Singularities and Higher Torsion in
Symplectic Cobordism I
Boris I. Botvinnik and Stanley O. Kochman
Abstract
In this paper we construct higher twotorsion elements of all orders in the
symplectic cobordism ring. We begin by constructing higher torsion ele
ments in the symplectic cobordism ring with singularities using a geometric
approach to the AdamsNovikov spectral sequence in terms of cobordism
with singularities. Then we show how these elements determine particular
elements of higher torsion in the symplectic cobordism ring.
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 an Sp
structure on their stable normal bundles. Although MSp* only has twotorsion,
its ring structure is far more complicated than any of the other cobordism rings
MG* for the classical Lie groups G = O, SO, Spin, U, SU which have been
completely computed. Over the past thirty years, these cobordism rings MG*
have had a major impact on differential topology and homotopy theory. On the
other hand, if the complexity of the ring MSp* were understood, then symplectic
cobordism theory MSp*(.) would have the potential to become a powerful tool
in algebraic topology.
The symplectic cobordism ring MSp* is still far from being computed and
understood despite much research on the subject over the past twenty years.
It seems beyond present methods to completely compute MSp* in the near
future. Nevertheless, we can try to determine some general structural properties
of this ring. The most striking example of such a result is the application of *
*the
Nilpotence Theorem [5] to MSp* which says that all of its torsion elements are
nilpotent. Another basic structural question is:
Do there exist elements of order 2k in the ring MSp* for all k(11?)
Note that the corresponding structural property is wellknown for all other cla*
*ssi
cal cobordism rings as well as for framed cobordism, the stable homotopy groups
of 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 a*
*nd Engineering
Research Council of Canada.
1
2
We begin by describing the background of our research. In the torsion of
MSp* there are the fundamental Ray elements [12]: OE0 = j 2 MSp1 which
comes from framed cobordism, and OEi 2 MSp8i3for 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. These
OEi determine basic patterns in all approaches to understanding the structure of
symplectic cobordism. In particular, projections of these elements to the Adams
and AdamsNovikov spectral sequences for MSp have had a major impact on
the 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 made
by D. Segal [13] in 1970. Subsequently the second author in [6], [7], [8] compu*
*t
ed the E2 and E3terms, showed that the spectral sequence does not collapse,
computed the image of MSp* in N*, found elements of order four beginning in
degree 111 and computed the first 100 stems.
The other approach based on the AdamsNovikov 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 order
four in MSp* occurs in degree 103 (unpublished).
It became apparent from both approaches that if there was torsion of order
greater than four in MSp* then it would occur in such a high degree that it wou*
*ld
not be reasonable to try to discover it through stem by stem computations. In
addition, there were no candidates for elements in E2 of either the ASS or ANSS
which might represent elements of higher torsion. (The only such family of
candidates 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* ! MSp1* ! MSp2* ! . ..!MSpn* ! . ..!MSp*
of cobordism rings MSpn* of symplectic manifolds with singularities which starts
with the mysterious ring MSp* and ends with MSp*, a polynomial ring over the
integers. In the twolocal category, the spectrum MSp splits as a wedge of sus
pensions of the spectrum BP . Here = (P1; . .;.Pn; .a.).nd n = (P1; :::; Pn)
are sequences of closed Spmanifolds which represent the Ray elements [P1] = j
and [Pi] = OE2i2for i 2. This led to the description of the AdamsNovikov
spectral sequences for the spectra MSpn in terms of cobordism with singular
ities [2], and, in particular, to a precise formula for the AdamsNovikov diffe*
*r
ential d1 that reduces the computation of the E2term to elementary algebraic
manipulations.
3
The opportunity that we have had to work together at York University has
led to the understanding that the geometry of manifolds with singularities can
be used to uncover the deep interaction between the Adams and AdamsNovikov
spectral sequences for MSp thereby constructing torsion elements of all orders
