Adams spectral sequence and higher torsion in MSp
Abstract
In this paper we study higher torsion in the symplectic cobordism ring. We use Toda brackets and manifolds with singularities to construct elements of higher torsion and use the Adams spectral sequence to determine an upper bound for the order of these elements.
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Publicacions Matem`atiques, Vol 40 (1996), 157–193. ADAMS SPECTRAL SEQUENCE AND HIGHER TORSION IN MSp∗ Boris I. Botvinnik and Stanley O. Kochman Abstract In this paper we study higher torsion in the symplectic cobordism ring. We use Toda brackets and manifolds with singularities to construct elements of higher torsion and use the Adams spectral sequence to determine an upper bound for the order of these elements. 1. Introduction The symplectic cobordism ring MSp∗is the homotopy of the Thom spectrum MSp and classifies up to cobordism the ring of smooth manifolds with a symplectic structure on their stable normal bundles. Although MSp∗only has two-torsion, its ring structure is very complicated and is only completely understood through the 100 stem [7],[13],[15]. In [2], we proved that there are nontrivial elements in MSp∗of all orders 2k. In this paper, we construct new elements of higher torsion by means of Toda brackets, and we study their properties using the Adams spectral sequence (ASS). The following result provides the geometrical input we use to construct higher torsion elements. Its proof in Section 5 uses low dimensional calculations in the Atiyah-Hirzebruch spectral sequence for π∗MSp. Let φ0=η∈MSp1, and let φk∈MSp8k−3for k≥1 denote the Ray elements [12]. The elements of MSp∗are built from the Ray elements using Toda brackets. The most elementary ones are φm,2,φ nfor 0 ≤m<n. Gorbounov [1, p. 139], [4] showed that these triple brackets contain zero when m= 0. On the other hand, it was shown in [6, Thm. 8.1 3(c)] that these triple brackets do not contain zero when (m, n)≥(3,5) in the lexicographical order. The following theorem resolves the situation when m= 1 leaving open only the case m=2. This research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
158 B. I. Botvinnik, S. O. Kochman Theorem 1. In MSp∗, the Toda brackets φ1,2,φ ncontain zero for all n≥0. Let J=(j1,... ,j s) with 0 <j 1<j 2<··· <j s. By induction on s≥1, we define elements a[J]∈MSp∗. The following theorem describes our elements of higher torsion a[J]. Although we show how the a[J] decompose in terms of Toda brackets, the a[J] will be defined by specific representative symplectic manifolds. Theorem 2. There exist elements a[J]∈MSp∗with the following properties: (a) a[j1]=a[j1,j 2]=a[j1,j 2,j 3]=0; (b) a[j1,j 2,j 3,j 4]∈φj1φj2φj3,2,φ j4+φj3φj4φj1,2,φ j2; (c) a[2j1,... ,2js]is indecomposable for s≥5; (d) φ1a[J]=0and a[j1,... ,j s]∈φjs,2,φ 1,a[j1,... ,j s−1]for s≥5; (e) for s≥7and 1≤i1<··· <i sthe element a2i1,... ,2is∈ MSp4∗+1 has order at least 2h(s)where h(s)=[(s+1)/2] −2. Our main tool for proving Theorem 2 in Section 6 is the ASS which we apply to the spectrum MSp and the spectra MSp Σn ∗. The latter spectra classify bordism classes of symplectic manifolds with singularities Σn=(P2,... ,P n) where [Pi]=φ2i−2for 2 ≤i≤n. The spectrum MSp Σ3 ∗is especially useful to us. Let MSp Σ3 ∗ β3 −→ MSp Σ2 ∗ β2 −→ MSp∗ be the Bockstein operators. Using the ASS we first construct higher torsion elements t[J] in the ring MSp Σ3 ∗using Toda brackets. Then we define the elements a[J]∈MSp∗as β2 β3(t[J]). We prove Theorem 2 by identifying the projections of elements t[J] and a[J] in the ASS. In particular, we show that the elements 2ka[J] for 0 ≤k≤s−4 determine towers of infinite cycles in E4∗+1,2k+4 2of the ASS for MSp∗. These towers are very interesting: their heights give upper bounds for the orders of our elements. However, we show that their top halves bound by higher differentials, so only their bottom halves survive. This explains why our elements of higher torsion only have half of their potential order. To analyze the lower bounds of the orders of the a[J] we use the results of [2] which were proved using the Adams-Novikov spectral sequence. Let MSpΣn ∗denote the bordism theory with singularities Σn=(P1,... ,P n) where [P1]=η.IfJ=(2 i1,... ,2is), let i=(i1,... ,i s). In [2]
Higher torsion in MSp∗159 we constructed the higher order elements τ3(i)∈MSpΣ3 ∗of order at least 2[(s+1)/2] which defined elements α(i)∈MSp∗of order at least 2[(s+1)/2]−3. We show that the elements 2a[2i1,... ,2is] may be identified with the α(i). Our analysis in the ASS and the ANSS is far from low-dimensional. For example, the first element of order eight in MSp∗given by Theorem 2 has degree 16,377. However, if the following conjecture were true then the first of these elements of order eight would be in degree 729. Conjecture. The elements a[J]∈MSp∗of Theorem 2 are indecomposable of order 2[(s+1)/2]−2for all sequences Jof distinct positive even integers of length at least 7. All groups, rings and spectra are two-local throughout this paper. By [14],[16], the theories MSpΣn ∗(·) and MSp Σn ∗(·) have admissible commutative and associative product structures. In particular, the associativity, commutativity and Toda bracket constructions as well as all the results of [2, Section 3] are valid for all of these theories. The authors thank the referee for his careful reading of this paper and his constructive suggestions. 2. May Spectral Sequence for MSpΣn Let MSpΣn,n≥1, be the spectrum defined in the Introduction with singularities Σn=(P1,... ,P n), and let MSpΣ0denote MSp. In this section we compute the E2-term of the Adams spectral sequence (ASS): (1) Es,t 2= Cotors AH∗MSpΣn,Z/2t=⇒MSpΣn ∗. Our approach is analogous to that used in [5] in the case n=0. In particular, we use a change of rings theorem to reduce the problem of calculating E2to computing (2) CotorB(n)(Z/2,Z/2) . Here B(n) is a truncated polynomial algebra which we define as a quotient of the dual of the Steenrod algebra below. Then we use the May spectral sequence to compute the algebra (2). We compute E2of these May spectral sequences using the resolution constructed by May in [11]. Then we construct filtered polynomial DGA algebras Pnas quotients of the cobar construction which induce these May spectral sequences. We prove this from the case n= 0 of [5] by using induction on nand
