Adams Spectral Sequence and Higher Torsion in MSp*

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.


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 2 k .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 φ 0 = η ∈ MSp 1 , and let φ k ∈ MSp 8k−3 for 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 φ m , 2, φ n for 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. Theorem 1.In MSp * , the Toda brackets φ 1 , 2, φ n contain zero for all n ≥ 0.
Let J = (j 1 , . . ., 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[j 1 ] = a[j 1 , j 2 ] = a[j 1 , j

May Spectral Sequence for MSp Σn
Let MSp Σn , n ≥ 1, be the spectrum defined in the Introduction with singularities Σ n = (P 1 , . . ., P n ), and let MSp Σ0 denote MSp.In this section we compute the E 2 -term of the Adams spectral sequence (ASS): (1) E s,t 2 = Cotor s A H * MSp Σn , Z/2 t =⇒ MSp Σn * .
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 E 2 to computing (2) Cotor B(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 E 2 of these May spectral sequences using the resolution constructed by May in [11].
Then we construct filtered polynomial DGA algebras P n as 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 n and a generalized Five Lemma.Then for n ≥ 1 we define representative cycles of the algebra generators of E 2 to 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 d 2 -differentials and nontrivial extensions.Consequently, for n ≥ 1 we can describe E 2 of 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 MSp Σn as a comodule over the dual of the Steenrod algebra A * = Z/2 [ξ 1 , . . ., ξ k , . . .].Let S be the A * -primitive polynomial algebra: where m = 2, 4, 5, . . ., m = 2 l − 1, and deg V m = 4m.V. Vershinin [14], [16] proves that there is an isomorphism of A * -comodules: (3) H * MSp Σn ∼ = Z/2 ξ 2 1 , . . ., ξ 2 n , ξ 4 n+1 , . . ., ξ 4 k , . . .⊗ S for n ≥ 0. Define the Z/2-Hopf algebra and n < k with coproduct ψ induced from the coproduct of A * .Note that in [5] the Hopf algebra B (0) is denoted as B. By (3), the problem of computing E 2 of 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: To compute the cohomology of the B (n) we use the May spectral sequence [11]: Recall that this spectral sequence is defined by giving B (n) the coproduct filtration where by induction on p ≥ 1 Here ψ denotes the reduced coproduct: and IB (n) denotes the augmentation ideal of B (n).The following lemma describes the structure of the Hopf algebra E 0 B (n).It is an immediate consequence of the coalgebra structure of A * and the definition of the B (n).

Lemma 2.1.
There is an isomorphism of Hopf algebras: where the elements ξ (1) k , n < k, are primitive and As in [5, Section 1], we compute the E 2 -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 In the notation of [5] and [11], we define the DGA 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: We will use the elements defined below to compute the homology of the D (n).