2k in MSp*.
First we construct higher torsion elements in MSpn* for n 3. The keys
to this construction are that the cobordism ring MSpn* has new elements
w1; . .;.wn that have the same degrees and behavior as the elements v1; . .;.vn*
* 2
BP* and that the Toda brackets 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 indecomposable
elements on (i1; . .;.is)2 MSpn4*+1with the following properties:
(i)on(i1) = OE2i12;
(ii)on(i1; . .;.is) 2 ,
(iii)on(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 symplectic
cobordism ring MSp*.
However, Theorem B is just an existence theorem. It does not give a particular
way to construct higher torsion elements in MSp*. The remainder of this paper is
devoted to the construction of elements ff(i1; . .;.is) 2 MSp* from the elements
o3(i1; . .;.is) 2 MSp3*. Consider the diagram below
MSp3* _________MSp2*fi3_________MSp1*fi2_________MSp*fi1
@@Iss
@
MSp*
We define the elements
ff0(i1; . .;.is) = fi2(fi3(o3(i1; . .;.is)))
in the ring MSp1*. Then we construct elements ff(i1; . .;.is) 2 MSp4*+1 such
that ss(ff(i1; . .;.is)) and 2ff0(i1; . .;.is) in MSp1* project to the same ele*
*ment
of E3;4*+12MSp1 in the AdamsNovikov spectral sequence. Finally we prove
that the ff(i1; . .;.is) 2 MSp4*+1 are elements of higher order in MSp*.
Main Theorem. The element ff (i1; . .;.is)2 MSp4*+1 has order at least
2[(s+1)=2]3for s 7 and 3 i1 < . .<.is.
4
We describe the contents of this paper in detail. In Section 2, we summarize
the basic facts about the spectra MSpn which we will be using. In Section
3, we give the definition and basic properties of threefold Toda brackets of
manifolds with singularities. These Toda brackets will be used to inductively
define the elements we construct. In Section 3, we study the AdamsNovikov
spectral sequence for the MSpn . The key technical and conceptual fact we use
is that the AdamsNovikov spectral sequence for the spectrum MSp* coincides
with the singularities spectral sequence which is defined in terms of cobordism
with singularities [2]. The singularities spectral sequence gives us a specif*
*ic
resolution M for computing E2 of the ANSS for MSpn . In particular, we
identify torsion elements tn (i)of all orders 2k in the first line of the ANSS
* *
E1;*2= Ext1;*ABPBP MSpn ; BP :
Let i = (i1; . .;.is). In Section 5, we use Toda brackets to construct elements
on (i)in MSpn* and prove Theorems A and B. In particular, the element on (i)
has order at least 2[(s+1)=2]because it projects in the AdamsNovikov spectral
sequence to the infinite cycle tn (i)that has order 2[(s+1)=2]in E1;*2(MSpn ). *
*In
section 6, we study the elements
fl (i)= fi3(o3(i))2 MSp24*+3andff0(i)= fi2(fl (i))2 MSp14*+1
and identify their projections to the third line of the ANSS. In Section 7 we
prove the Main Theorem by projecting the elements ff0(i1; . .;.is) to the third
line E3;*2(MSp1 ) of the ANSS. We use chromatic arguments to compute the
order of these projections in E3;*2, and we show that they can not be killed by
d3differentials.
In [3] we compute the E2terms of the Adams spectral sequences for the spec
tra MSpn and apply them to study the elements of higher torsion constructed
in this paper.
All groups, rings and spectra are twolocal throughout this paper.
The first author would like to thank the topology community at M.I.T. for
their warm hospitality during his visit in the spring term of 1991 as well as
the Department of Mathematics and Statistics at York University for their kind
support and hospitality. In addition he would like to thank Haynes Miller for
important discussions on the basic ideas of this paper.
2 Symplectic Cobordism with Singularities
In this section we collect basic constructions and theorems concerning the spec*
*tra
MSp* and MSpn* of symplectic cobordism with singularities. In particular, we
determine formulas for computing the Bockstein operators which will be used in
Sections 4 and 7 to make computations in the ANSS for MSpn .
Let = (P1; :::; Pn; :::) be a sequence of closed Spmanifolds representing *
*the
Ray elements such that [P1] = j and [Pi] = OE2i2for i 2. For n 1, let n
5
denote the sequence (P1; :::; Pn). The bordism theory of Spmanifolds with n,
singularities is denoted by MSpn*(.), MSp*(.) respectively. By [2], [16], the
theory MSpn*(.) has an admissible product structure, and the coefficient ring
MSpn* is polynomial up to dimension 2n+2  3. (See [16], [2, Theorem 3.3.3].)
The following theorem describes the structure of the ring MSp*.
