we study the Adams-Novikovspectral sequence for the MSpΣn . The key technical and conceptual fact we useis that the Adams-Novikov spectral sequence for the spectrum MSp∗ coincideswith the Σ-singularities spectral sequence which is defined in terms of cobordismwith singularities [2]. The Σ-singularities spectral sequence gives us a specificresolution M〈n〉 for computing E2 of the ANSS for MSpΣn . In particular, weidentify torsion elements tn (i) of all orders 2k in the first line of the ANSS

E1,∗2 = Ext1,∗

ABP

(BP ∗

(MSpΣn

), BP ∗

).

Let i = (i1, . . . , is). In Section 5, we use Toda brackets to construct elementsτn (i) in MSpΣn

∗ and prove Theorems A and B. In particular, the element τn (i)has order at least 2[(s+1)/2] because it projects in the Adams-Novikov spectralsequence to the infinite cycle tn (i) that has order 2[(s+1)/2] in E1,∗

2 (MSpΣn). Insection 6, we study the elements

γ (i) = β3 (τ3 (i)) ∈MSpΣ24∗+3 and α′ (i) = β2 (γ (i)) ∈MSpΣ1

4∗+1

and identify their projections to the third line of the ANSS. In Section 7 weprove the Main Theorem by projecting the elements α′(i1, . . . , is) to the thirdline E3,∗

2 (MSpΣ1) of the ANSS. We use chromatic arguments to compute theorder of these projections in E3,∗

2 , and we show that they can not be killed byd3–differentials.

In [3] we compute the E2–terms of the Adams spectral sequences for the spec-tra MSpΣn and apply them to study the elements of higher torsion constructedin this paper.

All groups, rings and spectra are two-local throughout this paper.

The first author would like to thank the topology community at M.I.T. fortheir warm hospitality during his visit in the spring term of 1991 as well asthe Department of Mathematics and Statistics at York University for their kindsupport and hospitality. In addition he would like to thank Haynes Miller forimportant discussions on the basic ideas of this paper.

2 Symplectic Cobordism with Singularities

In this section we collect basic constructions and theorems concerning the spectraMSpΣ

∗ and MSpΣn∗ of symplectic cobordism with singularities. In particular, we

determine formulas for computing the Bockstein operators which will be used inSections 4 and 7 to make computations in the ANSS for MSpΣn .

Let Σ = (P1, ..., Pn, ...) be a sequence of closed Sp-manifolds representing theRay elements such that [P1] = η and [Pi] = φ2i−2 for i ≥ 2. For n ≥ 1, let Σn

5

denote the sequence (P1, ..., Pn). The bordism theory of Sp-manifolds with Σn-,Σ-singularities is denoted by MSpΣn

∗ (·), MSpΣ∗ (·) respectively. By [2], [16], the

theory MSpΣn∗ (·) has an admissible product structure, and the coefficient ringMSpΣn

∗ is polynomial up to dimension 2n+2 − 3. (See [16], [2, Theorem 3.3.3].)The following theorem describes the structure of the ring MSpΣ

∗ .

Theorem 2.1 (V.Vershinin [16]) There exists an admissible product structurein the theory MSpΣ

∗ (·) such that its coefficient ring MSpΣ∗ is isomorphic to the

polynomial ring

MSpΣ∗∼= Z(2) [w1, . . . , wj , . . . , x2, x4, x5, . . . , xm, . . .]

where degwj = 2(2j − 1) for j = 1, 2, . . . and deg xm = 4m for m = 2, 3, 5, . . .,m 6= 2s − 1. The generators wj are represented by Sp-manifolds Wj such that∂Wj = 2Pj

In fact, the cobordism theory MSp∗Σ(·) splits as a sum of the theories BP ∗(·).

Theorem 2.2 [2, Corollary 3.5.3] The ring spectrum MSpΣ splits as

MSpΣ = BP ∧M(G)

where G = Z(2) [x2, . . . , xm, . . .], m = 2, 4, 5, . . ., m 6= 2l − 1, deg xm = 4m andM(G) is a graded Moore spectrum.

Note 2.1 Theorem 2.2 implies that the ANSS based on the cohomology theoriesBP ∗(·) and MSp∗Σ(·) are isomorphic.

There are Bockstein operators in the theory MSpΣ∗ (·) for i ≥ 1:

βi : MSpΣ∗ (·) −→MSpΣ

∗ (·).

They have the following properties:

βi βi = 0, and βi βj = βj βi.

In general a product formula for Bockstein operators acting on a bordism theorywith singularities is too complicated to write down. However in our case thisformula has the following simple form.

Theorem 2.3 [2, Theorem 4.2.4] The product structure and the elements wi inTheorem 2.1 may be chosen in such a way that the Bockstein operators βi, i ≥ 1,satisfy the product formula:

βi(x · y) = (βix) · y + x · (βiy)− wi · (βix) · (βiy) (2)

where x, y ∈MSpΣ∗ .

6

To describe the action of the Bockstein operators on the polynomial generatorsof MSpΣ

∗ we introduce the following notation. Let m+ 1 = 2i1−1 + · · · + 2is−1

be a binary decomposition of the integer m+ 1 where 1 ≤ i1 < i2 < · · · < is. Ifm is odd, the generator xm is denoted by xi1,...,is . If m = 2i−1 with 1 ≤ i thenthe generator xm is denoted by x1,i.

Theorem 2.4 [2, Theorem 4.5.1] There are generators xm of the ring MSpΣ∗

such that the formulas below describe the action of Bockstein operators βk on xmfor k ≥ 2.

1. If m = 2i−1 + 2j−1 − 1, 1 ≤ i < j then

βixi,j = wj , βjxi,j = wi, and βkxi,j = 0 if k 6= i, j. (3)

2. If m = 2i1−1 + · · ·+ 2is−1 − 1, 2 ≤ i1 < i2 < · · · < is and s ≥ 3, then

βkxi1,...,is =

w1 · xi1,...,it,...,is , if k = it

0 if k 6= i1, ..., is

. (4)

3. If m is even and not a power of two then

βkxm = 0 . (5)

Formulas (2)–(5) are the ones we will use in Sections 4, 5 and 6 to make compu-tations in the ANSS for the spectra MSpΣn . Note that (2) and (3) are invariantunder permutations π of the subscripts where π is a permutation of the set ofintegers greater than one.

3 Toda Brackets in MSpΣn∗

In this section we extend the construction of Alexander [1] to define triple Todabrackets in the ring MSpΣn

∗ . Note that we do not claim that all of the usualproperties of Massey products [10] and Toda brackets [14] generalize to cobor-dism with singularities. The problem is that the ring spectrum MSpΣn probablydoes not admit an H∞-structure. Therefore, we only study those Toda bracketswhich we will use, and we only prove those properties of Toda brackets whichwe will need.

We begin by recalling from [2, Chapter 2] the construction of an admissibleproduct µn in the bordism theory MSpΣn

∗ (·). We summarize the properties ofthe associativity construction An which measures the lack of associativity of µnand the properties of the commutativity construction Kn which measures thelack of commutativity of µn. We will use An in the definition of Toda brackets,and the properties of An and Kn will be used to prove the properties of theseToda brackets. In addition, Kn is a geometric cup-one product of manifolds with

7

singularities which satisfies the usual boundary and Hirsch formulas. We willuse it in Sections 5 and 6 in our constructions of higher torsion elements.

Let Pn be as above with [P1]Sp = η and [Pn]Sp = φ2n−2 for n ≥ 2. Considerthe manifold

P ′n = P (1)n × P (2)

n × I.

Here P(1)n , P

(2)n are two copies of the Σn–manifold Pn such that:

∂P ′n = βnP′n × Pn and

βnP′n = P (1)

n × 0 ∪ P (2)n × 1 .

Note that the Σn-manifold P ′n is Σn-bordant to a manifold without singularitiessince 2 [Pn]Sp = 0. The cobordism class of the Σn-manifold P ′n is the obstructionto the existence of a product structure. In our case, [P ′n]Σn = 0, and we let Qndenote a Σn-manifold such that δQn = P ′n as in [2, Theorem 4.2.4]. Thus, wehave the following product construction of [2, Theorem 2.2.2].

A product mn(Aa, Bb

)of two Σn-manifolds is defined by induction on n as

follows:

m1

(Aa, Bb

)= Aa ×Bb ∪ (−1)bβ1A

a × β1Bb ×Q1

and for n ≥ 2

mn(Aa, Bb

)= mn−1

(Aa, Bb

)∪ (−1)bmn−1

(mn−1

(βnA

a, βnBb), Qn

).

In particular, if C is an Sp-manifold without singularities then

mn (X,C) = X × C and mn (C,X) = C ×X.

We have the following diffeomorphisms of Σn-manifolds:

δmn(Aa, Bb) = mn(δA,B) ∪ (−1)amn(A, δB),

C ×mn(A,B) = mn(C ×A,B), mn(A,B)× C = mn(A,B × C), (6)

mn(A,C ×B) = mn(A× C,B).

By [2, Theorem 3.3.3], this product of Σn-manifolds mn determines an admis-sible product structure µn in the theory MSpΣn∗ (·) which is commutative andassociative. At the level of Σn–manifolds commutativity of µn means that for allΣn-manifolds Aa, Bb there exists a Σn-manifold Kn

(Aa, Bb

)called the canonical

commutativity construction which is functorial in the category of Σn-manifoldsand satisfies the formula

δKn(A,B)=mn(A,B)∪(−1)abmn(B,A)∪−Kn(δA,B)∪(−1)a+1Kn(A, δB). (7)

See [2, Definition 2.1.3].

8

Associativity of µn at the level of Σn–manifolds means that for all Σn-manifoldsAa, Bb Cc there exists a Σn-manifoldAn

(Aa, Bb, Cc

)called the canon-

ical associativity construction which is functorial in the category of Σn-manifoldsand satisfies the formula

δAn(Aa, Bb, Cc

)= mn(A,mn(B,C)) ∪ −mn(mn(A,B), C) ∪ −An(δA,B,C)

∪(−1)a+1An(A, δB,C) ∪ (−1)a+b+1An(A,B, δC). (8)

See [2, Definition 2.1.3].

Let Aa and Bb be Sp-manifolds without singularities. Then Kn(A,B) can betaken to be the cylinder

Kn(A,B) = I × A×B

with an Sp–structure such that there is a diffeomorphism preserving Sp–structures:

∂ (I ×A×B) = A×B ∪−(−1)abB ×A ∪ −I × ∂A×B ∪ (−1)a+1I ×A× ∂B.

In this case Kn(A,B) has been described [7, section 10] in the special categoryof manifolds as a “cup-one product of manifolds”. It projects in E1 of the ASSto an algebraic cup-one product. Moreover, the Sp–structure on Kn(A,B) canbe chosen so that Kn satisfies the Hirsch formula [9]. Using the definition ofKn(A,B), this property generalizes to the two cases of Σn–manifolds given inthe following lemma which suffice for the constructions of this paper. Statement(b) is nontrivial; it is essential that the Σn–manifolds Aa and Cc have only onecommon singularity, i.e. there is only one i ≤ n, such that both βiA 6= ∅ andβiC 6= ∅.