Definition 2.3. In the algebra Cotor
Note 2.1.We will also need the following degenerate cases of these elements: The homology of the D (n) can be computed as in [ A complete set of relations among these generators is given by the degeneracy relations of Note 2.1 and by: . ., g i , . . ., g t ).
We use the methods of [5, Section 3], to show that E 2 = 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 P n whose homology is Cotor B(n) (Z 2 , 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 P n , n ≥ 1, with the required properties from the P constructed 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 P and a Z/2-linear map λ : A − → P .
The map λ induces an algebra homomorphism λ : C (Z 2 , A, Z 2 ) − → P which we assume is a map of DGAs.We also assume that the algebra homomorphism induced by λ is an isomorphism.Suppose that we have a primitive element x in the center of A which is not a zero-divisor.Let A 1 = A/(x), y = λ(x) and P 1 = P/(y).Lemma 2.5 (Generalized Five Lemma).Assume that we have a Z/2-Hopf algebra A, a DGA P , a Z/2-linear map λ : A − → P and elements x, y as above which satisfy the following additional conditions: Then the following cobar constructions give a short exact sequence of DGAs: where j (a) = a + 0X and 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) = a and γ (a + bX) = π (a) where π : P → P 1 and π : A → A 1 are the canonical projection maps.By condition (iii), λ induces a map of DGAs λ 1 making 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 Cotor A = Cotor A (Z/2, Z/2) and Cotor A1 = Cotor A1 (Z/2, Z/2). (6) The short exact sequences on the top and bottom rows of this diagram induce the following long exact sequences in homology.( 7) We show that diagram (7) commutes.It then follows from the usual Five Lemma that λ 1 * is an isomorphism.Square 1 commutes because Square 2 commutes because the trapezoid in (6) commutes.Note that We will need the following generalization of the previous lemma which follows from it by induction on n ≥ 1.
Lemma 2.6.Let the Z/2-Hopf algebra A, the algebra P and the Z/2linear map λ : A → P be as above.Let x 1 , . . ., x n , . . .be a sequence of elements in the center of A. Let I 0 = 0, I n = (x 1 , . . ., x n ) for n ≥ 1 and A n = A/I n .Let y n = λ (x n ), J n = (y 1 , . . ., y n ), and P n = P/J n .Assume that: (i) the ideals I n are prime and invariant; A complete set of relations is given by: ( and the degeneracy relations Proof: Recall the DGA P constructed in [5, Section 3].P is the Z/2algebra with generators: Then Observe that since R 0 = 0 in P n for n ≥ 1, the algebra P n is the commutative polynomial algebra We check the hypotheses of Lemma 2.6. (i) Since ξ 2 n , n ≥ 1, is a regular sequence of primitive elements, the ideals We construct representative cycles in P n of the algebra generators of E 2 of the May spectral sequence for Cotor . Using these representative cycles of the algebra generators of E ∞ , it is straightforward to check that all four families of relations in E ∞ are also valid in Cotor B(n) (Z/2, Z/2).Thus, the structure of E 2 of the ASS follows from (4) and Lemma 2.4.
Observe that the commutativity of the P n is the reason why the elements Q 2 k and P (m 1 , . . ., m s ) are cycles in P n for n ≥ 1 while in P they are not cycles and support nonzero d 2 -differentials in the May spectral sequence for Cotor B (Z/2, Z/2) when s ≥ 3.

Adams Spectral Sequence for MSp Σn
In the preceding section we obtained a concise algebraic description of E 2 of the ASS (1) for MSp Σn , n ≥ 1.However, this algebraic description is not suitable for computing the differentials in the ASS or for understanding MSp Σn * which is determined by the topology of the spectrum MSp Σn .Thus, we begin this section with an alternate description of E 2 in terms of the projections Φ n of the Ray elements φ n .Although this description may seems algebraicly awkward, it enables us to compute all of the d 2 -differentials and some of the d 3 -differentials.These d 3 -differentials are used to prove a technical fact which we needed in [2, Section 6].In addition, we will use this description of E 2 in Section 6 to identify and analyze the elements of higher torsion we construct there.
Recall from [5,Theorem 5.3] that the Ray elements of the ASS for MSp.In the following definition, as in [5, Section 4], we rewrite all the elements of E 2 of the ASS for MSp Σn in terms of the Ray elements.
Definition 3.1.Let n ≥ 1. Define the following elements in E 2 of the ASS for MSp Σn : In terms of these elements, the following description of E 2 follows from Theorem 2.7.

Corollary 3.2. E 2 of the ASS for MSp
Σn is the algebra generated by: A complete set of relations is given by: (