160 B. I. Botvinnik, S. O. Kochman a generalized Five Lemma. Then for n≥1 we define representative cycles of the algebra generators of E2to show that these May spectral sequences collapse and that all the algebra extensions from E∞to (2) are trivial. Thus, when n≥1 the situation is much simpler than the case n= 0 where there are nonzero d2-differentials and nontrivial extensions. Consequently, for n≥1 we can describe E2of the ASS (1) in terms of five families of algebra generators and four families of relations while for n= 0 nine families of algebra generators and forty families of relations were required. We begin by recalling the structure of the homology of MSpΣnas a comodule over the dual of the Steenrod algebra A∗=Z/2[ξ1,... ,ξ k,...]. Let Sbe the A∗-primitive polynomial algebra: S=Z/2[V2,V 4,V 5,... ,V m,...] where m=2,4,5,...,m=2 l−1, and deg Vm=4m. V. Vershinin [14], [16] proves that there is an isomorphism of A∗-comodules: (3) H∗MSpΣn∼ =Z/2ξ2 1,... ,ξ2 n,ξ4 n+1,... ,ξ4 k,...⊗S for n≥0. Define the Z/2-Hopf algebra B(n)=A∗/ξ2 h,ξ 4 k|1≤h≤nand n<k with coproduct ψinduced from the coproduct of A∗. Note that in [5] the Hopf algebra B(0) is denoted as B. By (3), the problem of computing E2of the ASS (1) is greatly simplified by Liulevicius’s interpretation [10, Corollary I.5] of the Cartan-Eilenberg change of rings theorem [3, Proposition VI.4.1.3] which gives an isomorphism of Z/2-algebras: (4) E2= CotorA∗H∗MSpΣn,Z/2∼ =CotorB(n)(Z/2,Z/2) ⊗S. To compute the cohomology of the B(n) we use the May spectral sequence [11]: E2= CotorE0B(n)(Z/2,Z/2) =⇒CotorB(n)(Z/2,Z/2) . Recall that this spectral sequence is defined by giving B(n) the coproduct filtration F0B(n)⊂F−1B(n)⊂···⊂F−pB(n)⊂··· where by induction on p≥1 F0B(n)=Z/2,F −pB(n)=b∈B(n)|ψ(b)∈F−p+1B(n)⊗IB(n). Here ψdenotes the reduced coproduct: ψ(b)=ψ(b)−b⊗1−1⊗b, and IB(n) denotes the augmentation ideal of B(n). The following lemma describes the structure of the Hopf algebra E0B(n). It is an immediate consequence of the coalgebra structure of A∗and the definition of the B(n).
Higher torsion in MSp∗161 Lemma 2.1. There is an isomorphism of Hopf algebras: E0B(n)∼ =Eξ(1) j|1≤j⊗Eξ(2) k|n<k where the elements ξ(1) j,1≤j≤n+1,ξ(2) k,n<k, are primitive and ψξ(1) j=ξ(2) j−1⊗ξ(1) 1for j≥n+2. As in [5, Section 1], we compute the E2-term of these May spectral sequences by using the methods of May [11, Section 5] to construct a DGA D(n) whose homology is isomorphic to CotorE0B(n)(Z/2,Z/2) . In the notation of [5] and [11], we define the DGA D(n)=Z/2sξ(1) j,sξ (2) k|j≥1,k>n with differential: dsξ(1) j=0 for j≤n+1 sξ(1) 1sξ(2) j−1for j≥n+2, dsξ(2) k= 0 for k>n. The following lemma is a straightforward generalization of [5, Lemmas 1.4, 1.5 and Theorem 1.6]. Lemma 2.2. There is an isomorphism of algebras: H∗D(n)∼ =CotorE0B(n)(Z/2,Z/2) . We will use the elements defined below to compute the homology of the D(n). Definition 2.3. In the algebra CotorE0B(n)(Z2,Z 2)∼ =H∗D(n) define the following elements: h=sξ(1) 1,rk=sξ(2) k+1for k≥n, qj=sξ(1) j+2for 0 ≤j<n,q2 k=sξ(1) k+22for k≥n, p(m1,... ,m s)=s i=1 sξ(2) mi+1sξ(1) m1+2 ... sξ(1) mi+2 ...sξ(1) ms+2
162 B. I. Botvinnik, S. O. Kochman for 0 ≤m1<···<m s. Note 2.1. We will also need the following degenerate cases of these elements: rm= 0 for m<n, q2 m=q2 mfor m<n, p(m)=rm,p(m, m)=0,(5) p(m1,... ,m s,m,m)=p(m1,... ,m s)q2 mfor s≥1. The homology of the D(n) can be computed as in [10, Proposition I.11]. Lemma 2.4. For n≥1, the elements h, rk,q j,q2 k,p(m1,... ,m s) for k≥n,0≤j<n,0≤m1<···<m sare generators of the algebra CotorE0B(n)(Z/2,Z/2) . A complete set of relations among these generators is given by the degeneracy relations of Note 2.1 and by: (1) p(m, m1,... ,m s)=p(m1,... ,m s)qmfor m<nand s≥1; (2) hp (m1,... ,m s)=0; (3) s i=1 rmip(m1,... ,mi,... ,m s)=0; (4) p(m1,... ,m s)p(g1,... ,g t) =t i=1 rgip(m1,... ,m s,g 1,... ,gi,... ,g t). We use the methods of [5, Section 3], to show that E2=E∞and that all the extensions are trivial in the May spectral sequence of B(n) for n≥1. That is, we construct polynomial DGAs Pnwhose homology is CotorB(n)(Z2,Z 2). To avoid repeating an analogue of the proof given in [5, Section 3], we use the following lemma which shows how we automatically obtain the Pn,n≥1, with the required properties from the Pconstructed in [5, Section 3]. Let C(Z/2,A,Z/2) denote the cobar construction for A, a connected Z/2-Hopf algebra. Suppose we have a DGA Pand a Z/2-linear map λ:A−→ P. The map λinduces an algebra homomorphism λ:C(Z2,A,Z 2)−→ Pwhich we assume is a map of DGAs. We also assume that the algebra homomorphism λ∗: CotorA(Z/2,Z/2) −→ H∗P
Higher torsion in MSp∗163 induced by λis an isomorphism. Suppose that we have a primitive element xin the center of Awhich is not a zero-divisor. Let A1=A/(x),y=λ(x) and P1=P/(y). Lemma 2.5 (Generalized Five Lemma).Assume that we have a Z/2Hopf algebra A, a DGA P,aZ/2-linear map λ:A−→ Pand elements x,yas above which satisfy the following additional conditions: (i) x2=0; (ii) yis central in P; (iii) λ(IA·x)=0. Then λinduces a map of DGAs λ1:C(Z/2,A 1,Z/2) −→ P1such that λ1∗: CotorA1(Z/2,Z/2) −→ H∗P1 is an algebra isomorphism. Proof: Let M=Z/2⊕Z/2(X), with deg X= deg x, denote a comodule over the algebra A. The comodule structure on M,ψ:M−→ M⊗A, is induced by: ψ(X)=X⊗1+1⊗x. Then the following cobar constructions give a short exact sequence of DGAs: 0→C(Z/2,A,Z/2) j →C(M,A,Z/2) ρ →C(Z/2(X),A,Z/2) →0 where j(a)=a+0Xand ρ(a+bX)=bX for a,b∈Z/2. Consider the diagram (6) below. In this diagram, j=γ◦j,ρ=α◦ρ,α(aX)=aand γ(a+bX)=π(a) where π:P→P1and π:A→A1are the canonical projection maps. By condition (iii), λinduces a map of DGAs λ1making the trapezoid in (6) commute. By condition (i), the exterior algebra E(x) is a sub-Hopf algebra of A. Therefore, γ∗in (6) is an isomorphism by the change of rings theorem [10, I.5]. We use the abbreviations CotorA= CotorA(Z/2,Z/2) and CotorA1= CotorA1(Z/2,Z/2).