Lemma 3.1 (a) (Hirsch Formula) If Cc is an Σn–manifold and one of theΣn–manifolds Aa, Bb is an Sp–manifold without singularities then we have adiffeomorphism of Σn–manifolds preserving Sp–structures:

Kn(A×B,C) = (−1)amn (A,Kn(B,C)) ∪ (−1)bcmn(Kn(A,C), B). (9)

(b) (Generalized Hirsch Formula) If Aa and Cc are closed Σn–manifolds thathave only one nonempty common singularity and c is even then there is aΣn–cobordism between the manifolds

Kn(mn(C,A), C)∪An(C,A,C) and mn(C,Kn(A,C))∪mn(Kn(C,C), A). (10)

Comments on the Proof. (a) It is proved by comparison of the constructionson the left and right sides of equation (9).

(b) The obstruction to the associativity of the product structure µn hasorder three in the group MSpΣn∗ ; see [2, Lemma 2.4.2]. Since the group MSpΣn∗does not have any odd torsion, the associativity construction An may be taken

9

to be a cylinder. This gives a way to construct a cobordism between the Σn–manifolds in (10). The construction of this cobordism is straightforward whenthe manifolds A, C have only one common singularity.

Now we are ready to define the Toda bracket 〈a, b, c〉, where a, b, c ∈MSpΣn∗ .

Since the product of Σn–manifolds is not associate we need to use an associativityconstruction (8) to glue together the two usual pieces which define such a bracketin an associative context. We use the standard sign conventions of [10].

Definition 3.2 Let a, b, c ∈MSpΣn∗ such that ab = 0 and bc = 0. Let A, B, C

be a Σn-manifold which represents a, b, c, respectively. Let X, Y be Σn-manifoldssuch that δX = mn(A,B) and δY = mn(B,C). Then

δmn(X,C) = mn(mn(A,B), C), and δmn(A, Y ) = (−1)degA

mn(A,mn(B,C)).

Let the Toda brackets 〈a, b, c〉 be the set of all cobordism classes of Σn-manifoldsof

Z = (−1)1+degB

mn(X,C)∪ (−1)1+degB

An (A,B,C)∪ (−1)degA+degB

mn(A, Y ).

Note 3.1 Let the Toda bracket 〈a, b, c〉 be defined where the element a is repre-sented by a closed Sp-manifold A. In this case the Σn-manifold An (A,B,C) isjust the cylinder

An (A,B,C) = I ×A×mn(B,C);

see [2, Theorem 2.5.1]. Thus, the following Σn-manifold Z represents an elementof 〈a, b, c〉:

Z=(−1)1+degB

mn(X,C)∪(−1)1+degB

I×A×mn(B,C)∪(−1)degA+degB

mn(A, Y ).

By properties of mn, An (see [2, Section 2.2]), the Σn–manifold Z dependsonly on the cobordism classes a, b, c and on the choice of the Σn–manifolds Xand Y with δX = mn(A,B) and δY = mn(B,C). Therefore we have the usualindeterminacy

in〈a, b, c〉 =ay + xc | x, y ∈MSpΣn

∗.

We define a generalized quadratic construction which we use in the next lem-ma to identify Toda brackets of the form 〈a, b, a〉. Suppose M is a Σn–manifoldof dimension 2k. Define a closed Σn-manifold ∆(M) as follows:

∆(M) = mn(M (1),M (2)

)× I ∪ −Kn

(M (2),M (1)

)(11)

where we identify the following manifolds:

mn(M (1),M (2)) = mn(M (1),M (2))

?

?

mn(M (1),M (2))× 0 −δKn(M (2),M (1)),

10

−mn(M (2),M (1)) = −mn(M (2),M (1))

?

?

mn(M (1),M (2))× 1 −δKn(M (2),M (1))

Note that in the case where M is a manifold without singularities, the manifold∆(M) is just the quadratic construction.

Lemma 3.3 Let the Toda bracket 〈a, b, a〉 be defined in MSpΣn∗ , where a = [M ]

and b = [R] for M a Σn-manifold of dimension 2k and R a Sp-manifold. Then

〈a, b, a〉 =

(−1)1+deg b

b [∆(M)] + ax | x ∈MSpΣn∗

. (12)

Proof. Throughout this proof we ignore trivial associativity constructions inwhich one of the three entries has empty singularities. Let Y be a Σn–manifoldsuch that δY = R×M . Then

δ (Y ∪ (R×M × I) ∪ −Kn (R,M)) = M ×R,and the Σn-manifold

C = mn(Y,M) ∪mn(R×M,M)× I ∪ −mn (Kn (R,M) ,M) ∪−mn (M,Y )

is a representative of the Toda bracket (−1)1+deg b 〈M,R,M〉. Glue the Σn-

manifold Kn (Y,M) to the cylinder C× I by identifying the following manifolds:

mn(Y,M)× 1 = mn(Y,M)

?

?

C × 1 δKn (Y,M)

−mn(M,Y )× 1 = −mn(M,Y )

?

?

C × 1 δKn (Y,M)

mn (K (R,M) ,M)× 1 = mn (K (R,M) ,M)

?

?

C × 1 δKn (Y,M)

The boundary of the resulting Σn-manifold Z is given by

δZ = −R×∆(M) ∪C × 0since, using the Hirsch formula,

δKn (Y,M)∩δZ = −Kn (δY,M)∩δZ = −Kn (R×M,M)∩δZ = −R×Kn (M,M) .

See Figure 1. Thus, (−1)1+deg b 〈a, b, a〉 contains [∆ (M)] b and in〈a, b, a〉 =

aMSpΣn∗ . Therefore, (12) holds.

11' $

$' $'

mn(Y,M) mn (R×M,M)× I −mn (Kn (R,M) ,M) −mn (M,Y )

Kn (Y,M)

−R× Kn (M,M)

−R×∆(M)

@@@R

mn(Y,M)×1 R×mn (M,M)×I×1 −mn (Kn (R,M) ,M)×1 −mn (M,Y )×1

Figure 1: The Σn-manifold Z.

The next property is well known for manifolds without singularities. See [1,Definition 2.1(5)].

Lemma 3.4 Let 〈a, b, c〉 be a Toda bracket which is defined in the ring MSpΣn∗ .

Assume that a = [A] is represented by a closed Sp-manifold, and b = [B], c = [C]are represented by Σn-manifolds. Then the following inclusion holds in the ringMSpΣn

∗ :

b〈a, b, c〉 ⊂ (−1)deg a 〈b, a, b〉c.

Proof. Let X, Y be Σn–manifolds such that δX = A×B and δY = mn(B,C).

Then the following Σn–manifold Z represents an element of (−1)1+deg b 〈a, b, c〉:

Z = mn(X,C) ∪−I ×A×mn(B,C) ∪ (−1)1+deg a

A× Y.

Thus, the Σn–manifold mn(B,Z) represents an element of (−1)1+deg bb〈a, b, c〉,

and using (6):

mn(B,Z) = mn(B,mn(X,C)) ∪ −mn(B, I ×A×mn(B,C))∪

(−1)1+deg a

mn(B,A× Y )

= mn(B,mn(X,C)) ∪ (−1)1+deg b

I ×mn(B ×A,mn(B,C))∪

(−1)1+deg a

mn(B ×A, Y ).

12

−σbI×An(B×A,B,C)−An(B,X,C)

σamn(X, Y )

σbAn(X, B,C)

σbmn(X,mn(B,C))

@@@R

−σbmn(mn(X, B), C)

@@@R

mn(B,mn(X,C)) −σbI×mn(B×A,mn(B,C)) −σamn(B ×A, Y )

−σbI×mn(mn(B×A,B), C)mn(mn(B,X), C)

V1@

@@I

Figure 2: Σn-manifold V2, where σa = (−1)deg a, σb = (−1)deg b.

Let X be a Σn– manifold such that δX = B × A. Glue (−1)deg a

mn(X, Y ) tothe cylinder mn(B,Z)× I along their common boundary:

mn(B ×A, Y ) = mn(B ×A, Y )

?

?

δmn(B,Z)× 1 δmn(X, Y ) .

Denote the resulting Σn-manifold as V1; see Figure 2. Now consider the Σn-manifolds

An(B,X,C), I × An(B ×A,B,C), An(X, B,C)

with boundaries:

δAn(B,X,C)

= mn(B,mn(X,C)) ∪ −mn(mn(B,X), C) ∪ (−1)deg b+1

An(B, δX,C)

= mn(B,mn(X,C)) ∪ −mn(mn(B,X), C) ∪ (−1)deg b+1

An(B,A×B,C)

= mn(B,mn(X,C)) ∪ −mn(mn(B,X), C) ∪ (−1)deg b+1

An(B ×A,B,C);

13

δ (I × An(B ×A,B,C)) =

I × −mn(B ×A,mn(B,C)) ∪mn(mn(B ×A,B), C) ∪ ∂I × An(B ×A,B,C);

δAn(X, B,C) = mn(X,mn(B,C)) ∪ −mn(mn(X, B), C) ∪ −An(δX,B,C)

= mn(X,mn(B,C)) ∪ −mn(mn(X, B), C) ∪ −An(B × A,B,C).

Now we glue together the Σn-manifolds

V1, −An(B,X,C), (−1)1+deg bI×An(B×A,B,C), (−1)deg bAn(X, B,C).

The resulting Σn-manifold V2 gives a bordism between the Σn-manifolds

mn(B,Z)× 0 and

mn

(−mn(B,X) ∪ (−1)

deg bI ×mn(B ×A,B) ∪ (−1)

deg bmn(X, B), C

).

The latter Σn–manifold represents an element of (−1)1+deg a + deg b 〈b, a, b〉c.

Thus,(−1)deg b+1

b〈a, b, c〉 ⊂ (−1)deg a+deg b+1 〈b, a, b〉cand b〈a, b, c〉 ⊂ (−1)deg a 〈b, a, b〉c, as required.

4 Adams-Novikov Spectral Sequence for MSpΣn

Recall that our plan for determining elements of higher order in the ring MSp∗is to construct Σn–manifolds which project to infinite cycles in the E2-terms ofthe ANSS and ASS of MSpΣn∗ for n ≥ 3. Then we determine the order of theseprojections in E2 of the ANSS and bring back these Σn–manifolds to MSp∗. Inthis section, we accomplish the first part of our program by describing particulartorsion elements t(i1, . . . , i2s) of higher order in the first line

E1,4∗+12 (MSpΣn) = Ext1,4∗+1

ABP(BP ∗(MSpΣn), BP ∗)

of the ANSS which are the projections of the Σn–manifolds which we will con-struct in Section 5.