Theorem 2.1 (V.Vershinin [16]) There exists an admissible product structure
in the theory MSp*(.) such that its coefficient ring MSp* is isomorphic to the
polynomial ring
MSp* ~=Z(2)[w1; . .;.wj; . .;.x2; x4; x5; . .;.xm ; . .].
where degwj = 2(2j  1) for j = 1; 2; . .a.nd deg xm = 4m for m = 2; 3; 5; . .,.
m 6= 2s  1. The generators wj are represented by Spmanifolds Wj such that
@Wj = 2Pj
In fact, the cobordism theory MSp*(.) splits as a sum of the theories BP *(.).
Theorem 2.2 [2, Corollary 3.5.3] The ring spectrum MSp splits as
MSp = BP ^ M(G)
where G = Z(2)[x2; . .;.xm ;,.m.].= 2; 4; 5; . .,. m 6= 2l 1, degxm = 4m and
M(G) is a graded Moore spectrum.
Note 2.1 Theorem 2.2 implies that the ANSS based on the cohomology theories
BP *(.) and MSp*(.) are isomorphic.
There are Bockstein operators in the theory MSp*(.) for i 1:
fii: MSp*(.) ! MSp*(.):
They have the following properties:
fiiO fii= 0; and fiiO fij = fij O fii:
In general a product formula for Bockstein operators acting on a bordism theory
with singularities is too complicated to write down. However in our case this
formula has the following simple form.
Theorem 2.3 [2, Theorem 4.2.4] The product structure and the elements wi in
Theorem 2.1 may be chosen in such a way that the Bockstein operators fii, i 1,
satisfy the product formula:
fii(x . y) = (fiix) . y + x . (fiiy)  wi. (fiix) . (fiiy)(2)
where x; y 2 MSp*.
6
To describe the action of the Bockstein operators on the polynomial generators
of MSp* we introduce the following notation. Let m + 1 = 2i11+ . .+.2is1
be a binary decomposition of the integer m + 1 where 1 i1 < i2 < . .<.is. If
m is odd, the generator xm is denoted by xi1;:::;is. If m = 2i1 with 1 i then
the generator xm is denoted by x1;i:
Theorem 2.4 [2, Theorem 4.5.1] There are generators xm of the ring MSp*
such that the formulas below describe the action of Bockstein operators fik on *
*xm
for k 2.
1. If m = 2i1+ 2j1  1; 1 i < j then
fiixi;j= wj; fijxi;j= wi; and fikxi;j= 0 ifk 6= i; j: (3)
2. If m = 2i11+ . .+.2is1 1; 2 i1 < i2 < . .<.is and s 3; then
8
< w1 . xi1;:::;bit;:::;is;ifk = it
fikxi1;:::;is= : : (4)
0 if k 6= i1; :::; is
3. If m is even and not a power of two then
fikxm = 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 MSpn . Note that (2) and (3) are invariant
under permutations ss of the subscripts where ss is a permutation of the set of
integers greater than one.
3 Toda Brackets in MSpn*
In this section we extend the construction of Alexander [1] to define triple To*
*da
brackets in the ring MSpn*. Note that we do not claim that all of the usual
properties of Massey products [10] and Toda brackets [14] generalize to cobor
dism with singularities. The problem is that the ring spectrum MSpn probably
does not admit an H1 structure. Therefore, we only study those Toda brackets
which we will use, and we only prove those properties of Toda brackets which
we will need.
We begin by recalling from [2, Chapter 2] the construction of an admissible
product n in the bordism theory MSpn*(.). We summarize the properties of
the associativity construction An which measures the lack of associativity of n
and the properties of the commutativity construction Kn which measures the
lack of commutativity of n. 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 these
Toda brackets. In addition, Kn is a geometric cupone product of manifolds with
7
singularities which satisfies the usual boundary and Hirsch formulas. We will
use it in Sections 5 and 6 in our constructions of higher torsion elements.
Let Pn be as above with [P1]Sp= j and [Pn]Sp= OE2n2for n 2. Consider
the manifold
Pn0= Pn(1)x Pn(2)x I:
Here Pn(1), Pn(2)are two copies of the nmanifold Pn such that:
@Pn0= finPn0x Pn and
finPn0= Pn(1)x {0}[ Pn(2)x {1} :
Note that the nmanifold Pn0is nbordant to a manifold without singularities
since 2 [Pn]Sp= 0. The cobordism class of the nmanifold Pn0is the obstruction
to the existence of a product structure. In our case, [Pn0]n= 0, and we let Qn
denote a nmanifold such that ffiQn = Pn0as in [2, Theorem 4.2.4]. Thus, we
have the following product construction of [2, Theorem 2.2.2].
A product mn Aa; Bb of two nmanifolds is defined by induction on n as
follows:
m1 Aa; Bb = Aa x Bb [ (1)bfi1Aa x fi1Bb x Q1
and for n 2
a b a b b a b
mn A ; B = mn1 A ; B [ (1) mn1 mn1 finA ; finB; Qn:
In particular, if C is an Spmanifold without singularities then
mn (X; C)= X x C and mn (C; X)= C x X:
We have the following diffeomorphisms of nmanifolds:
ffimn(Aa; Bb) = mn(ffiA; B) [ (1)amn(A; ffiB);
C x mn(A; B) = mn(C x A; B); mn(A; B) x C = mn(A; B x C); (6)
mn(A; C x B) = mn(A x C; B):
By [2, Theorem 3.3.3], this product of nmanifolds mn determines an admis
sible product structure n in the theory MSpn*(.) which is commutative and
associative. At the level of nmanifolds commutativity of n means that for all
nmanifolds Aa, Bb there exists a nmanifold Kn Aa; Bb called the canonical
commutativity construction which is functorial in the category of nmanifolds
and satisfies the formula
ffiKn(A; B)=mn(A; B)[(1)abmn(B; A)[Kn(ffiA; B)[(1)a+1Kn(A; ffiB):(7)
See [2, Definition 2.1.3].
8
Associativity of n at the level of nmanifolds means that for all n
manifolds Aa, Bb Cc there exists a nmanifold An Aa; Bb; Cccalled the canon
ical associativity construction which is functorial in the category of nmanifo*
*lds
and satisfies the formula
a b c
ffiAn A ; B ; C = mn(A; mn(B; C)) [ mn(mn(A; B); C) [ An(ffiA; B; C)
[(1)a+1An(A; ffiB; C) [ (1)a+b+1An(A; B; ffiC):(8)
See [2, Definition 2.1.3].
Let Aa and Bb be Spmanifolds without singularities. Then Kn(A; B) can be
taken to be the cylinder
Kn(A; B) = I x A x B
with an Spstructure such that there is a diffeomorphism preserving Spstructur*
*es:
@ (I x A x B)= A x B [ (1)abB x A [ I x @A x B [ (1)a+1I x A x @B:
In this case Kn(A; B) has been described [7, section 10] in the special category
of manifolds as a "cupone product of manifolds". It projects in E1 of the ASS
to an algebraic cupone product. Moreover, the Spstructure on Kn(A; B) can
be chosen so that Kn satisfies the Hirsch formula [9]. Using the definition of
Kn(A; B), this property generalizes to the two cases of nmanifolds given in
the following lemma which suffice for the constructions of this paper. Statement
(b) is nontrivial; it is essential that the nmanifolds Aa and Cc have only one
common singularity, i.e. there is only one i n, such that both fiiA 6= ; and
fiiC 6= ;.
Lemma 3.1 (a) (Hirsch Formula) If Cc is an nmanifold and one of the
nmanifolds Aa, Bb is an Spmanifold without singularities then we have a
diffeomorphism of nmanifolds preserving Spstructures:
Kn(A x 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 nmanifolds that
have only one nonempty common singularity and c is even then there is a
ncobordism 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 constructions
on the left and right sides of equation (9).
(b) The obstruction to the associativity of the product structure n has
order three in the group MSpn*; see [2, Lemma 2.4.2]. Since the group MSpn*
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 n
manifolds in (10). The construction of this cobordism is straightforward when
the manifolds A, C have only one common singularity. 
Now we are ready to define the Toda bracket , where a, b, c 2 MSpn*.
Since the product of nmanifolds is not associate we need to use an associativi*
*ty
construction (8) to glue together the two usual pieces which define such a brac*
*ket
in an associative context. We use the standard sign conventions of [10].
Definition 3.2Let a, b, c 2 MSpn* such that ab = 0 and bc = 0. Let A, B, C
be a nmanifold which represents a, b, c, respectively. Let X, Y be nmanifolds
such that ffiX = mn(A; B) and ffiY = mn(B; C). Then
ffimn(X; C) = mn(mn(A; B); C); and ffimn(A; Y ) = (1)degAmn(A; mn(B; C)):
Let the Toda brackets be the set of all cobordism classes of nmanifo*
*lds
of
Z = (1)1+degBmn(X; C) [ (1)1+degBAn (A; B; C)[ (1)degA+degBmn(A; Y ):
Note 3.1 Let the Toda bracket be defined where the element a is repre
sented by a closed Spmanifold A. In this case the nmanifold An (A; B; C)is
just the cylinder
An (A; B; C)= I x A x mn(B; C);
see [2, Theorem 2.5.1]. Thus, the following nmanifold Z represents an element
of :
Z =(1) 1+degBmn(X; C)[(1)1+degBIxAxmn(B; C)[(1)degA+degBmn(A; Y ):
By properties of mn; An (see [2, Section 2.2]), the nmanifold Z depends
only on the cobordism classes a; b; c and on the choice of the nmanifolds X
and Y with ffiX = mn(A; B) and ffiY = mn(B; C). Therefore we have the usual
indeterminacy
in = ay + xc  x; y 2 MSpn*:
We define a generalized quadratic construction which we use in the next lem
ma to identify Toda brackets of the form . Suppose M is a nmanifold
of dimension 2k. Define a closed nmanifold (M) as follows:
i j i j
(M) = mn M(1); M(2) x 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)) x {0} ffiKn(M(2); M(1)),
10
mn(M(2); M(1)) = mn(M(2); M(1))
_ _
 