164 B. I. Botvinnik, S. O. Kochman (6) 0PP P 10 ✲ ✲ y✲ π✲ ❄ λ ✲ ✲ j0C(M,A,Z/2)0 C(Z/2,A,Z/2) ✲ ρ✲ C(Z/2(X),A,Z/2) C(Z/2,A,Z/2) ❄ α ∼ = C(Z/2,A 1,Z/2) ❄ γ ❄ λ1 ❍❍❍❍❍❍❍ ❍❥ j❍❍❍❍❍❍❍ ❍❥ ρ The short exact sequences on the top and bottom rows of this diagram induce the following long exact sequences in homology. (7) ··· H∗PH ∗P ··· CotorA ✲ ✲ ∂ CotorA✲ j ∗CotorA1 ✲ ˜ρ∗CotorA ❄ λ∗ ∼ = ✲ ✲ yH∗P1 ✲ π∗H∗P ✲ ∂ ✲··· ✲··· 1 ✍✌ ✎ 2 ✍✌ ✎ 3 ✍✌ ✎ ❄ λ∗ ∼ = ❄ λ1∗ ❄ λ∗ ∼ = In this diagram ∂=∂◦α−1 ∗and ˜ρ∗=ρ ∗◦γ−1 ∗. We show that diagram (7) commutes. It then follows from the usual Five Lemma that λ1∗is an isomorphism. Square 1 commutes because λ∗∂{Z}=λ∗j−1dρ−1(XZ)=λ∗j−1d(XZ) =λ∗j−1(xZ)=λ∗{xZ}=λ∗{x}λ∗{Z}=yλ∗{Z}. Square 2 commutes because the trapezoid in (6) commutes. Note that d(Z+XZ)=d(Z)+xZ +Xd(Z) in C(M,A,Z/2). Thus, if Z+XZ is a cycle then Z is a cycle and d(Z)=xZ. Therefore, Square 3 commutes because λ∗˜ρ∗γ∗{Z+XZ}=λ∗α∗{XZ}=λ∗{Z}and ∂λ1∗γ∗{Z+XZ}=∂λ1∗{Z}=∂{πλ(Z)}={dλ(Z)/y}={λd(Z)/y} ={λ(xZ)/y}={λ(x)λ(Z)/y}=λ∗{Z}. We will need the following generalization of the previous lemma which follows from it by induction on n≥1.
Higher torsion in MSp∗165 Lemma 2.6. Let the Z/2-Hopf algebra A, the algebra Pand the Z/2linear map λ:A→Pbe as above. Let x1,... ,x n,... be a sequence of elements in the center of A.LetI0=0,In=(x1,... ,x n)for n≥1 and An=A/In.Letyn=λ(xn),Jn=(y1,... ,y n), and Pn=P/Jn. Assume that: (i) the ideals Inare prime and invariant; (ii) x2 n∈In−1for n≥1; (iii) the ynare central in P; (iv) λ(IA·xn)=0for n≥1. Then for n≥1,λinduces maps of DGAs λn:C(Z2,A n,Z 2)→Pnsuch that the λn∗: CotorAn(Z2,Z 2)→H∗Pn are algebra isomorphisms. We apply this lemma to complete our analysis of the May spectral sequence for CotorB(n)(Z/2,Z/2) thereby computing E2of the ASS (1). Recall from Lemma 4 that E2∼ =CotorB(n)(Z/2,Z/2) ⊗S, where Sis the polynomial algebra with generators Va,a=2 k−1. Let |c|denote the degree of c. Theorem 2.7. Let n≥1. Then E2of the ASS for MSpΣn ∗is the algebra generated by Va,|Va|=(0,4a),a=2 l−1; h0,|h0|=(0,0); Rk,|Rk|=(1,2k+2 −3),k≥n; Qj,|Qj|=(1,2j+2 −2),0≤j<n; Q2 k,|Q2 k|=(2,2k+3 −4),k≥n; P(m1,... ,m s),|P(m1,... ,m s)|=(s, 2m1+2 +···+2 ms+2 −2s−1), 0≤m1<···<m s. A complete set of relations is given by: (1) P(m, m1,... ,m s)=P(m1,... ,m s)Qmfor m<nand s≥1; (2) h0P(m1,... ,m s)=0; (3) s i=1 RmiP(m1,... , mi,... ,m s)=0; (4) P(m1,... ,m s)P(g1,... ,g t) =t i=1 RgiP(m1,... ,m s,g 1,... ,gi,... ,g t);
172 B. I. Botvinnik, S. O. Kochman It follows from the change of rings theorem [10, Corollary I.5] that for n≥1 there is an isomorphism of Z/2-algebras: (12) E2= CotorA∗H∗MSp Σn,Z/2∼ =Cotor B(n)(Z/2,Z/2) ⊗S. Define the sub-Hopf algebra C(n)of B(n)by C(n)=Z/2[ξ1,ξ k|n<k]/ξ4 1,ξ 4 k|n<k . Since the ξh,2≤h≤n, are primitive in B(n), B(n)∼ = C(n)⊗E(ξ2,... ,ξ n) as Hopf algebras. Let Qh−1denote the homology class of [ξh] for 2 ≤ h≤n. We thus have the following lemma. Lemma 4.1. For n≥2, there is an isomorphism of Z/2-algebras: (13) Cotor B(n)(Z/2,Z/2)∼ =Cotor C(n)(Z/2,Z/2)⊗Z/2[Q1,... ,Q n−1]. We compute Cotor C(n)(Z/2,Z/2) thereby determining E2of the ASS for MSp Σn. Recall from [5, Theorem 3.7] that CotorB(Z/2,Z/2) can be described as the algebra generated by h0and by seven families F(k1,... ,k t) of generators with forty familites of relations. In particular, Fis one of the following symbols: q(t= 1), Q(t≥1), R(t= 1), P(t= 2), P2(t≥3), Y(t≥7) or Zs(t≥s+2≥4). Proposition 4.2. For n≥2, let Cndenote the subalgebra of CotorB(Z/2,Z/2) generated by h0Rk1 qk1Q(k1,... ,k t)(t≥1) P(k1,k 2)P2(k1,... ,k t)(t≥3) Y(k1,... ,k t)(t≥7) Zs(k1,... ,k t)(t≥s+2≥4) where each of the kiis either zero or greater than or equal to n. Then E2of the ASS for MSp Σnis given by E2∼ =Cn⊗Z/2[Q1,... ,Q n−1]⊗S.