Throughout this section, let n ≥ 0 be a fixed integer, and let MSpΣ0 denoteMSp. In [2, section 1.6], the ANSS for each of the spectra MSpΣn is described interms of geometrical constructions on manifolds with singularities. In particular,the ANSS for the spectrum MSpΣn is identified with the Σ-singularities spectralsequence (Σ-SSS) associated with the exact couple:

MSpΣn∗

@@@@R

π(0)n

MSpΣΓ(1)n∗

@@@@R

π(1)n

MSpΣΓ(2)n∗

@@@@R

π(2)n

· · · γ(1)n γ(2)n γ(3)n

MSpΣ(1)n∗

∂(1)n

MSpΣ(2)n∗

∂(2)n

· · ·-β(1)n -β(2)n

(13)

14

Here MSpΣΓ(k)n∗ , MSp

Σ(k)n∗ are the coefficient groups of particular bordism the-

ories closely related to the theories MSpΣn∗ (·), MSpΣ∗ (·); see [2, section 1.4].

The E∗,∗2 -term of the Σ-SSS (or the ANSS) is described as follows. Consider thebigraded commutative algebra:

M〈n〉 = MSpΣ∗ [un+1, un+2, . . .un+k, . . .]

where |un+k| = (1, 2(2n+k − 1)), and |x| = (0,deg x) for x ∈MSpΣ∗ . Let

M〈n〉s = z ∈M〈n〉 | |z| = (s, ∗) .

As we shall see, M〈n〉s is isomorphic to the s-th line Es,∗1 of the ANSS forMSpΣn . We have the following complex:

M〈n〉0D〈n〉−−−→M〈n〉1

D〈n〉−−−→M〈n〉2D〈n〉−−−→ · · · D〈n〉−−−→M〈n〉k

D〈n〉−−−→ · · · . (14)

The differential D〈n〉 is defined as

D〈n〉(xua1

i1· · ·uajij

)=

j∑t=1

(−1)εt(α)(

(βitx)ua1i1· · ·uat+1

it· · ·uajij

)where n < i1 < . . . < ij , α = (a1, ..., aj) is a sequence of nonnegative integers

and εt(α) =∑ti=1 ai. It follows from the product formula (2) for Bockstein

operators, that the subalgebra of cycles of the algebraM〈n〉 is a DGA. Therefore,the homology H∗(M〈n〉) of the complexM〈n〉 has an induced algebra structurefromM〈n〉.

Note 4.1 The elements ui, i = n+ 1, n+ 2, . . . are the projections of the “basicRay elements” u1 = η, ui = φ2i−2 for i ≥ 2. We use the same notation for theseelements and their projections to the ANSS.

Theorem 4.1 [2, Theorems 3.4.1, 4.4.5]

(i) The exact couple (13) is an Adams resolution of the spectrum MSpΣn inthe theory BP ∗(·).

(ii) There is an isomorphism of algebras

E2(MSpΣn) = Ext∗,∗ABP

(BP ∗(MSpΣn), BP ∗) ∼= H∗(M〈n〉).

In particular, there is a ring isomorphism

E0,∗2 = Hom∗ABP (BP ∗(MSpΣn)), BP ∗)

∼= H0(M〈n〉) =⋂∞k=n+1 Ker(βk : MSpΣ

∗ −→MSpΣ∗ ).

Note 4.2 The complex M〈n〉 in (14) is the bottom line of the diagram (13). Inparticular the first Adams-Novikov differential D〈n〉 :M〈n〉0 −→M〈n〉1 is givenby

D〈n〉 = β(1)n =∞⊕

k=n+1

βk.

15

Now we are ready to use the above results to set up the environment in whichwe will do our chromatic calculations of E2(MSpΣn

∗ ). Since the ring MSpΣ∗ is

polynomial and wm is one of its generators, we have the following exact sequence:

0 −→MSpΣ∗·wkm−−→MSpΣ

∗π(k)

−−→MSpΣ∗ /(w

km) −→ 0 (15)

where (wkm) is the principal ideal generated by wkm and π(k) is the natural projec-tion. Since D〈n〉(wm) = 0, the exact sequence (15) induces the exact sequenceof complexes:

0 −→M〈n〉 ·wkm−−→M〈n〉 π(k)

−−→M〈n〉/(wkm) −→ 0 . (16)

We paste the sequences (16) together to obtain the following commutative dia-gram:

0−−−−→M〈n〉 ·wm−−−−→M〈n〉 π(1)

−−−−→M〈n〉/(wm)−−−−→ 0

Id

∣∣∣∣y ·wm

∣∣∣∣y ·wm

∣∣∣∣y0−−−−→M〈n〉 ·w2

m−−−−→M〈n〉 π(2)

−−−−→M〈n〉/(w2m)−−−−→ 0

Id

∣∣∣∣y ·wm

∣∣∣∣y ·wm

∣∣∣∣y...

......

Id

∣∣∣∣y ·wm

∣∣∣∣y ·wm

∣∣∣∣y0−−−−→M〈n〉 ·wkm−−−−→M〈n〉 π(k)

−−−−→M〈n〉/(wkm)−−−−→ 0

Id

∣∣∣∣y ·wm

∣∣∣∣y ·wm

∣∣∣∣y...

......

(17)

Taking the direct limit of the rows of (17), we obtain the following short sequenceof complexes:

0 −→M〈n〉 −→ w−1m M〈n〉

π−→M〈n〉/(w∞m ) −→ 0 (18)

where

w−1m M〈n〉 = lim

−→

(M〈n〉 ·wm−−→M〈n〉 ·wm−−→ · · ·

)and

M〈n〉/(w∞m ) = lim−→

(M〈n〉/(wm)

·wm−−→M〈n〉/(w2m)

·wm−−→ · · ·).

The short exact sequence of complexes (18) induces the following long exactsequence in homology:

0→ H0(M〈n〉) −→ H0(w−1m M〈n〉) −→ H0(M〈n〉/w∞m )

δn−→ H1(M〈n〉) −→

H1(w−1m M〈n〉) −→ H1(M〈n〉/w∞m )

δn−→ H2(M〈n〉)→ · · ·(19)

The key point about (19) is that the complex w−1m M〈n〉 is acyclic.

16

Lemma 4.2 For n ≥ 1, n ≥ m ≥ 0 and s ≥ 1:

Hs(w−1m M〈n〉) = 0. (20)

It follows that we have the exact sequence

0→ H0(M〈n〉) −→ H0(w−1m M〈n〉) −→ H0(M〈n〉/w∞m )

δm−→ H1(M〈n〉)→ 0,(21)

and for s ≥ 1 we have group isomorphisms

Hs(M〈n〉/w∞m ) ∼= Hs+1(M〈n〉).

Proof. Consider the following subalgebra of w−1m M〈n〉:

L(i1, . . . , ik) = w−1m MSp∗ [ui1 , . . . , uik ] ,

where n < i1 < · · · < ik. This subalgebra is closed under the differential D〈n〉,so it is also a subcomplex of w−1

m M〈n〉. To prove (20) it is enough to show that

Hs(L(i1, . . . , ik)) ∼= 0 for all k and s ≥ 1. (22)

We prove (22) by induction on k.

The case k = 1: Hs(L(i)) ∼= 0 for s ≥ 1.

By Theorem 2.4, βixm,i = wm. In terms of the algebra L(i) this means that

D〈n〉(w−1m xm,i

)= ui.

Let xuai ∈ L(i), a ≥ 1, be any element such that D〈n〉 (xuai ) = (βix)ua+1i = 0.

Then βix = 0 and

D〈n〉(w−1m xm,ixu

a−1i

)= βi

(w−1m xm,ix

)ua−1i = xuai .

The induction step: Hs(L(i1, . . . , ik−1)) ∼= 0 =⇒ Hs(L(i1, . . . , ik)) ∼= 0.

We have the following exact sequences of complexes:

0 −→ L(i1, . . . , ik−1) −→ L(i1, . . . , ik) −→ L(ik) −→ 0 . (23)

The long exact sequence determined by (23) implies that Hs(L(i1, . . . , ik)) ∼= 0for s ≥ 1.

Now let n ≥ 3. We describe the structure of the subring

H0(w−1n−1M〈n〉) ⊂ w−1

n−1MSpΣ∗ .

Let wi, xi,j , xi1,...,is be the polynomial generators of the ring MSpΣ∗ described

in Theorem 2.4. Define the following polynomial generators of w−1n−1MSpΣ

∗ :

Zj = xj,n−1, j = 1, 2, . . . , n− 2, Zn = xn−1,n; (24)

17

Xi =2xn−1,i

wn−1− wi, Yi =

xn−1,i

w2n−1

(xn−1,i − wiwn−1), i ≥ n+ 1. (25)

Note thatX2i = 4Yi + w2

i .

Let 1 ≤ i < j; i, j 6= n− 1. Then define

Xi,j = xi,j −wixn−1,j − wjxn−1,i

wn−1+

2xn−1,ixn−1,j

w2n−1

. (26)

Note that if 1 ≤ i < n− 1, j 6= n− 1, then we could have chosen the polynomialgenerators

X ′i,j = xi,j −wixn−1,j

wn−1. (27)

We can also choose polynomial generators Xi1,...,is of w−1n−1MSpΣ

∗ for s ≥ 3as xi1,...,is + · · ·. We only need their existence. Their exact definition, is notnecessary for our computations. However, for completeness, we define them asfollows.

If 1 < i1 < · · · < is, i1, . . . , is 6= n− 1 then define

Xi1,...,is = xi1,...,is

+s−2∑k=1

(−1)kw1

wkn−1

∑1≤t1<···<tk≤s

xn−1,it1· · ·xn−1,itk

xi1,...,it1 ,...,itk ,...,is

+ (−1)s−1w1

(∑st=1witxn−1,i1 · · · xn−1,it · · ·xn−1,is

ws−1n−1

− 2xn−1,i1 · · ·xn−1,is

wsn−1

).

If i1 < · · · < is and im = n− 1 then define

Xi1,...,is = xi1,...,is

+s−m−2∑k=1

(−1)kw1

wkn−1

∑m+1≤t1<···<tk≤s

xn−1,it1· · ·xn−1,itk

xi1,...,it1 ,...,itk ,...,is

+ (−1)s−m−1

(w1

ws−1n−1

s∑t=m+1

xn−1,i1 · · · xn−1,it · · ·xn−1,isxi1,...,im,it

)

+ (−1)s−1δ1m

w1

wsn−1

xn−1,i2 · · ·xn−1,is

+ (−1)s−m(1−δ1m)

w1

ws−1n−1

xn−1,im+1 · · ·xn−1,isxi1,...,im

where δ1m is the Kronecker delta.

We have the following two polynomial subrings of the polynomial ringMSpΣ∗ :

P∗ = Z(2)

[xm| m even & m 6= 2t

], W 〈n〉∗ = Z(2) [w1, . . . , wn−2] .