? ?
mn(M(1); M(2)) x {1} ffiKn(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 be defined in MSpn*, where a = [M]
and b = [R]for M a nmanifold of dimension 2k and R a Spmanifold. Then
n o
= (1)1+deg bb [(M)] + ax  x 2 MSpn* : (12)
Proof. Throughout this proof we ignore trivial associativity constructions in
which one of the three entries has empty singularities. Let Y be a nmanifold
such that ffiY = R x M. Then
ffi (Y [ (R x M x I)[ Kn (R; M))= M x R;
and the nmanifold
C = mn(Y; M) [ mn(R x M; M) x I [ mn (Kn (R; M); M)[ mn (M; Y )
is a representative of the Toda bracket (1)1+deg b. Glue the n
manifold Kn (Y; M)to the cylinder C x I by identifying the following manifolds:
mn(Y; M) x {1} = mn(Y; M) mn(M; Y ) x {1} = mn(M; Y )
_ _ _ _
   
? ? ? ?
C x {1} ffiKn (Y; M) C x {1} ffiKn (Y; M)
mn (K (R; M); M)x {1} = mn (K (R; M); M)
_ _
 
? ?
C x {1} ffiKn (Y; M)
The boundary of the resulting nmanifold Z is given by
ffiZ = R x (M) [ C x {0}
since, using the Hirsch formula,
ffiKn (Y; M)\ffiZ = Kn (ffiY;\M)ffiZ = Kn (R x M; M)\ffiZ = RxKn (M; M) :
See Figure 1. Thus, (1)1+deg b contains [ (M)] b and in =
aMSpn*. Therefore, (12) holds. 
11
' _____________________________________________________________ $
 