. V I(ms,j(s)) ; and the degeneracy relations
Using the description of the elements of E 2 given in Corollary 3.2, we compute the d 2 -differentials.
Proof: Using the canonical map from the ASS for MSp to the ASS for MSp Σn , the first three parts of this theorem follow from [5,Theorem 6.1].It remains to prove (d).Clearly h 0 is an infinite cycle converging to 2. Since E * ,2k−1 2 = 0 for 2k − 1 < 2 n+2 − 3, the Ψ i are infinite cycles.It remains to prove that the ρ (m 1 , . . ., m s ) are infinite cycles.
Proof: We construct the elements r n (m 1 , . . ., m t ) by induction on t ≥ 1.When t = 1 we use (i) to define r n (m 1 ).Assume that t ≥ 2 and that this proposition is true for t−1.Select any element r n (m 1 , . . ., m t ) of the Toda bracket φ mt , 2, r n (m 1 , . . ., m t−1 ) .By [2, Lemma 3.4 and Note 3.1] we have: Note that the Σ n -manifold ∆(2) is a representative of η, and η = 0 in the ring MSp Σn * for n ≥ 1.Thus, by [2, Lemma 3.3 and Note 3.1] and by our induction hypothesis we have: Proof of Theorem 3.3 continued: Clearly the element r n (m 1 , . . ., m s ) projects to ρ (m 1 , . . ., m s ) in E 2 of the ASS for MSp Σn .
Next we compute the d 3 -differentials on some of the polynomial generators of E 0,4 * 3 of the ASS for MSp Σn .Recall from [6,Theorem 8.7(d)] the following d 3 -differentials in the ASS for MSp: where ( 9) Applying the canonical map from MSp to MSp Σn , we obtain the following result.Proposition 3.5.In E 3 of the ASS for MSp Σn : In order to identify the projection to the Adams Novikov spectral sequence of the elements of higher torsion which we constructed in [2, Section 6] we used the following technical fact.Proof: Recall that we defined ∆ (W 2 ) in [2, Section 3] as ) is either φ 2 or zero.By Proposition 3.5(b), 2 n which can be defined as m 2 (V 2 n , V 2 n ) union several manifolds of positive Adams filtration degree including K 2 (V 2 n , W 2 )×φ 2 n .Using the Hirsch formula, Lemma 3.1(a) and Note 3.1 of [2], δV [2] 2 n has as part of its boundary: This unionand of δV [2] 2 n is the only one which could possibly project to Φ 2 Φ 2 2 n in the ASS for MSp Σ2 .Thus, ∆ (W 2 ) must equal φ 2 and not zero.

Adams Spectral Sequence for MSp Σn
Let MSp Σn , n ≥ 2, be the spectrum with singularities Σ n = (P 2 , . . ., P n ) defined in the Introduction, and let MSp Σ1 denote MSp.
In this section we compute E 2 of the ASS: As in Section 2, we use a change of rings theorem to reduce the problem of calculating E 2 to computing where B (n) is the truncated polynomial algebra which is defined as the following quotient Hopf algebra of the dual of the Steenrod algebra: Note that in [5] the Hopf algebra B (1) is denoted as B. We compute the algebra (11) by showing that it is the tensor product of a polynomial algebra and a direct summand of Cotor B (Z/2, Z/2) which was computed in [5].Thus, E 2 of the ASS (10) for n ≥ 2 has all the complexity of the E 2 -term of the ASS for MSp: it has nine families of algebra generators and forty families of relations.As in Section 3, we give an alternate description of E 2 in terms of the projections Φ n of the Ray elements φ n into the ASS (10).In Section 6, we use this description to identify and analyze the elements of higher torsion which we construct there.V. Vershinin [14], [16] showed that for n ≥ 2 there is an isomorphism of A * -comodules: It follows from the change of rings theorem [10, Corollary I.5] that for n ≥ 1 there is an isomorphism of Z/2-algebras: Define the sub-Hopf algebra C (n) of B (n) by We thus have the following lemma.
Lemma 4.1.For n ≥ 2, there is an isomorphism of Z/2-algebras: where each of the k i is either zero or greater than or equal to n.Then E 2 of the ASS for MSp Σn is given by Proof: By ( 12), ( 13), Let ι n : C (n) → B denote the inclusion map.Define a map σ n : B → C (n) of Hopf algebras which splits ι n by Then σ n * is a splitting of the inclusion Thus, we view Cotor C(n) (Z/2, Z/2) as a subalgebra of Cotor B (Z/2, Z/2).
The effect of σ n * on the algebra generators ) is given by σ n * (h 0 ) = h 0 and ( 14) Note 4.1.The map π r , r ≥ 2, of ASS induced by the canonical map of spectra π : MSp → MSp Σn does not induce the projection map We conclude with an alternate description of E 2 in terms of the projections Φ n of the Ray elements φ n to the ASS.If in E 2 of the ASS for MSp and F (k 1 , . . ., k t ) is one of the above seven families of algebra generators of E 2 of the ASS for MSp Σn then define A complete set of relations for E 2 is given by the forty relations listed in [5,Theorem 3.7] as well as the following relations: (c) if F is any of the above ten families except h 0 or V a then From now on we only use the description of E 2 in terms of the F (k 1 , . . ., k t ), and we abuse notation by denoting them as F (k 1 , . . ., k t ).