18

We will also need the following polynomial subrings of w−1n−1MSpΣ

∗ :

R〈n〉1 = Z(2) [Zj,Xi, Yi | j = 1, . . . , n− 2, n, i ≥ n+ 1] ,

R〈n〉2 = Z(2) [Xi,j | 1 < i < j, i, j 6= n− 1] ,

R〈n〉3 = Z(2) [Xi1,...,is | s ≥ 3, 1 < i1 < · · · < is] ,

R〈n〉 = R〈n〉1 ⊗R〈n〉2 ⊗R〈n〉3.

Lemma 4.3 There is a ring isomorphism:

H0(w−1n−1M〈n〉) ∼= Z(2)

[wn−1, w

−1n−1

]⊗W〈n〉∗ ⊗P∗ ⊗R〈n〉

Proof. There is a ring isomorphism:

w−1n−1MSpΣ

∗∼= Z(2)

[wn−1, w

−1n−1

]⊗W〈n〉∗ ⊗P∗ ⊗ T ⊗R〈n〉2 ⊗R〈n〉3,

where

T = Z(2) [wn−1, . . . , wn+k, . . . , x1,n−1, . . . , xn−2,n−1, xn−1,n, . . . , xn−1,n+k, . . .]

Since the subring T ofMSpΣ∗ is closed under the action of the Bockstein operators

βj for j ≥ n+ 1, T generates the subcomplex

T = T [un+1, . . . , un+k, . . .] (28)

of M〈n〉. To prove this lemma, it suffices to establish the isomorphism:

H0(T ) ∼= w−1n−1R〈n〉1. (29)

For k ≥ 1, define the following subrings of w−1n−1MSpΣ

∗ :

T (k) = Z(2) [wn−1, . . . , wn+k, x1,n−1, . . . , xn−2,n−1, xn−1,n, . . . , xn−1,n+k] ,

T(k)0 = Z(2) [wn−1, wn+k, xn−1,n+k] ,

R(k) = Z(2) [wn−1, Zj ,Xi, Yi | j = 1, . . . , n− 2, n, i = n+ 1, . . . , n+ k] ,

R(k)0 = Z(2) [wn−1,Xn+k, Yn+k] .

Since the rings T (k), T(k)0 are also closed under the action of the Bockstein

operators βj for j ≥ n + 1, we can define the subcomplexes T (k) and T (k)0 of

M〈n〉 as in (28). Direct computation shows that

H0(w−1n T

(k)0 ) ∼= w−1

n−1R(k)0 . (30)

19

In particular, we have the isomorphisms:

H0(w−1n−1T (1)) ∼= H0(w−1

n−1T(1)

0 )⊗Z(2) [x1,n−1, . . . , xn−2,n−1, xn−1,n]

∼= w−1n−1R

(1)0 ⊗Z(2) [x1,n−1, . . . , xn−2,n−1, xn−1,n] = w−1

n−1R(1).

By induction on k ≥ 1, we have the a homomorphism of short exact sequencesof complexes

0 w−1n−1T (k) w−1

n−1T (k+1) w−1n−1T

(k+1)0 0- - - -

0 w−1n−1R

(k)

6

w−1n−1R

(k+1)

6

w−1n−1R

(k+1)0

6

0- - - -

where the complexes on the bottom line have zero differential. The left and rightvertical maps induce isomorphisms in homology. Thus, the Five Lemma com-pletes the induction proof of the isomorphisms (30). Taking the direct limit overk of the isomorphisms (30) establishes the isomorphism (29).

Define

tn(i1, . . . , i2s) = δn

(xn−1,i1 · · ·xn−1,i2s

wn−1

)∈ H1(M〈n〉) (31)

where 3 ≤ n < i1 < · · · is, δn is the boundary homomorphism of (21) and

H1(M〈n〉) ∼= Ext1,∗ABP

(BP ∗(MSpΣn), BP ∗).

These are the required elements of higher torsion in the first line of the ANSS.

Proposition 4.4 The element

tn(i1, . . . , is) ∈ Ext1,4∗+1ABP

(BP ∗(MSpΣn), BP ∗)

has order 2[(s+1)/2] for any sequence (i1, . . . , is), s ≥ 1, 3 ≤ n < i1 < · · · < is.

For convenience we prove Proposition 4.4 assuming that s is even. The prooffor the case when s odd is a slight modification of the even case. The followingtechnical lemma will be used to show that 2s−1tn(i1, . . . , i2s) 6= 0.

Lemma 4.5 There does not exist an element Y ∈MSpΣ∗ , such that the element

Z = 2s−1xn−1,i1 · · ·xn−1,i2s − wn−1Y, (32)

belongs to the ring H0(M〈n〉).

20

Proof. By the definition of Xi,j in (26),

ξ(i, j) = 2xn−1,ixn−1,j − wn−1(wixn−1,j − wjxn−1,i) + w2n−1xi,j

= w2n−1Xi,j ∈MSpΣ

∗ .

Thus, ξ(i, j) ∈ H0(M〈n〉), and there is an element a ∈MSpΣ∗ such that

ξ(i1, i2) · · · ξ(i2s−1, i2s) = 2sxn−1,i1 · · ·xn−1,i2s − wn−1a ∈ H0(M〈n〉).

Suppose that the element Z from (32) does exist. Then we have:

2Z = 2sxn−1,i1 · · ·xn−1,i2s − 2wn−1Y or

ξ(i1, i2) · · · ξ(i2s−1, i2s)− 2Z = wn−1(2Y − a).

In particular, in the polynomial ring H0(w−1n−1M〈n〉)⊗Z/2 we have

ξ(i1, i2) · · · ξ(i2s−1, i2s) = w2sn−1Xi1,i2 · · ·Xi2s−1,i2s = wn−1(2Y − a).

Thus,2Y − a = w2s−1

n−1 Xi1,i2 · · ·Xi2s−1,i2s .

It remains to observe that the element w2s−1n−1 Xi1,i2 · · ·Xi2s−1,i2s does not belong

to the ring H0(M〈n〉)⊗Z/2, while the element 2Y − a does.

Now we can prove Proposition 4.4 from Lemma 4.5 and the exact sequence (21).

Proof of Proposition 4.4. The element

w−1n−1xn−1,i1 · · ·xn−1,i2s ∈ H0(M〈n〉/w∞n−1)

is a D〈n〉–cycle. Suppose that

2s−1tn(i1, . . . , i2s) = 0

in H1(M〈n〉). By the exact sequence (21), there is an element Y ∈MSpΣ∗ such

that

Y +2s−1xn−1,i1 · · ·xn−1,i2s

wn−1

is a cycle in H0(w−1n−1M〈n〉). Then the element

wn−1Y + 2s−1xn−1,i1 · · ·xn−1,i2s ∈MSpΣ∗

is a cycle in M〈n〉, which contradicts Lemma 4.5.

Note 4.3 Proposition 4.4 fails for n = 2 since the generators Xi1,...,is for thering H0(w−1

1 M〈2〉) are essentially different from the case n > 2 because theelements x1,j1,...,jt are not defined in the ring MSpΣ

∗ .

21

5 Existence of Higher Torsion Elements

This section is devoted to the proof of Theorem A. In particular, we constructelements τn(i1, . . . , is) ∈ MSpΣn

∗ which project to the elements tn(i1, . . . , is) ofhigher torsion in the one line of the ANSS which we studied in Section 4.

Theorem A. For each i = (i1, . . . , is), 3 ≤ n < i1 < · · · < is, there existindecomposable elements τn(i) ∈MSpΣn

4∗+1 with the following properties:

(i) τn(i1) = φ2i1−2;

(ii) τn(i1, . . . , is) ∈ 〈φ2is−2 , wn−1, τn(i1, . . . , is−1)〉;

(iii) τn(i1, . . . , is) has order at least 2[(s+1)/2] for s ≥ 1.

We then prove Theorem B by using Bocksten long exact sequences to deducethe existence of higher torsion in MSp∗ of all orders.

Theorem B. For each k ≥ 1 there exist elements of order 2k in the symplecticcobordism ring MSp∗.

The motivation for defining the τn(i1, . . . , is) by induction on s ≥ 1 so that theysatisfy conditions (i) and (ii) of Theorem A is that their projections tn(i1, . . . , is)into the one line of E1,4∗+1

2 (MSpΣn) of the ANSS satisfy the correspondingalgebraic conditions:

(i) tn(i1) = ui1 ;

(ii) tn(i1, . . . , is) ∈ 〈uis , wn−1, tn(i1, . . . , is−1)〉.

Note that the complex M〈n〉 has a nice product structure which enables us todefine Massey products in E2 = H∗ (M〈n〉) in the usual way.

Our construction begins with the following result. We let MSpΣ0∗ denote

MSp and φ−1 denote φ0 = η.

Theorem 5.1 (V.Gorbunov, [2, Theorem 4.3.5]) For j > 2n−2 and n ≥ 1 the

Toda bracket 〈φ2n−2 , 2, φj〉 contains zero in the ring MSpΣn−1∗ .

Recall from Section 2 that MSpΣn∗ is a polynomial ring in degrees less than

or equal to 2n+2 − 4 with w1, . . . , wn as polynomial generators. P1 is an Sp–manifold which represents u1 = η and Pk is an Sp–manifold which representsthe basic Ray element uk = φ2k−2 for k ≥ 2. We also chose Sp–manifolds Wi

such that ∂Wi = 2Pi. Let j ≥ n ≥ 2. By Theorem 5.1, the Toda bracket

〈φ2n−3 , 2, φ2j−2〉 contains zero in the ring MSpΣn−2∗ . In other words, there exist

Σn−2-manifolds Xn−1,j , W(n−1)j and W

(j)n−1 such that

δW(j)n−1 = Pn−1 × 2, δW

(n−1)j = 2× Pj and

22

δXn−1,j = W(j)n−1 × Pj ∪ Pn−1 ×W (n−1)

j .

As Σn–manifolds we have

δXn−1,j = W(j)n−1 × Pj . (33)

Note that cobordism classes of the manifolds W(j)n−1 depend, in general, on j.

Lemma 5.2 For j ≥ n ≥ 2, there exist Σn-manifolds Wn−1 and Xn−1,j, suchthat

δXn−1,j = Wn−1 × Pj (34)

where Wn−1 does not depend on j.

Proof. We prove this lemma by induction on n ≥ 2. Let n = 2. For each j ≥ 2

we have that β1W(j)1 = 2 by construction. For j ≥ 2, all the Σ1-bordism classes[

W(j)1

]equal the same element w1 ∈ MSpΣ1

2 since w1 is the unique cobordism

class such that β1w1 = 2.