Higher torsion in MSp∗173 Proof: By (12), (13), E2=Cotor C(n) (Z/2,Z/2)⊗Z/2[Q1, . . .,Qn−1]⊗S. Let ιn: C(n)→Bdenote the inclusion map. Define a map σn:B→ C(n) of Hopf algebras which splits ιnby σn(ξk)=ξkif k=1ork>n 0if2≤k≤n. Then σn∗is a splitting of the inclusion ιn∗: Cotor C(n)(Z/2,Z/2) :→CotorB(Z/2,Z/2) . Thus, we view Cotor C(n)(Z/2,Z/2) as a subalgebra of CotorB(Z/2,Z/2). The effect of σn∗on the algebra generators F(k1,... ,k t)of CotorB(Z/2,Z/2) is given by σn∗(h0)=h0and (14) σn∗(F(k1,... ,k t)) = F(k1,...,k t)if{k1,...,k t}∩{1,... ,n−1}=∅ 0 otherwise for Fone of q,Q,R,P,P2,Yor Zs. Observe that Cn= Image ιn∗is the subalgebra of CotorB(Z/2,Z/2) spanned by all F(k1,... ,k t) with {k1,... ,k t}⊂{1,... ,n−1}. Thus by (14), σn∗:Cn→Image σn∗is an isomorphism. Therefore, Cotor C(n)(Z/2,Z/2) = Image σn∗∼ =Cn. Note 4.1. The map πr,r≥2, of ASS induced by the canonical map of spectra π:MSp →MSp Σndoes not induce the projection map σn∗⊗1:E2∼ =CotorB(Z/2,Z/2) ⊗S→E2= Cotor C(n)(Z/2,Z/2) ⊗S. For example, π2(P(1,n)) = Q1Rnwhile (σn∗⊗1) (P(1,n))=0. We conclude with an alternate description of E2in terms of the projections Φnof the Ray elements φnto the ASS. If Φk= j≥0 RjVI(k,j)∈E1,8k−3 2 in E2of the ASS for MSp and F(k1,... ,k t) is one of the above seven families of algebra generators of E2of the ASS for MSp Σnthen define F(k1,... ,k t)= j1≥0 ··· jt≥0 F(j1,... ,j t)VeF I(k1,j1)...VeF I(kt,jt) where eFequals 1, 4, 2, 1, 2, 1, 2 if Fequals R,q,Q,P,P2,Y,Zs, respectively. We denote Qkas Ψk. We do not explicitly specify the forty relations in E2of the ASS for MSp Σninduced from E2of the ASS for MSp because we do not use them in this paper.
174 B. I. Botvinnik, S. O. Kochman Corollary 4.3. For n≥2,E2of the ASS for MSp Σnis the Z/2algebra generated by Ψk(1 ≤k<2n−1)Va,a=2 r−1 h0Rk qkQ(k1,... ,k t)(t≥1) P(k1,k 2)P2(k1,... ,k t)(t≥3) Y(k1,... ,k t)(t≥7) Zs(k1,... ,k t)(t≥s+2≥4). A complete set of relations for E2is given by the forty relations listed in [5, Theorem 3.7] as well as the following relations: (a) if 0<k t<2n−1and Fis Q,P2,Yor Zsthen F(k1,... ,k t)=F(k1,... ,k t−1)Ψ eF kt; (b) if 0<k 1<2n−1then Rk1=0;qk1=Ψ 4 k1;P(k1,k 2)=Ψ k1Rk2; (c) if Fis any of the above ten families except h0or Vathen F(k1,... ,k t)= j1≥0 ··· jt≥0 F2j1−1,... ,2jt−1VeF I(k1,j1)...VeF I(kt,jt). From now on we only use the description of E2in terms of the F(k1,... ,k t), and we abuse notation by denoting them as F(k1,... ,k t). 5. Construction of Higher Torsion Elements In this section we prove Theorem 1 and use it to construct elements of higher torsion. The vanishing of the Toda brackets φ1,2,φ nof Theorem 1 allows us to construct specific Sp-manifolds Vnwith no singularities in Lemma 5.3 such that ∂Vnis the canonical element in this Toda bracket. Let J=[j1,... ,j s] with s≥1 and J=[j1,... ,j s−1] with s≥2 throughout this section. In Proposition 5.4 we use the Vnto generalize the constructions of Section 5 of [2] to construct the elements t[J]∈MSp Σ3 ∗which define the elements g[J]= β3(t[J]) ∈MSp Σ2 ∗ and a[J]= β2( β3(t[J])) ∈MSp∗described in the Introduction for J=[j1,... ,j s]. We give the basic properties of the t[J] and g[J] including their Toda bracket decompositions and their projection in the
Higher torsion in MSp∗175 ASS. We abbreviate those constructions which are analogous to those of [2]. We begin with the proof of Theorem 1. Its proof relies on decomposing φ1as a triple Toda bracket based upon the smash product. Recall from [8] the definition of this type of Toda bracket. We are given three maps of spectra α:S→E,β:S→F,γ:S→Gand associative pairings of spectra ωEF :E∧F→M, ωFG :F∧G→N, ωMG :M∧G→P, ωEN :E∧N→P such that ωEF (α∧β)=0andωFG (β∧γ)=0. Letξ:D→Mbe an extension of ωEF ◦(α∧β) to a disc and let ζ:D→Nbe an extension of ωFG ◦(β∧γ) to a disc. Then α, β, γis defined as the set of homotopy classes of all maps (ωMG ◦(ξ∧γ)) ∪(ωEN ◦(α∧ζ)) : S=(D∧S)∪(S∧D)→P for all choices of ξand ζ. We identify such a Toda bracket in the case E=F=Sand G=MSp which decomposes φ1. We also give a similar decomposition of φ2in terms of a four-fold Toda bracket. Recall that MSp8=Zwith the generator q0. Lemma 5.1. Let µ:S→MSp denote the unit of the spectrum MSp. Then (a) φ1=η,ν,µ; (b) φ2∈η,ν,σ,µ={φ2,φ 2+φ1q0}. Proof: The proof of this lemma is based upon the analysis of the following Atiyah-Hirzebruch spectral sequence. (15) E2 ∗,∗=H∗MSp⊗πS ∗=⇒MSp∗. This spectral sequence was analyzed through degree 50 in [9]. Fortunately, we only require its structure through degree 5 which is depicted in Figure 1. We use the notation H∗MSp =Z[b1,... ,b n,...] where H∗HP∞has the Z-basis {b1,... ,b n,...}. The only differential in our range is d4(b1)=ν. (a) Since MSp5=Z/2φ1and the only infinite cycle in E2 ∗,∗of degree 5 is ηb1, the only possibility for the projection of φ1to E∞ ∗,∗is ηb1. The fact that ηb1is an infinite cycle of (15) means that if B1:D→MSp represents b1such that B1|S=νand ξ:D→Ssuch that ξ|S=η∧ν then φ1is represented by: η∧B1∪ξ∧µ∈η,ν,µ.