 
 Kn (Y; M) 
 
 
 R x K (M; M) 
 '' ________________$$n 
 
   
   
   
  R x (M)  
   
  @  
   
________________________________________________________________@@R
    
mn(Y;M)x{1}Rxmn (M;M) xIx{1} mn (Kn (R;M);M)x{1} mn (M;Y )x{1}
    
    
    
    
__________________________________________________________________
mn(Y; M) mn (R x M; M) x I mn (Kn (R; M); M) mn (M; Y )
Figure 1: The nmanifold Z.
The next property is well known for manifolds without singularities. See [1,
Definition 2.1(5)].
Lemma 3.4 Let be a Toda bracket which is defined in the ring MSpn*.
Assume that a = [A] is represented by a closed Spmanifold, and b = [B], c = [C]
are represented by nmanifolds. Then the following inclusion holds in the ring
MSpn*:
b (1)deg ac:
Proof. Let X, Y be nmanifolds such that ffiX = A x B and ffiY = mn(B; C).
Then the following nmanifold Z represents an element of (1)1+deg b:
Z = mn(X; C) [ I x A x mn(B; C) [ (1)1+deg aA x Y:
Thus, the nmanifold mn(B; Z) represents an element of (1)1+deg bb,
and using (6):
mn(B; Z) = mn(B; mn(X; C)) [ mn(B; I x A x mn(B; C))[
(1)1+deg amn(B; A x Y )
= mn(B; mn(X; C)) [ (1)1+deg bI x mn(B x A; mn(B; C))[
(1)1+deg amn(B x A; Y ):
12 


aeaeaeae
aeae 
oebmn(mn(Xe; B); C) aeae 
aeae 
@ aeae 
@ aeae 
@R aeae 
m (m (B; X); C) oe Ixm (m (BxA; B); C) aeae 
_________________________________________aeaeoebAn(Xe;bB;C)nnnn
   
   
  oe m (Xe; m (B; C))
   b n n 
   @ aeae
   ae 
   @@Rae 
An(B; X; C)  oebIxAn(BxA; B; C)  ae 
   ae 
   ae 
   ae @I
   ae oeamn(Xe; Y )@
_________________________________________________________ae@
    V1
   
   
   
   
   
__________________________________________________________
mn(B; mn(X; C)) oebIxmn(BxA; mn(B; C)) oeamn(B x A; Y )
Figure 2: nmanifold V2, where oea = (1)dega, oeb = (1)degb.
Let eXbe a n manifold such that ffiXe = B x A. Glue (1)deg amn(Xe; Y ) to
the cylinder mn(B; Z) x I along their common boundary:
mn(B x A; Y ) = mn(B x A; Y )
_ _
 