Now assume that this lemma is true for n−1. Let Wn−1 be any Sp-manifold

such that ∂Wn−1 = 2Pn−1. Let j ≥ n with Xn−1,j and W(j)n−1 Σn−2-manifolds as

above. We will define an Σn-manifold Xn−1,j that satisfies (34). Since MSpΣn∗

is a polynomial ring in degrees less than or equal to 2n+2 − 4, we have that

γ =[W

(j)n−1

]− [Wn−1] ,

is a polynomial in the generators w1, . . . , wn−1, xr, r ≤ 2n−3, r 6= 2l−1, deg xr =4r, as in the statement of Theorem 2.1. Let γ be the sum of k momomials:γ =

∑ki=1 γi. Since dimWn−1 = 2(2n−1 − 1), each monomial γi contains at

least one factor wmi , mi ≤ n − 2; write γi = γiwmi . By induction, there is a

Σmi-manifold Xmi,j , mi < n − 1, such that δXmi,j = Wmi × Pj . Let Γi be aΣn-manifold which represents γi. Define a Σn-manifold Xn−1,j as the disjointunion of the following Σn-manifolds:

Xn−1,j = Xn−1,j

k⋃i=1

−mn(

Γi,Xmi,j × Pj)

where the Σn-manifolds Xn−1,j are as in (33). Clearly δXn−1,j=Wn−1×Pj.

Lemma 5.3 There exist elements τn (i1, . . . , is), in the ring MSpΣn4∗+1 for s ≥ 1

and n < i1 < · · · < is such that:

(i) τn(i) = ui;

(ii) τn (i1, . . . , is) ∈ 〈uis , wn−1, τn (i1, . . . , is−1)〉 for s ≥ 2;

(ii) wn−1τn (i1, . . . , is) = 0.

23

Proof. We construct the elements τn (i1, . . . , is) by induction on s ≥ 1. Fors = 1, Theorem 5.1 gives that wn−1uj = 0. Assume that this lemma is true fors− 1. Select any element τn (i1, . . . , is) of the Toda bracket

〈uis , wn−1, τn (i1, . . . , is−1)〉.

We must show that wn−1τn (i1, . . . , is) = 0. Let Wn−1 be a Σn-manifold as inLemma 5.2 which represents wn−1 so that there exists a Σn-manifold Xn−1,is

with δXn−1,is = Wn−1 × Pis . By Lemma 3.4 we have that

wn−1τn (i1, . . . , is) ∈ wn−1〈uis , wn−1, τn (i1, . . . , is−1)〉

⊂ 〈wn−1, uis , wn−1〉τn (i1, . . . , is−1) .

Note that the Σn-manifold ∆(Wn−1) of (11) has dimension 2(2n−1 − 1) − 1 =dimun. The element un is the unique nontrivial element of this degree in

MSpΣn−1∗ . Thus,

[∆(Wn−1)]Σn−1= κun, where κ = 0 or 1. (35)

Therefore, in MSpΣn∗ , [∆(Wn−1)] = 0. Thus in MSpΣn

∗ , Lemma 3.3 and theinduction hypothesis give

〈wn−1, uis , wn−1〉τn (i1, . . . , is−1)=(κunuis+wn−1a) τn (i1, . . . , is−1)=0.

Next we determine the projection of τn (i1, . . . , is) into the E2-term of theANSS for MSpΣn to be tn(i1, . . . , is) which we defined in (31) and studied inProposition 4.4. We do this by constructing a Σn–manifold which representsτn (i1, . . . , is) and a Σn–manifold whose boundary equals wn−1τn (i1, . . . , is)modulo the Adams–Novikov filtration. These manifolds will be used in the con-structions of the next section. We denote as m, K, A the constructions mn, Kn,An from Section 3.

Lemma 5.4 There exists an element τn(i1, . . . , is) of the Toda bracket

〈uis , wn−1, τn (i1, . . . , is−1)〉

such that the projection of τn(i1, . . . , is) into E1,4∗+12 (MSpΣn) of the ANSS

equals

tn(i1, . . . , is) =s∑r=1

uirxn−1,i1 · · · xn−1,ir · · ·xn−1,is .

Proof. Lemma 5.3 gives us a Σn-manifold T (i1, . . . , is) which represents theelement τ (i1, . . . , is) and a Σn-manifold H (i1, . . . , is) with

δH (i1, . . . , is) = m (Wn−1, T (i1, . . . , is)) .

We use induction on s ≥ 1 to define specific Σn-manifolds Ts and Hs such that:

24

(i) δHs = m (Wn−1, Ts) ∪ Ls;

(ii) Ts projects in the one line of the ANSS to tn(i1, . . . , is) ∈ H1 (M〈n〉);

(iii) Hs projects in the zero line of the ANSS to xn−1,i1 · · ·xn−1,is ∈ H0 (M〈n〉);

(iv) Ls has Adams–Novikov filtration degree two.

If s = 1 let T1 = Pi1 , H1 = Xn−1,i1 and L1 = ∅. By induction suppose thatwe have Σn-manifolds Ts−1 and Hs−1 which satisfy the above four conditions.Consider the Σn-manifold

H(0) = m (Hs−1,Xn−1,is)

with

δ(H(0)) = m(m(Wn−1, Ts−1),Xn−1,is)∪m(Hs−1,Wn−1×Pis)∪m(Ls−1,Xn−1,is).

Glue the Σn-manifolds H(0) and A(Wn−1, Ts−1,Xn−1,is) together along theircommon boundary m(m(Wn−1, Ts−1),Xn−1,is) to obtain the Σn-manifold

H(1) = H(0) ∪ A (Wn−1, Ts−1,Xn−1,is)

with

δ(H(1)) = m(Wn−1,m(Ts−1,Xn−1,is)) ∪m(Hs−1,Wn−1 × Pis)∪

A(Wn−1, Ts−1,Wn−1 × Pis) ∪m(Ls−1,Xn−1,is)

= m (Wn−1,m (Ts−1,Xn−1,is)) ∪m(Hs−1,Wn−1)× Pis∪

An(Wn−1, Ts−1,Wn−1)× Pis ∪m(Ls−1,Xn−1,is).

Next glue the Σn-manifolds H(1) and −K(Hs−1,Wn−1)×Pis together along theircommon boundary mn(Hs−1,Wn−1)× Pis to obtain the Σn-manifold

Hs = H(1) ∪ −K(Hs−1,Wn−1)× Piswith

δ(Hs) = m(Wn−1,m(Ts−1,Xn−1,is)) ∪m(Wn−1,Hs−1)× Pis∪

K (m(Wn−1, Ts−1) ∪ Ls−1,Wn−1)× Pis∪

A(Wn−1, Ts−1,Wn−1)× Pis ∪m(Ls−1,Xn−1,is)

= m(Wn−1,m(Ts−1,Xn−1,is) ∪Hs−1 × Pis)∪

K(m(Wn−1, Ts−1) ∪ Ls−1,Wn−1) ∪A (Wn−1, Ts−1,Wn−1) × Pis∪

m(Ls−1,Xn−1,is)

= m(Wn−1, Ts) ∪ Ls

25

where

Ts = m(Ts−1,Xn−1,is) ∪Hs−1 × Pis .

and

Ls = K (m(Wn−1, Ts−1) ∪ Ls−1,Wn−1) ∪ A(Wn−1, Ts−1,Wn−1) × Pis∪

m(Ls−1,Xn−1,is).

Since Ts−1, Ls−1 has Adams–Novikov filtration degree one, two, respectively,the projection of the Σn-manifold Ls to the one line of the ANSS are trivial.Therefore, Ls has Adams–Novikov filtration degree two. By construction andthe induction hypothesis, the projection of Ts to the one line of the ANSS equals

tn (i1, . . . , is−1) xn−1,is + xn−1,i1 · · ·xn−1,is−1uis = tn (i1, . . . , is) .

Since Ts−1 and Pis have Adams–Novikov filtration degree one, the projections ofA (Wn−1, Ts−1,Xn−1,is) and K(Hs−1,Wn−1)× Pis to the zero line of the ANSSare trivial. Thus, the projection of Hs to the zero line of the ANSS equals theprojection of H(0) which is xn−1,i1 · · ·xn−1,is−1 · xn−1,is .

We complete the proof of Theorem A. By the previous lemma, the elementτn (i1, . . . , is) ∈MSpΣn

∗ can be defined as required so that it projects to

tn(i1, . . . , is) ∈ Ext1,∗ABP (BP ∗(MSpΣn), BP ∗)

which has order 2[(s+1)/2] by Lemma 4.4. Therefore, τn (i1, . . . , is) has ordergrater or equal to 2[(s+1)/2] in MSpΣn

∗ . Finally, note that τn (i1, . . . , is) is in-decomposable in MSpΣn∗ because its projection tn(i1, . . . , is) into the algebraE∗,∗2 (MSpΣn) is indecomposable.

Proof of Theorem B. By Theorem A, there are torsion elements of ordergreater than or equal to 2k for all k ≥ 1 in the ring MSpΣ3 . Consider theBockstein-Sullivan exact sequences:

· · · −→MSpΣ2∗·φ2−−→MSpΣ2

∗π2−→MSpΣ3

∗β3−→MSpΣ2

∗ −→ · · ·

· · · −→MSpΣ1∗·φ1−−→MSpΣ1

∗π1−→MSpΣ2

∗β2−→MSpΣ1

∗ −→ · · ·

· · · −→MSp∗·η−→MSp∗

π0−→MSpΣ1∗

β1−→MSp∗ −→ · · ·

We show that exponents of the groups Tors MSpΣ2∗ , Tors MSpΣ1∗ and Tors MSp∗must be infinite since the exponent of Tors MSpΣ3

∗ is infinite. Assume, tothe contrary, that all torsion of MSpΣ2

∗ has exponent 2k. We take an elementa ∈ MSpΣ3∗ of order 22k+1. Then the element a1 = β2(a) has order no morethan 2k. From the above Bockstein-Sullivan exact sequence,

2ka ∈ ImMSpΣ2

∗π1−→MSpΣ3

∗

⊂MSpΣ3

∗ .

26

Let π2(a2) = 2ka. Then 2k+1a2 ∈ Ker π2 = Im (·φ2), so 2k+1a2 = φ2x. Conse-quently 2k+2a2 = 0, and a2 has finite order. Since

π2(2ka2) = 22ka 6= 0 ,

the element a2 ∈ MSpΣ2∗ has order grater than or equal to 2k+1, contradictingthe assumption that Tors MSpΣ2

∗ has exponent 2k. Thus, the exponent ofTors MSpΣ2

∗ is infinite.

Note 5.1 Theorem B is just an existence theorem, and its proof does not givea specific way to construct torsion elements of higher order in MSp∗. However,for each n ≥ 3 the family τn(i) ∈MSpΣn

∗ determines a different family of higherorder torsion elements in MSp∗. In the next two sections and in [3] we studythe family that is determined by τ3(i) ∈MSpΣ3∗ . The analysis of the cases n ≥ 4requires more topological information about MSp∗ than we have at present.