176 B. I. Botvinnik, S. O. Kochman Note that we have suppressed the canonical pairings of spectra involved in the previous statement. Since µ∗πS 5= 0 and η·MSp4= 0, the indeterminacy of η,ν,µis zero. ✲ H∗(MSp;Z) ✻ πS ∗ t t t t t t t b1 η η2 ν 0 0 ηb1 ❆ ❆ ❆ ❆ ❆ ❆❑ d3 Figure 1: The Atiyah-Hirzebruch Spectral Sequence for MSp∗ (b) Observe that η,ν,σ⊂πS 12 = 0 and ν, σ, µ⊂MSp11 = 0. Thus, the Toda bracket η,ν,σ,µ⊂MSp13 is defined. Consider any defining system of λ∈η,ν,σ,µand let ξbe the element of this defining system whose boundary is an element of ν, σ, µ. Then ξprojects to a nonzero element X∈H12 (MSp;Z) in the zero row of the Atiyah-Hirzebruch spectral sequence (15) which is not divisible by two. If bωis a monomial summand of Xwith a coefficient that is nonzero modulo two then ηξ is a unionand of λand sω(λ)=η. Since MSp13 =Z2φ2⊕Z2φ1q0 and sω(φ1q0) = 0, it follows that λ=φ2+kφ1q0for some k∈Z/2. Note that the indeterminacy of η,ν,σ,µcontains η,ν,MSp8which contains φ1q0by (a). Thus, η,ν,σ,µ={φ2,φ 2+φ1q0}as asserted. Our representative of φ1can be described in terms of (Sp,fr)- manifolds as η×Y4∪W5 where Y4is an Sp-manifold with ∂Y 4=νand W5is a framed manifold with ∂W5=η×ν. Recall from [12] that the Ray elements φnare closed under the action of the Landweber-Novikov operations. In particular, s∆2k(φm)=φm−k if 1 ≤k<m.By[6, Theorem 11.4], the action of the LandweberNovikov operations on the Toda brackets φm,2,φ nsatisfies the Cartan
Higher torsion in MSp∗177 formula: sωφm,2,φ n⊂ ω=ω1+ω2 sω1(φm),2,s ω2(φn). We thus have the following formula for the action of the s∆2kon our Toda brackets. Lemma 5.2. For m>k≥1,s∆2kφ1,2,φ m⊂φ1,2,φ m−k. We use the action of the s∆2kon our Toda brackets and the decomposition of φ1to prove Theorem 1. Proof of Theorem 1: By Lemma 5.1, φn,2,φ 1=φn,2,η,ν,µ which contains an element which is also an element of φn,2,η,ν,µ+φn,2,η,ν,µ=φn,0,µ+ηA,ν,µ. The last equality uses Gorbunov’s Theorem [1, Theorem 4.3.5] which says that 0 ∈φn,2,η. Therefore, any element of φn,2,ηis of the form ηA. By Lemma 5.1 and the observation that ηA ·MSp4=0,we see that φn,2,φ 1contains an element which is also an element of φn·MSp6+Aφ1modulo Image µ∗. This sum is contained in the ideal spanned by φ1modulo Image µ∗. Thus, for all n, we conclude that φn,2,φ 1contains an element which is in Image µ∗. By Lemma 5.2, s∆2nφ2n,2,φ 1⊂φn,2,φ 1. Recall that an element in the image of the unit µ∗of MSp is annihilated by all Landweber-Novikov operations. It follows that φn,2,φ 1contains zero. The main technique which we use in constructing the t[J] is the existence of Sp-manifolds Vjas in the following lemma. The proof of this lemma is based upon Theorem 1. We abuse notation below by denoting a cobordism class φnand an Sp-manifold representing φnby the same symbol φn. Lemma 5.3. There are Sp-manifolds ψnfor n≥0and Vnfor n≥2 such that ∂ψ1=φ1×2, ∂ψn=2×φnfor n=1, ∂Vn=ψ1×φn∪φ1×ψn.
178 B. I. Botvinnik, S. O. Kochman In particular, ψ1does not depend on n. Proof: By Theorem 1, there are Sp-manifolds ψ(n) 1,ψnfor n≥1 and V nfor n≥2 such that ∂ψ(n) 1=φ1×2, ∂ψn=2×φnand ∂V n= ψ(n) 1×φn∪φ1×ψn. Let ψ1=ψ(2) 1. Since MSp6=Z/2ηφ1,ψ1∪−ψ(n) 1 is bordant to knηφ1for some kn∈Z/2. By Theorem 1, 0=φn,2,φ 12=φn2,φ 1,2=φnφ1η in MSp∗noting that 2,φ 1,2=φ1[∆(2)] = φ1ηby [2, Lemma 3.3 and Note 3.1]. Thus, there exists an Sp-manifold Ynwith ∂Yn=ψ1×φn∪−ψ(n) 1×φn. Define Vn=V n∪Yn. Then ∂Vn=φ1×ψn∪ψ1×φnas required. We are now ready to construct t[J]∈MSp Σ3 ∗. We denote the product construction of Σ3-manifolds by m3, the associativity construction by A3 and the commutativity construction by K3. Proposition 5.4. For each J=[j1,... ,j s], there exists an element t[J]∈MSp Σ3 ∗with the following properties. (a) t[j1]=φj1. (b) ψ1t[J]=0. (c) t[j1,... ,j s]∈φjs,ψ 1,t[j1,... ,j s−1]for s≥2. (d) If jk=2 ik−2for 1≤k≤sand i=(i1,... ,i s)then λ3∗(t[J])= τ3(i)under the canonical map λ3:MSp Σ3→MSpΣ3. (e) t[J]projects to t[J]= s k=1 ΦjkV1,j1... V1,jk...V 1,js. in both E1,4∗+1 2of the ASS for MSpΣ3 ∗and E1,4∗+1 2of the ASS for MSp Σ3 ∗. (f) There are ν0(j)∈MSp Σ2 ∗which project to the infinite cycles h0V1,j in E1,4∗ 2of the ASS for MSp Σ2such that for s≥2, 2t[J]= m3(ν0(js),t[J]) .