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 φ 1 , 2, φ n of Theorem 1 allows us to construct specific Sp-manifolds V n with no singularities in Lemma 5.3 such that ∂V n is the canonical element in this Toda bracket.Let J = [j 1 , . . ., j s ] with s ≥ 1 and J = [j 1 , . . ., j s−1 ] with s ≥ 2 throughout this section.In Proposition 5.4 we use the V n to generalize the constructions of Section 5 of [2] to construct the elements t We give the basic properties of the t[J] and g[J] including their Toda bracket decompositions and their projection in the 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 φ 1 as 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 → G and associative pairings of spectra for all choices of ξ and ζ.We identify such a Toda bracket in the case E = F = S and G = MSp which decomposes φ 1 .We also give a similar decomposition of φ 2 in terms of a four-fold Toda bracket.Recall that MSp 8 = Z with the generator q 0 .Lemma 5.1.Let µ : S → MSp denote the unit of the spectrum MSp.
Proof: The proof of this lemma is based upon the analysis of the following Atiyah-Hirzebruch spectral sequence.This spectral sequence was analyzed through degree 50 in [9].Fortunately, we only require its structure through degree 5 which is depicted in (a) Since MSp 5 = Z/2φ 1 and the only infinite cycle in E 2 * , * of degree 5 is ηb 1 , the only possibility for the projection of φ 1 to E ∞ * , * is ηb 1 .The fact that ηb 1 is an infinite cycle of (15) Note that we have suppressed the canonical pairings of spectra involved in the previous statement.Since µ * π S 5 = 0 and η • MSp 4 = 0, the indeterminacy of η, ν, µ is zero.
Our representative of φ 1 can be described in terms of (Sp, f r)manifolds as where Y 4 is an Sp-manifold with ∂Y 4 = ν and W 5 is a framed manifold with ∂W 5 = η × ν.
Recall from [12] that the Ray elements φ n are closed under the action of the Landweber-Novikov operations.In particular, Theorem 11.4], the action of the Landweber-Novikov operations on the Toda brackets φ m , 2, φ n satisfies the Cartan formula: We thus have the following formula for the action of the s ∆ 2k on our Toda brackets.
We use the action of the s ∆ 2k on our Toda brackets and the decomposition of φ 1 to prove Theorem 1.
Proof of Theorem 1: By Lemma 5.1, φ n , 2, φ 1 = φ n , 2, η, ν, µ which contains an element which is also an element of The last equality uses Gorbunov's Theorem [1,Theorem 4.3.5]which says that 0 ∈ φ n , 2, η .Therefore, any element of φ n , 2, η is of the form ηA. By Lemma 5.1 and the observation that ηA • MSp 4 = 0, we see that φ n , 2, φ 1 contains an element which is also an element of This sum is contained in the ideal spanned by φ 1 modulo Image µ * .Thus, for all n, we conclude that φ n , 2, φ 1 contains an element which is in Image µ * .By Lemma 5.2, Recall that an element in the image of the unit µ * of MSp is annihilated by all Landweber-Novikov operations.It follows that φ n , 2, φ 1 contains zero.
The main technique which we use in constructing the t[J] is the existence of Sp-manifolds V j as in the following lemma.The proof of this lemma is based upon Theorem 1.We abuse notation below by denoting a cobordism class φ n and an Sp-manifold representing φ n by the same symbol φ n .