6 Construction of Elements in MSpΣ2∗ and MSpΣ1

∗

In Lemma 5.3 we constructed elements of higher torsion

τ (i1, . . . , is) = τ3 (i1, . . . , is) ∈MSpΣ34∗+1.

In this section we study the elements:

γ (i1, . . . , is) = β3τ (i1, . . . , is) ∈MSpΣ24∗+3,

α′ (i1, . . . , is) = β2γ (i1, . . . , is) ∈MSpΣ14∗+1.

In particular we compute their projection to the three line of the ANSS. Through-out this section, let m and K denote the canonical constructions m2 and K2 ofSection 3.

We begin by interpreting β3 (τ3 (i1, . . . , is)) in terms of manifolds with sin-gularities. Recall that by Lemma 5.3 there is a representative Σ3-manifoldT (i1, . . . , is) of τ3 (i1, . . . , is) and a Σ3-manifold H (i1, . . . , is) such that

δH (i1, . . . , is) = m(W2, T (i1, . . . , is)).

We can consider the manifold T (i1, . . . , is), H (i1, . . . , is) as a Σ2-manifold

T (i1, . . . , is), H (i1, . . . , is), respectively, with

δH (i1, . . . , is) = m(W2, T (i1, . . . , is)) ∪ P3 ×E (i1, . . . , is) ,

δT (i1, . . . , is) = P3 ×G (i1, . . . , is)

(36)

where G (i1, . . . , is) = β3T (i1, . . . , is) represents the Σ2–cobordism classγ (i1, . . . , is). Note that

δE (i1, . . . , is) = m (W2, G (i1, . . . , is)) .

27

To determine the projection of E (i1, . . . , is) to the ANSS we need to identify thequadratic construction ∆ (W2) which was defined in Section 3. The followinglemma is an easy computation in the ASS for MSpΣ2∗ . We defer its proof toLemma 4.3 of [3].

Lemma 6.1 The cobordism class of the Σ2-manifold ∆(W2) equals P3 in MSpΣ2∗ .

We are now ready to compute the projection of E (i1, . . . , is) into the two lineof the ANSS for MSpΣ2 . This will lead directly to the identification of theprojection of the γ (i1, . . . , is) into the three line of the ANSS for MSpΣ2 .

Lemma 6.2 (a) E (i1) = ∅.

(b) For s ≥ 2, E (i1, . . . , is) projects in E2,4∗+21 of the ANSS for MSpΣ2 to

e (i1, . . . , is) =∑

1≤t1<t2≤suit1uit2x2,i1 · · · x2,it1

· · · x2,it2· · ·x2,is .

Proof. (a) Since T (i) = Pi, H (i) can be taken to be the Σ2-manifold X2,i ofLemma 5.2 with boundary

δX2,i = W2 × Piwhere W2 does not depend on i. Thus, E (i) = ∅.

(b) By induction on s ≥ 2, we construct Σ2-manifolds Ts, Hs, Es and Lssuch that:

(i) Ts represents τ (i1, . . . , is);

(ii) δHs = m(W2, Ts

)∪ P3 ×Es ∪ Ls;

(iv) Ts projects to t (i1, . . . , is) in the one line of the ANSS;

(v) Hs projects to x2,i1 · · ·x2,is in the zero line of the ANSS;

(vi) Es projects to ∑1≤t1<t2≤s

uit1uit2x2,i1 · · · x2,it1· · · x2,it2

· · ·x2,is

in the two line of the ANSS;

(vii) Ls has Adams–Novikov filtration degree four.

The case s = 2 will be proved as a special case of the induction step withT1 = Pi1 , H1 = X2,i1 , E1 = ∅ and L1 = ∅. Clearly statements (i)–(vii) are valid

28

%&

"!#

m(Hs−1,X2,is)

m(W2,K(Ts−1,W2))×Pis×I

(Y ∪m(Z, Ts−1))× Pis

A(W2,Ts−1,X2,is) −K(Hs−1,W2)×Pis

m

(W2,m(Ts−1,X2,is) ∪ K(Ts−1,W2)×Pis×0 ∪ Hs−1×Pis

)

P3×Ts−1×Pis

Figure 3: The Σn-manifold Hs.

for s = 1. Thus, assume that s ≥ 2 and that our seven assertions are true in thecase s− 1. As in the proof of Lemma 5.4, define the Σ2–manifold

H ′s = m

(Hs−1,X2,is

)∪A

(W2, Ts−1,X2,is

)∪

−K(Hs−1,W2

)× Pis ∪m

(W2,K(Ts−1,W2)

)× Pis × I.

See Figure 3. Let F p denote the set of Σ2-manifolds of Adams–Novikov filtrationdegree p. Then, as in the proof of Lemma 5.4, we have modulo F 4 that

δH ′s ≡ m

(W2,m(Ts−1,X2,is) ∪ Hs−1×Pis ∪ K(Ts−1,W2)× Pis×0

)∪ P3 ×m (Es−1,X2,is) ∪ K

(m

(W2, Ts−1

),W2

)× Pis

∪ A(W2, Ts−1,W2

)× Pis ∪ −m

(W2,K(Ts−1,W2)

)× Pis × 1 .

Observe that δTs−1 ≡ δW2 ≡ ∅ module F 3. Thus by the generalized Hirschformula (10), we have modulo F 3 that there is a Σ2–manifold Y such that

δY ≡ −K(m

(W2, Ts−1

),W2

)∪ −A

(W2, Ts−1,W2

)∪ m

(W2,K

(Ts−1,W2

))∪ m

(K (W2,W2) , Ts−1

).

29

By Lemma 6.1 there is a Σ2–manifold Z such that

δZ = −K (W2,W2) ∪ P3.

Let Hs = H ′s ∪ Y × Pis ∪ m(Z, Ts−1

)× Pis . Then

δHs = m(W2, Ts

)∪ P3 ×Es ∪ Ls

where

Ts = m

(Ts−1,X2,is

)∪ Hs−1 × Pis ∪ K

(Ts−1,W2

)× Pis ,

Es = m (Es−1,X2,is) ∪ Ts−1 × Pis

and Ls has Adams–Novikov filtration degree four. By the induction hypothesis,Ts projects in the one line of the ANSS to

t (i1, . . . , is−1)x2,is + x2,i1 · · ·x2,is−1 · uis = t (i1, . . . , is) ,

Hs projects in the zero line of the ANSS to

x2,i1 · · ·x2,is−1 · x2,is ,

and Es projects in the two line of the ANSS to

γ (i1, . . . , is−1)x2,is + t (i1, . . . , is−1)uis = γ (i1, . . . , is) .

This completes the induction step. Observe that Es and E (i1, . . . , is) differ by aΣ2–manifold of Adams–Novikov filtration degree five. Therefore, they have thesame projection to the three line of the ANSS.

We can now determine the basic properties of the γ (i1, ..., is).

Proposition 6.3 The elements γ (i1, ..., is) = β3τ (i1, ..., is) ∈MSpΣ24∗+3 satisfy

the following conditions:

(a) γ (i1) = γ (i1, i2) = 0;

(b) γ (i1, i2, i3) = ui1ui2ui3 ;

(c) For s ≥ 4, γ (i1, ..., is) ∈ 〈uis , w2, γ (i1, ..., is−1)〉;

(d) For s ≥ 3, γ (i1, ..., is) projects in E3,∗1 (MSpΣ2) of the ANSS to

g (i1, . . . , is)=∑

1≤t1<t2<t3≤suit1uit2uit3x2,i1 · · ·x2,it1

· · ·x2,it2· · ·x2,it3

· · ·x2,is .

30

Proof. (a) Since τ (i1) = ui1 and τ(i1, i2) is represented by

Pi1 ×X2,i2 ∪ K (Pi1 ,W2)× Pi2 ∪X2,i1 × Pi2

it follws that G (i1) = G (i1, i2) = ∅.(b), (c) Let s ≥ 3. By Lemma 5.3(ii) and (36), we have the boundary of

Σ2–manifolds

δT (i1, . . . , is) = m (Xis,2, P3 ×G (i1, . . . , is−1)) ∪ Pis × P3 ×E (i1, . . . , is−1) .

Therefore, γ (i1, . . . , is) = β3τ (i1, . . . , is) is represented by the Σ2–manifold

G (i1, . . . , is) = m (Xis,2, G (i1, . . . , is−1)) ∪ Pis ×E (i1, . . . , is−1) . (37)

If s = 3 then the first unionand is vacuous. Observe that the constructions inthe proof of Lemma 6.2 give E2 = Pi1 × Pi2 and L2 = ∅. Thus, γ (i1, i2, i3) isrepresented by Pi2 × Pi2 × Pi3 . If s ≥ 4 then the element in (37) is an elementof the Toda bracket 〈uis , w2, γ (i1, · · · , is−1)〉.

(d) We use induction on s ≥ 3. The case s = 3 follows from (b). Assumethe case s− 1. By the description of the Σ2–manifold G (i1, . . . , is) in the proofof (c), γ (i1, . . . , is) projects in the three line of the ANSS to

g (i1, . . . , is) = xis,2g (i1, . . . , is−1) ∪ uis × e (i1, . . . , is−1) .

By the induction hypothesis and the previous lemma,

g (i1, . . . , is) = xis,2∑

1≤t1<t2<t3≤s−1

uit1uit2uit3x2,i1 · · ·x2,it1· · ·x2,it2

· · ·x2,it3· · ·x2,is−1

+ uis∑

1≤t1<t2≤s−1

uit1uit2x2,i1 · · · x2,it1· · · x1,it2

· · ·x2,is−1 .

This is the asserted value of g (i1, . . . , is) in (d).

We complete this section by computing the projection of the element

α′ (i1, . . . , is) = β2γ (i1, . . . , is) ∈MSpΣ1∗ .

to the ANSS E3,∗1 (MSpΣ1) where β2 : MSpΣ2

∗ →MSpΣ1∗ is the Bockstein oper-

ator. To describe this projection we introduce the notation

p(j1, . . . , jq) =∑

1≤t1<t2<t3≤qujt1ujt2ujt3wj1 · · · wjt1 · · · wjt2 · · · wjt3 · · ·wjq

in E3,∗1 (MSpΣ1) for q ≥ 3.

31

Proposition 6.4 The elements α′ (i1, . . . , is) = β2γ (i1, . . . , is) ∈ MSpΣ14∗+1

satisfy the following conditions.

(a) α′ (i1) = α′ (i1, i2) = α′ (i1, i2, i3) = 0.

(b) For s ≥ 4, α′ (i1, . . . , is) projects in E3,∗1 (MSPΣ1) of the ANSS to

a′ (i1, . . . , is)

=s∑

k=4

(−1)kwk−42

∑1≤t1<···<tk≤s

p(it1 , . . . , itk)x2,i1 · · · x2,it1· · · x2,itk

· · ·x2,is .