Higher torsion in MSp∗179 Proof: (a)-(c) We construct the t[j1,... ,j s] by induction on s≥1to satisfy (a)-(c) as in the proof of [2, Lemma 5.3]. (d) To insure that the t[J] map to the τ3(i) under λ3∗we must be careful how we choose t[J] in the Toda bracket of (c). In particular, for each sequence [j1,... ,j s] we proceed as in the proof of [2, Lemma 5.4] to use induction on s≥1 to define Σ3-manifolds Hsand Tssuch that: (1) T1=φj1and H1=Vj1; (2) δHs= m3(ψ1,T s); (3) For s≥2, Ts=φjs×Hs−1∪ m3V js,T s−1, Hs= m3(Vjs,H s−1)∪− A3(Vjs,ψ 1,T s−1)∪− m3( K3(Vjs,ψ 1),T s−1) ∪ m3(B×φjs,T s−1)∪ A3(ψ1,V js,T s−1) where V js=Vjs∪ K3(φjs,ψ 1) and Bis a Σ3-manifold with δ(B)= K3(ψ1,ψ 1). Such a Σ3-manifold Bexists because MSp Σ3 13 =0. By[2, Lemma 5.4], t[J] defined as the Σ3-cobordism class of Tsmaps under λ3∗to τ3(i). (e) By induction on s≥1, we prove that Tsprojects to t[j1,... ,j s]∈ E1,4∗+1 2and Hsprojects to V1,j1...V 1,js∈E0,4∗ 2in the ASS for MSp Σ3. The case s= 1 follows from (1). If s≥2, the induction hypothesis and (3) show that the projection of Tsto the one line of the ASS equals φjsV1,j1...V 1,js−1+V1,jst[j1,... ,j s−1]=t[j1,... ,j s]. Since ψ1,Ts−1and φjshave Adams filtration degree one, the projections of the manifolds A3(V1,js,ψ 1,T s−1), m3( K3(Vjs,ψ 1),T s−1), m3(B×φjs,T s−1) and A3(ψ1,V 1,js,T s−1) to the zero line of the ASS are trivial. Thus by (3), the projection of Hsto the zero line of the ASS equals the projection of m3(Vjs,H s−1) which by the induction hypothesis is V1,js·V1,j1...V 1,js−1. (f) The element 2t[J] is represented by the manifold 2φjs×Hs−1∪2 m3(V js,T s−1)∪−δ(ψjs×Hs−1) which is bordant to m32V js∪ψjs×ψ1,T s−1)= m3(ν0(js),T s−1
180 B. I. Botvinnik, S. O. Kochman where ν0(js) is defined as the Σ2-cobordism class of 2V js∪ψjs×ψ1which projects to h0V1,jsin E1,4∗ 2of the ASS for MSp Σ2. Consider the T[J], H[J] constructed above as a Σ2-manifold T[J], H[J], respectively. Then δ H[J]= m2ψ1, T[J]∪φ2×E[J], δ T[J]=φ2×G[J], δE[J]= m2(ψ1,G[J]) where G[J]= β3(T[J]) represents the Σ2-cobordism class g[J]. To identify the projection of g[J] into the ASS we need to know the projection of E[J] into the ASS. Lemma 5.5. (a) E[j1]=∅and E[j1,j 2]=φj1φj2. (b) For s≥2,E[J]projects in E2,4∗+2 2of the ASS for MSpΣ2and in E2,4∗+2 2of the ASS for MSp Σ2to e[j1,... ,j s]= 1≤t1<t2≤s Φjt1Φjt2V1,j1... V1,jt1... V1,jt2...V 1,js. Proof: (a) We can take T[j1]=φj1and H[j1]=Vj1as a Σ2-manifold with δVj1=ψ1×φj1. Thus, E[j1]=∅. It will follow from (iii) below that E[j1,j 2]=φj2φj1. (b) Observe that just as in the proof of [2, Lemma 6.2(b)], we can use induction on s≥2 to construct Σ2-manifolds Ts, Hs,Esand Lssuch that: (i) Tsrepresents t[J]; (ii) δ Hs= m2W2, Ts∪φ2×Es∪ Ls; (iii) E[J]= m2V js,E[J]∪φjs× T(j1,j 2); (iv) Tsprojects in the one line of the ASS for MSp Σ2and in the one line of the ASS for MSpΣ2to t[J]∈E1,4∗+1 2; (v) Hsprojects in the zero line of the ASS for MSp Σ2and in the zero line of the ASS for MSpΣ2to V1,j1...V 1,js∈E0,4∗ 2; (vi) Esprojects in the two line of the ASS for MSp Σ2and in the two line of the ASS for MSpΣ2to 1≤t1<t2≤s Φjt1Φjt2V1,j1... V1,jt1... V1,jt2...V 1,js∈E2,4∗+2 2;
Higher torsion in MSp∗181 (vii) Lshas Adams filtration degree four. Using this lemma, we determine the basic properties of the g[J]. Proposition 5.6. The elements g[J]= β3(t[J]) ∈MSp Σ2 ∗satisfy the following conditions. (a) g[j1]=g[j1,j 2]=0. (b) g[j1,j 2,j 3]=φj1φj2φj3. (c) ψ1g[J]=0. (d) g[J]∈φjs,ψ 1,g[J]for s≥4. (e) For s≥3,g[J]projects in E3,4∗+3 2of the ASS for MSpΣ2and in E3,4∗+3 2of the ASS for MSp Σ2to g[J]= 1≤t1<t2<t3≤s Φjt1Φjt2Φjt3V1,j1... V1,jt1... V1,jt2... V1,jt3...V 1,js. (f) 2g[J]= m2(ν0(js),g[J]) for s≥2. Proof: (a)-(d) These statements are proved in the same way as the analogous statements in [2, Proposition 6.3(a)-(c)]. In particular, g[J]= β3(t[J]) is represented by the Σ2-manifold (16) G[J]= m2V js,G[J]∪φjs×E[J]. (e) We use induction on s≥3. The case s= 3 follows from (b). Assume the case s−1. By (16), g[J] projects in the three line of the ASS to g[J]=V1,jsg[J]+Φ jse[J]. By the induction hypothesis and the previous lemma, g[J]=V1,js 1≤t1<t2<t3≤s−1 Φjt1Φjt2Φjt3V1,j1... V1,jt1... V1,jt2... V1,jt3...V 1,js−1 +Φ js 1≤t1<t2≤s−1 Φjt1Φjt2V1,j1... V1,jt1... V1,jt2...V 1,js−1. This is the asserted value of g[J] in (e). (f) By (16), 2g[J] is represented by the manifold 2G[J]=2 m2V js,G[J]∪2φjs×E[J]∪−δ(ψjs×E[J]) which is bordant to m22V js∪ψjs×ψ1,G[J]= m2(ν0(js),G[J]).