Lemma 5.3.
There are Sp-manifolds ψ n for n ≥ 0 and V n for n ≥ 2 such that In particular, ψ 1 does not depend on n.
Proof: By Theorem 1, there are Sp-manifolds ψ We are now ready to construct t[J] ∈ MSp Σ3 * .We denote the product construction of Σ 3 -manifolds by m 3 , the associativity construction by A 3 and the commutativity construction by K 3 .Proposition 5.4.For each J = [j 1 , . . ., j s ], there exists an element t Proof: (a)-(c) We construct the t [j 1 , . . ., j s ] by induction on s ≥ 1 to satisfy (a)-(c) as in the proof of [2,Lemma 5.3].
(e) By induction on s ≥ 1, we prove that T s projects to t in 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 T s to the one line of the ASS equals Since ψ 1 , T s−1 and φ js have Adams filtration degree one, the projections of the manifolds A 3 (V 1,js , ψ 1 , T s−1 ), m 3 ( K 3 (V js , ψ 1 ), T s−1 ), m 3 (B × φ js , T s−1 ) and A 3 (ψ 1 , V 1,js , T s−1 ) to the zero line of the ASS are trivial.Thus by (3), the projection of H s to the zero line of the ASS equals the projection of m 3 (V js , H s−1 ) which by the induction hypothesis is which is bordant to where ν 0 (j s ) is defined as the Σ 2 -cobordism class of 2V js ∪ψ js ×ψ 1 which where To identify the projection of g [J] into the ASS we need to know the projection of E[J] into the ASS.
(b) Observe that just as in the proof of [2, Lemma 6.2(b)], we can use induction on s ≥ 2 to construct Σ 2 -manifolds T s , H s , E s and L s such that: (i) T s represents t[J]; (iv) T s projects in the one line of the ASS for MSp Σ2 and in the one line of the ASS for MSp Σ2 to t[J] ∈ E 1,4 * +1 2 ; (v) H s projects in the zero line of the ASS for MSp Σ2 and in the zero line of the ASS for (vi) E s projects in the two line of the ASS for MSp Σ2 and in the two line of the ASS for (vii) L s has Adams filtration degree four.
Using this lemma, we determine the basic properties of the g[J].
Proposition 5.6.The elements g of the ASS for MSp Σ2 and in 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 (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 By the induction hypothesis and the previous lemma, This is the asserted value of g[J] in (e).

Elements of Higher Torsion and the ASS
In this section we analyze the elements In particular, we give their decomposition in terms of four-fold Toda brackets and identify their projections to the ASS.These results are summarized by Theorem 2. In addition, we shall see that the projections of the 2 k a[J], k ≥ 0, to E 2 of the ASS for MSp determine towers whose top halves are zero in E ∞ .Throughout this section J = [j 1 , . . ., j s ] with s ≥ 1 and J = [j 1 , . . ., j s−1 ] with s ≥ 2. We begin by determining the projection of the a[J] to the ASS for MSp.We will use the following notation from [5,Definition 7.12(19a)].Let H = (h 1 , . . ., h k ).Assume that r ≥ k, s ≥ 2r − k + 3 and s − k is even.Then the following elements of E 2 are d 2 -cycles in the ASS for MSp: where this sum is taken over all sequences (t 1 , . . ., t 2r−k ) of distinct integers between 1 and s such that We introduce the following notation for the particular elements of this family which we will be studying.
To describe the projections of the 2 k a[J] in E 2 of the ASS for MSp we introduce the following notation.For 0 ≤ k ≤ s − 4, let Note that a 0 [J] = a[J].
The annihilator ideal of {Φ N | N ≥ 0} in E 2 of the ASS for MSp is the ideal spanned by h 0 , and the latter ideal is zero in E 2 * ,4 * +1 2 .Thus, twice a k [J], the projection of 2 k+1 a[J], equals a k+1 [J] by a nontrivial extension of degree one.∂ψ js = φ js × 2; where A[J] is an Sp-manifold which represents a Thus, φ 1 a[j] = 0 and we have the following defining system for φ js , 2, φ 1 , a[J ] : Using (iii) from the proof of Lemma 5.5, we have Thus, (17) (e) Let j k = 2 i k −2 for 1 ≤ k ≤ s, let i = (i 1 , . . ., i s ) and let J = [j 1 , . . ., j s ].Consider the following commutative diagram.