(c) For s ≥ 4, there are elements α (i1, . . . , is) ∈ MSp∗ such that the elementπ∗ (α (i1, . . . , is)) ∈ MSpΣ1

∗ projects to 2a′ (i1, . . . , is) ∈ E3,4∗+12 (MSpΣ1),

where π : MSp→MSpΣ1 is the natural map.

(d) The projection a (i1, . . . , is) ∈ E3,4∗+12 (MSp) of α (i1, . . . , is) ∈ MSp∗ is a

nonzero infinite cycle.

Proof. (a) This follows from Proposition 6.3(a),(b).

(b) The Bockstein operator β2 : MSpΣ2∗ → MSpΣ1

∗ induces the homomor-phism

b2 : E∗,∗2 (MSpΣ2)→ E∗,∗2 (MSpΣ1).

Then α′ (i1, . . . is) = β2 (γ (i1, . . . , is)) projects in E3,∗1 (MSpΣ1) to b2 of the

projection of γ (i1, . . . , is) to E3,∗1 (MSpΣ2). In Proposition 6.3(d) we determined

the latter projection g (i1, . . . is). Direct computations establishes formula (6.4)for a′ (i1, . . . , is).

(c), (d) Since 2η = 0, the element 2α′(i) − w1β1 (α′(i)) is in the ker-nel of the Bockstein operator β1 : MSpΣ1

∗ →MSp∗ which equals the image ofπ∗ : MSp∗ →MSpΣ1

∗ . Let π∗(α(i)) = 2α′(i) − w1β1 (α′(i)). Observe that theelement β1 (α′(i)) has Adams-Novikov filtration at least four since β1a

′ (i) = 0in E3,∗

2 (MSpΣ1) from (b). Therefore, the element 2α′(i) − w1β1 (α′(i)) projectsto 2a′ (i1, . . . , is) ∈ E3,4∗+1

2 (MSpΣ1).

7 Proof of the Main Theorem

In Sections 5 and 6 we constructed the elements

τ (i1, . . . , is) ∈MSpΣ3∗ ,

τ (i1, . . . , is) = β3σ (i1, . . . , is) ∈MSpΣ2∗ ,

α′ (i1, . . . , is) = β2γ (i1, . . . , is) ∈MSpΣ1∗ ,

α (i1, . . . , is) ∈MSp∗

32

such that the element π∗ (α (i1, . . . , is)) = 2α′ (i1, . . . , is) − w1β1 (α′ (i1, . . . , is))projects to a′(i1, . . . , is) in E3,4∗+1

2 of the ANSS for MSpΣ1 . In this section, weprove the main theorem of this paper.

Main Theorem. The element α (i1, . . . , is) ∈ MSp∗ has order at least2[(s+1)/2]−3 if s ≥ 7 and 3 ≤ i1 < · · · < is.

In particular, the lowest degree element of order at least eight which we haveidentified in MSp∗ is α(3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) in degree 32,769.

We use the notation i for (i1, · · · , is) below. Let t (i), g (i), a′ (i) denote theprojection of τ (i), γ (i), α′ (i) to E∗,∗2

(MSpΣ3

), E∗,∗2

(MSpΣ2

), E∗,∗2

(MSpΣ1

),

respectively. Chromatic technique was developed to make computations in theE2–term of the ANSS for spheres. See [11]. In this section, we use a chromaticargument to prove the following proposition.

Proposition 7.1 Let i = (i1, . . . , is) with 3 ≤ i1 < · · · < is and s ≥ 6.

(i) The element g (i) ∈ E3,∗2

(MSpΣ2

)has order at least 2[(s+1)/2]−1.

(ii) The element a′ (i) ∈ E3,∗2

(MSpΣ1

)has order at least 2[(s+1)/2]−2.

Before proving Proposition 7.1, we show how the Main Theorem follows from it.

Proof of the Main Theorem Using Proposition 7.1.Recall from Proposition 6.4(d) that the infinite cycle a(i) ∈ E3,4∗+1

2 (MSp) is theprojection of α(i) ∈MSp∗. By Proposition 6.4(c), π∗(a(i)) = 2a(i)′ where π∗ isthe homomorphism

E3,∗1 (MSp)

π∗−→ E3,∗1 (MSpΣ1).

By Proposition 7.1, the element

2a′(i) = π∗ (a(i))

has order at least 2[(s+1)/2]−3 in E3,4∗+12 (MSpΣ1). Thus a(i) has order at least

2[(s+1)/2]−3 in E3,4∗+12 (MSp). Since E1,4∗+2

2 (MSp) = E0,4∗+22 (MSp) = 0, the

element 2ta(i) cannot be killed by differentials for t = 1, . . . , [(s + 1)/2] − 4.Thus, 2[(s+1)/2]−4α(i) projects to a nonzero element of E3,∗

∞ (MSp) and must benonzero.

Note 7.1 This argument can not be used to prove directly that 2[(s+1)/2]−3a′(i)is nonzero in E∞(MSpΣ1) because E0,4∗+2

2 (MSp) is nonzero which raises thepossibility of hitting 2[(s+1)/2]−3a′(i) by a d3–differential.

The following lemma shows that we can assume that i1 ≥ 4 in proving Proposi-tion 7.1(i).

Lemma 7.2 Let s ≥ 6. If g′(i1, i2, . . . , is) ∈ E3,∗2 (MSpΣ1) has order 2[(s+1)/2]−1

for all 4 ≤ i1 < · · · < is then g′(3, i2, . . . , is) ∈ E3,∗2 (MSpΣ1) has order 2[(s+1)/2]−1

for all 3 < i2 < · · · < is .

33

Proof. Let t = [(s+ 1)/2]− 2. Suppose there is x ∈ E2,∗1 (MSpΣ1) such that

d1(x) = 2tg(3, i2, . . . , is).

Choose n > is. The element x depends on the generators xj1,...,jk , wj anduj . In particular, the formula for the first differential is invariant under thetransposition (3, n) in all entries of the elements x and g(3, i2, . . . , is). Applyingthis permutation we obtain an element x′, such that d1(x′) = 2tg(i2, . . . , is, n), acontradiction.

We give the proof of Proposition 7.1 in the case s even and 4 ≤ i1 < · · · < is.The proof for the case s odd is obtained by a slight modification. Thus, i willdenote i1, · · · , i2s for the remainder of this section. We prove Proposition 7.1(i)

by showing that g(i) has order at least 2s−1. Let MSpΣ2π−→MSpΣ3 denote the

canonical map which induces the homomorphism

E∗,∗2 (MSpΣ2)π∗−→ E∗,∗2 (MSpΣ3).

Let g(i) = π∗(g(i)) ∈ E3,∗2 (MSpΣ3). By Proposition 6.3(d),

g(i) =∑

1≤t1<t2<t3≤2s

uit1uit2uit3x2,i1 · · · x2,it1· · · x2,it2

· · · x2,it3· · ·x2,i2s .

We use chromatic methods to determine the order of g(i). Consider the followingexact sequences of complexes:

0→M〈3〉 → w−12 M〈3〉 →M〈3〉/(w∞2 )→ 0,

0→M〈3〉/(w∞2 )→ w−11 M〈3〉/(w∞2 )→M〈3〉/(w∞2 , w∞1 )→ 0,

0→M〈3〉/(w∞2 , w∞1 )→ w−13 M〈3〉/(w∞2 , w∞1 )→M〈3〉/(w∞3 , w∞2 , w∞1 )→ 0.

Recall that by Theorem 5.1 there exist elements x1,k ∈MSpΣ1∗ , x2,k ∈MSpΣ2

∗ ,and x3,k ∈MSpΣ3∗ such that

βk (x1,k) = w1, βk (x2,k) = w2, βk (x3,k) = w3.

The arguments used to prove Lemma 4.2 can be used to prove the followinglemma.

Lemma 7.3 The complexes

w−12 M〈3〉, w−1

1 w−13 M〈3〉, w−1

1 M〈3〉/(w∞2 ),

w−13 M〈3〉/(w∞2 ), w−1

3 M〈3〉/(w∞2 , w∞1 )

are acyclic, i.e. their nth homology groups are zero for n ≥ 1.

34

Consider the following composition of boundary homomorphisms:

H0 (M〈3〉/(w∞2 , w∞3 , w∞1 ))δ(0)−−→ H1 (M〈3〉/(w∞2 , w∞1 ))

δ(1)−−→

H2 (M〈3〉/(w∞2 ))δ(2)−−→ H3 (M〈3〉) .

By Lemma 7.3, δ(0) is an epimorphism and δ(1), δ(2) are isomorphisms.

Define the following elements:

g2(i) ∈ H2 (M〈3〉/(w∞2 )) , g1(i) ∈ H1 (M〈3〉/(w∞2 , w∞1 )) ,

g0(i) ∈ H0 (M〈3〉/(w∞2 , w∞3 , w∞1 )) ,

where

g2(i) = w−12

∑1≤t1<t2≤2s

uit1uit2x2,i1 · · · x2,it1· · · x2,it2

· · ·x2,i2s ,

g1(i) = w−12 w−1

1

∑1≤t1<t2≤2s

x1,it1uit2x2,i1 · · · x2,it1

· · · x2,it2· · ·x2,i2s ,

g0(i) = w−12 w−1

1 w−13

∑1≤t1<t2≤2s

x1,it1x3,it2

x2,i1 · · · x2,it1· · · x2,it2

· · ·x2,i2s .

A direct calculation gives that

δ(0) (g0(i)) = g1(i), δ(1) (g1(i)) = g2(i), δ(2) (g2(i)) = g(i).

Proposition 7.1(i) is a consequence of the following lemma.

Lemma 7.4 For s ≥ 3, the element g1(i) ∈ H1 (M〈3〉/(w∞2 , w∞1 )) has order atleast 2s−1.

Proof. Assume that 2s−2 g1(i) = 0. Then δ(0)(2s−2g0(i)

)= 2s−2g1(i) = 0.

Therefore, there is an element Z1 ∈ H0

(w−1

3 M〈3〉/(w∞1 , w∞2 ))

such that

π1(Z1) = 2s−2g0(i)

where π1 is the homomorphism in the diagram below. In the following diagramthe row and both columns are exact.

35

H0

(w−1

3 M〈3〉/w∞2)-

6

0

H0

(w−1

1 w−13 M〈3〉/w∞2

)-H0

(w−1

3 M〈3〉/(w∞1 , w∞2 ))π2 - 0

6

H0

(w−1

2 w−11 w−1

3 M〈3〉)

H0

(w−1

1 w−13 M〈3〉

)6

6

0

H0 (M〈3〉/(w∞1 , w∞2 w∞3 ))

H1 (M〈3〉/(w∞1 , w∞2 ))

H0 (M〈3〉/(w∞1 , w∞2 ))

0

?

?

?

π1π3

?

δ(0)

?