188 B. I. Botvinnik, S. O. Kochman Proposition 6.4. (a) For s≥3, a[1,j 1,... ,j s]=d2 1≤h<k≤s ΦjhΦjkV1,j1... V1,jh... V1,jk...V 1,js . (b) Let j1,... ,j sbe distinct even natural numbers with s≥4. Then the element a[j1,... ,j s]=0in E4,4∗+1 ∞if and only if s=4. Proof: (a) Let D denote the sum of all distinct elements of the given form. Observe that a[1,j 1,... ,j s]= D Φj1Φj2P(1,j 3)V1,j4...V 1,js =d2 D Φj1Φj2V1,j3...V 1,js. (b) When s=4,a[j1,j 2,j 3,j 4]=d2(P(j1,j 2)Vj3,j4+P(j3,j 4)Vj1,j2). If s≥5 write jr=2j r. Then a[j1,...,j s]= 1≤t1<t2<t3<t4≤s Y(jt1,j t2,j t3,j t4)V1,j1... V1,jt1... V1,jt4...V 1,js which can not be a d2-boundary because V1,j1=V1,2j 1=V2j1+1,. . ., V1,js= V1,2j s=V2js+1 are distinct indecomposable elements of E0,4∗ 2. Theorem 2 implies that when the entries of Jare distinct powers of two then the elements in the bottom half of the tower in Figure 2 represent nonzero elements of MSp∗. We will show that all of the remaining elements in the top half of the tower in Figure 2 are boundaries in the ASS. We begin by introducing notation that we will need to describe specific elements in the ASS for MSp. Recall the d2-cycles Σ(a, b, c)∈ E1,4∗+1 2which were defined in (9). Let A1,B 1,... ,A n,B nbe a sequence such that each (Ak,B k) equals either (1) a pair of non-negative integers, (2) (1,Σ(1,x,y)) or (3) (0,Σ(1,x,y)). Let V1,Σ(1,x,y)=V1,xV1,y and V0,Σ(1,x,y)=V0,1Vx,y + V0,xV1,y +V0,yV1,x +V1,x,y. Thus, in all three cases we have elements VAk,Bkin E2of the ASS for MSp such that: d2(VAk,Bk)∈Ak,h 0,B k
Higher torsion in MSp∗189 where this Toda bracket is defined in E2=H∗(P⊗S). Thus, as in [5, Definition 7.12(19a)], we can define the following elements of E2n−2k,4∗+1 2: ζkY(A1,B 1,... ,A n,B n) = 1≤j1<···<jk≤n YA1,B 1,... , Aj1, Bj1,... , Ajk, Bjk,... ,A n,B n VAj1,Bj1...V Ajk,Bjk where 2 ≤nand 0 ≤k≤n−2. Lemma 6.5. All of the elements ζkY(A1,B 1,... ,A n,B n)are infinite cycles which are zero in E∞of the ASS for MSp where 2≤nand 0≤k≤n−2. Proof: A proof analogous to that of Proposition 6.1(b), shows that for 1 ≤k≤n−2, twice ζkY(A1,B 1,... ,A n,B n) equals ζk−1Y(A1,B 1,... ,A n,B n) by a nontrivial extension of degree one. Observe that ζn−2Y(A1,B 1,...,A n,B n)=d2 1≤i≤n ΦAiΦBiVA1,B1... VAi,Bi...V An,Bn . By [6, Theorems 12.1,12.4], all the ζkY(A1,B 1,... ,A n,B n), 0 ≤k≤ n−2, are boundaries in the ASS. Thus, they are infinite cycles which are zero in E∞. In the next two lemmas we identify the elements in the top half of the tower in Figure 2 in terms of various ζkY(A1,B 1,... ,A n,B n). We will use the following notation. Let i=[i1,... ,i p] and k=[k1,... ,k 2q]. Define i=[1,i 1,... ,1,i p] and k =[1,Σ(1,k 1,k 2),... ,1,Σ(1,k 2q−1,k 2q)] . Lemma 6.6. Let J=[j1,... ,j 2t+']with 3≤t,0≤s≤t−2and E=0,1. Then a2t−s+'−4[J] = s α=0 t β=t−α ζs−αY(i,j 2t−2α+'−3,j 2t−2α+'−2,k,j 2β+'−1,j 2β+')
190 B. I. Botvinnik, S. O. Kochman where i=[j1,... ,j 2t−2α+'−4]and k=j2t−2α+'−1,... ,j2β+'−1,j2β+',... ,j 2t+'. Proof: Let Λkdenote either Φkor Ψk. Consider the above double sum as an element Dof P⊗S. Write Das a polynomial in the canonical generators h0,Φ k,Ψ k,Va,b and V2nof P⊗S. Then each monomial summand of Din P⊗Shas a factor of maximal length of the following form: (18) Λj2i1+−1Λj2i1+···Λj2ip+−1Λj2ip+Vj2k1+−1,j2k1+···Vj2kq+−1,j2kq+ where i1<···<i p,k1<···<k q. Each of the factors Λj2ir+−1Λj2ir+ in (18) comes from either i,j2t−2α+'−3,j2t−2α+'−2or j2β+'−1,j2β+'. Thus, p≤t−α. The remaining t−α−ppossible sources for factors Λj2ir+−1Λj2ir+in (18) must be producing factors V1,j2ir+−1V1,j2ir+. The total number of such factors VAh,Bhin the summand with the factor (18) is s−α. Thus, t−α−p≤s−α≤t−α−2 and 2 ≤p. Each of the factors Vj2kr+−1,j2kr+in (18) comes from either j2t−2α+'−3,j2t−2α+'−2, k or j2β+'−1,j 2β+'. Thus, (1) k1≥t−α−1; (2) ip−1≤t−α−1; (3) if ip−1=t−α−1 then ip=βand ip−1<k 1. Therefore, there are three types of factors (18): (I) q≥1 and ip−1<k 1<i p; (II) q≥1 and ip<k 1; (III) q=0. Observe that each factor of type I occurs twice: t−α−1=ip−1,β=ipand t−α−1=k1,β=ip. Observe that each factor of type II occurs 2qtimes: t−α−1=ip,β=kr(1 ≤r≤q) t−α−1=ip−1,β=ip t−α−1=k1,β=kr(2 ≤r≤q). Observe that each factor of type III occurs once: t−α−1=ip−1,β=ip.