Our canonical representative
By [6,Thm. 8.13], where (m, n) is a closed symplectic manifold of Adams filtration degree three which projects to (m, n) in the Adams spectral sequence.By (17), is a closed symplectic manifold of Adams filtration degree five which projects to the infinite cycle Φ j1 Φ j2 (j 3 , j 4 ) + Φ j3 Φ j4 (j 1 , j 2 ) in E 5,4 * +1 3 of the Adams spectral sequence.
When we multiply the Toda brackets for a[J] by two, their length decreases.
where ν (j s ) projects to the infinite cycle h of the ASS for MSp.
Proof: Using the manifold A[J] which we constructed in Theorem 2(d) to represent a[J], we represent 2a[J] by 2A Since ψ js = ψ js ∪ −K(φ js , 2), the Hirsch formula shows that K(ψ js , 2) gives a cobordism between B js and −φ js × K(2, 2) = −φ js × η.In addition, ∂ (C js ) = −B js ×φ 1 .Thus, 2a[J] is represented by A 2 [J] which is in φ js η, φ 1 , a[J ] .Then we can represent 4a[J] by the manifold From the definition of C js , we see that the cobordism class ν (j s ) of D js projects to h of the ASS for MSp.
We conclude this section by showing that certain a k [J] = 0 in E ∞ of the ASS for MSp.Note that all the a k [j 1 , . . ., j s ] are nonzero in E 2 for 0 ≤ k ≤ s − 4. We begin by determining when a Theorem 2 implies that when the entries of J are 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 d 2 -cycles Σ(a, b, c) ∈ E 1,4 * +1 2 which were defined in (9).Let A 1 , B 1 , . . ., A n , B n be a sequence such that each (A k , 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 V 1,Σ(1,x,y) = V 1,x V 1,y and V 0,Σ(1,x,y) = V 0,1 V x,y + V 0,x V 1,y + V 0,y V 1,x + V 1,x,y .Thus, in all three cases we have elements V A k ,B k in E 2 of the ASS for MSp such that: where this Toda bracket is defined in E 2 = H * (P ⊗ S).Thus, as in [5,Definition 7.12(19a)], we can define the following elements of E 2n−2k,4 * +1 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. and the "j 0 " should be deleted from the last sum when = 0.
We combine the previous three lemmas to show that the elements of the top half of the tower of Figure 2  Proof: (a) Each summand of the decomposition of a s+t+ −2 [j 1 ,. .., j 2t+ ] in Lemma 6.6 is a boundary in the ASS by Lemma 6.5.
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 ≥ 7 and 0 ≤ j 1 < • • • < j s , the projection of the element 2 [(s+1)/2]−2 a [j 1 , . . ., j s ] into E
3 and Note 3.1].Thus, there exists an Sp-manifold Y n with

Proposition 6 . 1 .
Let s ≥ 4 and 0 ≤ k ≤ s − 4. Then (a) a[J] projects to the infinite cycle a[J] in E 4,4 * +1 2 of the ASS for MSp; (b) 2 k a[J] projects to the infinite cycle a k [J] in E 2k+4,4 * +1 2 of the ASS for MSp.Proof: (a) Let G 0 [J] denote G[J] viewed as an Sp-manifold.Since G 0 [J] is a representative manifold of g[J] and a[J] = β 2 (g[J]), ∂G 0 [J] = φ 1 × A[J]where A[J] is a representative manifold of a[J].By Proposition 5.6(e), G[J] projects in E 3,4 * +3 2 of the ASS for MSp to

Since multiplication by Φ 1
is a monomorphism on E 4,4 * +1 2 of the ASS for MSp and d 2 -boundaries project to zero in E 3 , a[J] projects to a[J].(b) We prove (b) by induction on k.(a) gives the case k = 0. Assume that (b) is true for some k with 0 ≤ k ≤ s − 5. We show that in E ∞ of the ASS of MSp twice a k [J] is equal to a k+1 [J] by a nontrivial extension of degree one.We apply [6, Theorem 12.2] to
Let j 1 , . . ., j s be distinct even natural numbers with s ≥ 4. Then the element a [j 1 , . . ., j s ] = 0 in E 4,4 * +1 ∞ if and only if s = 4. Proof: (a) Let D denote the sum of all distinct elements of the given form.Observe that
λ induces maps of DGAsλ n : C (Z 2 , A n , Z 2 ) → P n such that the λ n * : Cotor An (Z 2 , Z 2 ) → H * P n ⊗ S,where S is the polynomial algebra with generators V a , a = 2 k − 1.Let |c| denote the degree of c.Let n ≥ 1.Then E 2 of the ASS for MSp Σn * , respectively.We denote Q k as Ψ k .We do not explicitly specify the forty relations in E 2 of the ASS for MSp Σn induced from E 2 of the ASS for MSp because we do not use them in this paper.For n ≥ 2, E 2 of the ASS for MSp Σn is the Z/2algebra generated by k1,j1) ...V e FI(kt,jt)where e F equals 1, 4, 2, 1, 2, 1, 2 if F equals R, q, Q, P , P 2 , Y , Z s