0

From the diagram above, we see that there exist

Z2 ∈ H0

(w−1

1 w−13 M〈3〉/w∞2

)and Z3 ∈ H0

(w−1

2 w−11 w−1

3 M〈3〉)

such that π3(Z3) = Z2 and π2(Z2) = Z1. Let

S = 2s−2

∑1≤t1<t2≤2s

x1,it1x3,it2

x2,i1 · · · x2,it1· · · x2,it2

· · ·x2,i2s

.

By definition of the elements Z1, Z2, Z3 we have that

Z1 = w−11 w−1

2 w−13 S + w−p1 w−b12 Y1,

Z2 = w−11 w−1

2 w−13 S + w−p1 w−b12 Y1 + w−b22 Y2,

Z3 = w−11 w−1

2 w−13 S + w−p1 w−b12 Y1 + w−b22 Y2 + Y3

(38)

where Y1, Y2, Y3 ∈ MSpΣ∗ and p, b1, b2 ≥ 1. Let q = max b1, b2. Note that we

36

can define Y1 and Y2 so that q ≥ 3. It follows from (38) that

wp1wq2w3Z3 = wp−1

1 wq−12 S + w3

(wq−b12 Y1

)+ wp1

(wq−b22 w3Y2

)+ wq2 (wp1w3Y3) .

Since wp1wq2w3Z3 ∈ H0 (M〈3〉), the following lemma produces the contradiction

which proves our lemma.

Lemma 7.5 For p ≥ 1 and q, s ≥ 3, there are no elements B1, B2, B3 ∈MSpΣ∗

such thatC = wp−1

1 wq−12 S + wp1B1 + wq2B2 + w3B3

belongs to the ring H0 (M〈3〉).

Proof. Recall from Lemma 4.3 that H0

(w−1

2 M〈3〉)

is a polynomial ring andthat there is a ring monomorphism:

0 −→ H0 (M〈3〉) π∗−→ H0

(w−1

2 M〈3〉).

Assume that we have chosen Y1 and Y2 in (38) to make q so large that the cycle Ccan be written as a polynomial in the polynomial generators of H0

(w−1

2 M〈3〉).

Also, we can always choose integral polynomial generators for H0

(w−1

2 M〈3〉),

i.e. from the image π∗ (H0(M〈3〉)), which include

ξ1,j = w2x1,j − w1x2,j = w2X′1,j ,

ξ2,j = 2x2,j − w2wj = w2Xj ,

ξ3,j = w2x3,j − w3x2,j = w2X′3,j ,

ξi,j = 2x2,ix2,j − w2(wix2,j − wjx2,i) + w22xi,j = w2

2Xi,j

(39)

where 1 ≤ i < j, i, j /∈ 1, 2, 3 and X ′1,j, X′3,j, Xi,j , Xj are the polynomial

generators of H0

(w−1

2 M〈3〉)

defined in (25), (26), (27).

Define

X(j1, . . . , j2t) = Xj1,j2 · · ·Xj2t−1,j2t ∈ H0(w−12 M〈3〉),

ξ(j1, . . . , j2t) = ξj1,j2 · · · ξj2t−1,j2t = w2t2 X(j1, . . . , j2t) ∈ H0(M〈3〉).

It follows from (39) that

w22x1,it1

x3,it2= ξ1,it1 ξ3,it2 + w1x2,it1

ξ3,it2 + w3x2,it2ξ1,it1 + w1w3x2,it1

x2,it2

= ξ1,it1 ξ3,it2 + w1a1 + w3a2

(40)where a1, a2 ∈MSpΣ

∗ . Consider the monomial

l = 2s−1x1,it1x3,it2

x2,i1 · · · x2,it1· · · x2,it2

· · ·x2,i2s .

37

It follows from (39) and (40) that

w22l =

(w2

2x1,it1x3,it2

) (2s−1x2,i1 · · · x2,it1

· · · x2,it2· · ·x2,i2s

)=

(ξ1,it1 ξ3,it2 + w1a1 + w3a2

) (ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s) + w2D

)= ξ1,it1 ξ3,it2 ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s)

+ (w1a1(it1 , it2) + w3a3(it1 , it2)) ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s)

+w32x1,it1

x3,it2Dk(i1, . . . , it1 , . . . , it2 , . . . , i2s).

Assume that B1, B2, B3 exist which make C a cycle in M〈3〉. Then

2C = wp−11 wq−3

2

(2w2

2S)

+ 2 (wp1B1 + wq2B2 + w3B3)

= wp−11 wq−3

2

∑1≤t1<t2≤2s

ξ1,it1 ξ3,it2 ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s)

+wp1wq−32 A1 + w3w

p−11 A3 + wp−1

1 wq2D

+wp12B1 + wq22B2 + w32B3

= wp−11 wq−3

2

∑1≤t1<t2≤2s

ξ1,it1 ξ3,it2 ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s)

+wp1

(wq−3

2 A1 + 2B1

)+ w3

(wp−1

1 A3 + 2B3

)+ 2wq2B2

+wp−11 wq2D.

LetΞ =

∑1≤t1<t2≤2s

ξ1,it1 ξ3,it2 ξ(i1, . . . , it1 , . . . , it2 , . . . , i2s). (41)

Then in H0 (M〈3〉):

2C − wp−11 wq−3

2 Ξ = wp1K1 + wq2K2 + w3K3 = K. (42)

Let βl1,...,lkX denote the composition βl1 (βl2 (· · · (βlkX) · · ·)) for an element Xof MSpΣ

∗ . Recall that βi (βjX) = βj (βiX). Thus βl1,...,lkX does not depend onthe order of l1, . . . , lk. The proof of the following lemma is straightforward.

Lemma 7.6 Let X ∈MSpΣ∗ with degX < 2(2n − 1). Then

2nX +n∑k=1

(−1)k2n−k∑

4≤l1<···<lk≤nwl1 · · ·wlkβl1,...,lkX

belongs to the ring H0 (M〈3〉) ⊂MSpΣ∗ .

38

Proof of Lemma 7.5 Continued: Choose n so that 2(2n− 1) > degK. Then

Si = 2nKi +n∑k=1

(−1)k2n−k∑

4≤l1<···<lk≤nwl1 · · ·wlkβl1,...,lkKi

are cycles for i = 1, 2, 3. Since K is a cycle,

βl1,...,lkK = wp1βl1,...,lkK1 + wq2βl1,...,lkK2 + w3βl1,...,lkK3 = 0.

Therefore,

2nK = 2nK +n∑k=1

(−1)k2n−k∑

4≤l1<···<lk≤nwl1 · · ·wlkβl1,...,lkK

−n∑k=1

(−1)k2n−k∑

4≤l1<···<lk≤nwl1 · · ·wlkβl1,...,lkK

= 2nwp1K1 + 2nwq2K2 + 2nw3K3

+n∑k=1

(−1)k2n−k∑

4≤l1<···<lk≤nwl1 · · ·wlk

(wp1βl1,...,lkK1 + wq2βl1,...,lkK2 + w3βl1,...,lkK3)

= wp1S1 + wq2S2 + w3S3.

Multiplying (42) by 2n we get the following equality in the ring H0 (M〈3〉):

2n+1C − 2nwp−11 wq−3

2 Ξ = wp1S1 + wq2S2 + w3S3. (43)

Observe that we can choose Y1 and Y2 in (38) to make q is so large that C,S1, S2, S3 all belong to H0

(w−1

2 M〈3〉). Then the equality (43) occurs in the

polynomial ring H0

(w−1

2 M〈3〉). By the definition of Ξ in (41), wp−1

1 wq−32 Ξ is

a linear combination of monomials in the canonical polynomial generators withodd coefficients. Moreover, these monomials do not use the generators w1, w3

and are not divisible by w2 in H0 (M〈3〉). Thus, (43) is a nontrivial relation inthe polynomial ring H0

(w−1

2 M〈3〉), a contradiction. This completes the proof

of Lemmas 7.4, 7.5 and Proposition 7.1(i).

Let i = (i1, . . . , i2s) for s ≥ 6 and 3 ≤ i1 . . . < i2s. We turn our attention tothe elements α′(i) = β2 (γ(i)) and prove Theorem 7.1(ii). Recall that we alreadyknow a projection of α′(i) into the E3,∗

2 (MSpΣ1) from Proposition 6.4(d). InE3,∗

2 (MSpΣ3), for 4 ≤ k ≤ 2s define

a(k−4)(i) = wk−42

∑1≤t1<···<tk≤2s

p(it1 , . . . , itk)x2,i1 · · · x2,it1· · · x2,itk

· · ·x2,i2s

39

where

p(j1, . . . , jt) =∑

1≤t1<t2<t3≤tujt1ujt2ujt3wj1 · · · wjt1 · · · wjt2 · · · wjt3 · · ·wjt .

Then we can rewrite the projection a′(i) of α′(i) into E3,∗2 (MSpΣ3) as

a′(i) =2s∑k=4

(−1)ka(k−4)(i).

Direct computation shows that the elements a(k−4)(i) are d1–cycles. In the fol-lowing lemma, we obtain a convenient description of a′(i).

Lemma 7.7 In the algebra E∗,∗2 (MSpΣ3) for 4 ≤ k ≤ 2s− 1 and s ≥ 6:

2a(k−4)(i) = a(k−3)(i).

In particular, there is an odd number λ such that

a′(i) = λ a(0)(i).

Proof. In E∗,∗1 (MSpΣ3), for t ≥ 3 define

h(j1, . . . , jt) =∑

1≤q1<q2≤tujq1ujq2wj1 · · · wjq1 · · · wjq2 · · ·wjt

and

c(k−4)(i) = wk−42

∑1≤t1<···<tk≤2s

h(it1 , . . . , itk)x2,i1 · · · x2,it1· · · x2,itk

· · ·x2,i2s .

Thend1c

(k−4)(i) = 2a(k−4)(i)− a(k−3)(i).

Thus, a′(i) = λ a(0)(i) where λ = 1− 2 + 4 + · · ·+ 2(2s−4) is odd.

The following lemma indicates the relationship between g(i) and a′(i).

Lemma 7.8 In the algebra E∗,∗2 (MSpΣ3) for s ≥ 6:

2λg(i) = w2a′(i).

Proof. Define

c(i) =∑

1≤t1<···<t3≤2s

h(it1 , . . . , it3)x2,i1 · · · x2,it1· · · x2,it3

· · ·x2,i2s .

Thend1(λc(i)) = 2λg(i)− w2λa

(0)(i) = 2λg(i) − w2a′(i).

Proof of Proposition 7.1(ii): Suppose that 2s−3a′(i) = 0. By the previouslemma,

0 = 2s−3w2a′(i) = 2s−2g(i).

This contradicts Proposition 7.1(i).

40

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Department of Mathematics and Statistics, York University, 4700 Keele Street,North York, Ontario M3J 1P3, Canadae-mail: borisb@nexus.yorku.ca, kochman@nexus.yorku.ca.