Higher torsion in MSp∗191 The sum of the summands with factors of type III give exactly 1≤h1<···<hs≤2t−s+' Y12t−s+'−4,j 1,... ,jh1,... ,jhs,... ,j 2t−s+' V1,jh1...V 1,jhs =a2t−s+'−4(j1,... ,j 2t+'). The next lemma gives the obstruction to extending Lemma 6.6 to s=t−1 and thereby bounding one more element of the tower in Figure 2. Lemma 6.7. Let J=[j1,... ,j 2t+']with 4≤tand E=0,1. Then at+'−3[J]= t−2 α=0 t β=t−α ζt−α−1 Y(i,j2t−2α+'−3,j 2t−2α+'−2,k,j 2β+'−1,j 2β+') + t k=1 Y1t+'−3 ,j ',Σ(1,j '+1,j '+2),. . ., Σ(1,j 2k+'−1,j 2k+'), ... ,Σ(1,j 2t−1+',j 2t+'),j 2k+'−1,j 2k+' where i=[j1,... ,j 2t−2α+'−4], k=j2t−2α+'−1,... ,j2β+'−1,j2β+',... ,j 2t+' and the “j0” should be deleted from the last sum when E=0. Proof: We apply Lemma 6.6 to the element at+'−1[j1,... ,j 2t+'+2] where j2t+'+1 and j2t+'+2 are large powers of two. Applying the Landweber-Novikov operation s∆2j2t++1−2+∆2j2t++2−2and dividing by q1, we obtain this lemma. We combine the previous three lemmas to show that the elements of the top half of the tower of Figure 2 are boundaries in the ASS for MSp. Proposition 6.8. (a) For E=0,1,3≤tand 0≤s≤t−2, the element as+t+'−2[j1,... ,j 2t+'] is zero in E∞of the ASS for MSp. (b) For t≥3, the following element is also zero in E∞of the ASS for MSp: at−2[0,j 2,... ,j 2t+1].
192 B. I. Botvinnik, S. O. Kochman Proof: (a) Each summand of the decomposition of as+t+'−2[j1,...,j 2t+'] in Lemma 6.6 is a boundary in the ASS by Lemma 6.5. (b) When E= 1 and j1= 0 in Lemma 6.7, the last sum in the decomposition of at−2[0,j 2,... ,j 2t+1] equals t k=1 Y0,Σ(1,j 2,j 3),1,Σ(1,j 4,j 5),... ,1, Σ(1,j 2k,j 2k+1),... , 1,Σ(1,j 2t,j 2t+1),j 2k,j 2k+1. Thus, by Lemma 6.5 each summand of the decomposition of at[0,j 2,... ,j 2t+1] in Lemma 6.7 is a boundary in the ASS. Note 6.1. If j1,... ,j 2t+'is an increasing sequence of non-negative even integers then the only apparent way that the last sum in Lemma 6.7 can bound is as in (b) when E= 1 and j1=0. It follows from this proposition that the order of a[J] given in Theorem 2 is exact after projection into E∞of the ASS. Corollary 6.9. For s≥7and 0≤j1<···<j s, the projection of the element 2[(s+1)/2]−2a[j1,... ,j s]into E2[(s+3)/2],∗ ∞of the ASS for MSp is zero. References 1. B. Botvinnik,“Manifolds with singularities and the Adams-Novikov spectral sequence,” Lecture Notes Series of the London Math. Soc. 170, Cambridge University Press, 1992. 2. B. Botvinnik and S. O. Kochman, Singularities and higher torsion in symplectic cobordism, Canad. J. Math. 46 (1994), 485–516. 3. H. Cartan and S. Eilenberg,“Homological algebra,” Princeton Math. Series 19, Princeton Univ. Press, 1956. 4. V. Gorbounov and N. Ray, Orientation of spin bundles and symplectic cobordism, Publication of RIMS, Kyoto Univ. 28 (1992), 39–55. 5. S. O. Kochman, The symplectic cobordism ring I, Mem. Amer. Math. Soc. 228 (1980). 6. S. O. Kochman, The symplectic cobordism ring II, Mem. Amer. Math. Soc. 271 (1982).
Higher torsion in MSp∗193 7. S. O. Kochman, The symplectic cobordism ring III, Mem. Amer. Math. Soc. 496 (1993). 8. S. O. Kochman, Uniqueness of Massey products on the stable homotopy of spheres, Canad. J. Math. 32 (1980), 576–589. 9. S. O. Kochman, The symplectic Atiyah-Hirzebruch spectral sequence for spheres, Bol. Soc. Mat. Mexicana 37 (1992), 317–338. 10. A. Liulevicius, Notes on homotopy of Thom spectra, Amer. J. Math. 86 (1964), 1–16. 11. J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3(1966), 123–146. 12. N. Ray, Indecomposable in Tors Ω∗ Sp,Topology 10 (1971), 261–270. 13. V. Vershinin, Computation of the symplectic cobordism ring below the dimension 32 and nontriviality of the majority of triple products of the Ray elements, Siberian Math. J. 24 (1983), 41–51. 14. V. Vershinin, Symplectic cobordism with singularities, Izv. Akad. Nauk. SSSR Ser. Mat. 24 (1983), 230–247. 15. V. Vershinin and A. Anismov, A series of elements of order 4 in the symplectic cobordism ring, Canad. Math. Bull. 38 (1995), 373–381. 16. V. Vershinin, On bordism ring with principal torsion ideal, Algebric Topology Pozna´n 1989 Proceedings,, Lecture Notes in Math. 1474, Springer-Verlag, 1991, pp. 295–309. 1991 Mathematics subject classifications: Primary 55N22, 57R77. Boris I. Botvinnik: Department of Mathematics University of Oregon Eugene, OR 97403 U.S.A. e-mail: [email protected] Stanley O. Kochman: Department of Mathematics and Statistics York University 4700 Keele Street North York, Ontario M3J 1P3 Canada e-mail: ko[email protected]orku.ca Primera versi´o rebuda el 29 de Novembre de 1995, darrera versi´o rebuda el 17 de Gener de 1996