? ?
ffimn(B; Z) x {1} ffimn(Xe; Y ) :
Denote the resulting nmanifold as V1; see Figure 2. Now consider the nmanifol*
*ds
An(B; X; C); I x An(B x A; B; C); An(Xe; B; C)
with boundaries:
ffiAn(B; X; C)
= mn(B; mn(X; C)) [ mn(mn(B; X); C) [ (1)deg b+1An(B; ffiX; C)
= mn(B; mn(X; C)) [ mn(mn(B; X); C) [ (1)deg b+1An(B; A x B; C)
= mn(B; mn(X; C)) [ mn(mn(B; X); C) [ (1)deg b+1An(B x A; B; C);
13
ffi (I x An(B x A; B; C))=
I x {mn(B x A; mn(B; C)) [ mn(mn(B x A; B); C)}[ @I x An(B x A; B; C);
ffiAn(Xe; B; C)= mn(Xe; mn(B; C)) [ mn(mn(Xe; B); C) [ An(ffiXe; B; C)
= mn(Xe; mn(B; C)) [ mn(mn(Xe; B); C) [ An(B x A; B; C):
Now we glue together the nmanifolds
V1; An(B; X; C); (1)1+deg bI xAn(B xA; B; C); (1)deg bAn(Xe; B; C):
The resulting nmanifold V2 gives a bordism between the nmanifolds
mn(B; Z) x {0} and
i j
mn mn(B; X) [ (1)deg bI x mn(B x A; B) [ (1)deg bmn(Xe; B);:C
The latter nmanifold represents an element of (1)1+deg a + degc.
Thus,
(1)deg b+1b (1)deg a+deg b+1c
and b (1)deg ac, as required. 
4 AdamsNovikov Spectral Sequence for MSpn
Recall that our plan for determining elements of higher order in the ring MSp*
is to construct nmanifolds which project to infinite cycles in the E2terms of
the ANSS and ASS of MSpn* for n 3. Then we determine the order of these
projections in E2 of the ANSS and bring back these nmanifolds to MSp*. In
this section, we accomplish the first part of our program by describing particu*
*lar
torsion elements t(i1; . .;.i2s) of higher order in the first line
E1;4*+12(MSpn ) = Ext1;4*+1ABP(BP *(MSpn ); BP *)
of the ANSS which are the projections of the nmanifolds which we will con
struct in Section 5.
Throughout this section, let n 0 be a fixed integer, and let MSp0 denote
MSp. In [2, section 1.6], the ANSS for each of the spectra MSpn is described in
terms of geometrical constructions on manifolds with singularities. In particul*
*ar,
the ANSS for the spectrum MSpn is identified with the singularities spectral
sequence (SSS) associated with the exact couple:
MSpn* ____________MSp(1)n*oefl(1)n_____MSp(2)n*oefl(2.).n._oefl(3)n
@ @(1)n @ @(2)n @
ss(0@) ss(@1) ss(@2)
@@Rn @@Rn @@Rn
fi(1)n ______________fi(2)n
MSp(1)n* ___________MSp(2)n* . . .
(13)
14
Here MSp(k)n* , MSp(k)n*are the coefficient groups of particular bordism the
ories closely related to the theories MSpn*(.), MSp*(.); see [2, section 1.4].
The E*;*2term of the SSS (or the ANSS) is described as follows. Consider the
bigraded commutative algebra:
M = MSp* [un+1; un+2; . .u.n+k; . .].
where un+k = (1; 2(2n+k  1)), and x = (0; degx) for x 2 MSp*. Let
M s= {z 2 M  z = (s;:*)}
As we shall see, M s is isomorphic to the sth line Es;*1of the ANSS for
MSpn . We have the following complex:
M 0D!M1D!M2D!.D.<.n>!MkD!.:.(.14)
The differential D is defined as
i j Xj i j
D xua1i1.u.a.jij= (1)fflt(ff)(fiitx)ua1i1.u.a.t+1it.u.a.jij
t=1
where n < i1 < .P.<.ij, ff = (a1; :::; aj) is a sequence of nonnegative integers
and fflt(ff) = ti=1ai. It follows from the product formula (2) for Bockstein
operators, that the subalgebra of cycles of the algebra M is a DGA. Therefor*
*e,
the homology H*(M ) of the complex M has an induced algebra structure
from M .
Note 4.1 The elements ui, i = n + 1; n + 2; . .a.re the projections of the "bas*
*ic
Ray elements" u1 = j, ui= OE2i2for i 2. We use the same notation for these
elements 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 MSpn in
the theory BP *(.).
(ii) There is an isomorphism of algebras
E2(MSpn ) = Ext*;*ABP(BP *(MSpn ); BP *) ~=H*(M ):
In particular, there is a ring isomorphism
E0;*2 = Hom*ABP(BP *(MSpn )); BP *)
~= H0(M ) = T1k=n+1Ker(fik : MSp* ! MSp*): 
Note 4.2 The complex M in (14) is the bottom line of the diagram (13). In
particular the first AdamsNovikov differential D : M 0! M 1is given
by
M1
D= fi(1)n = fik:
k=n+1
15
Now we are ready to use the above results to set up the environment in which
we will do our chromatic calculations of E2(MSpn*). Since the ring MSp* is
polynomial and wm is one of its generators, we have the following exact sequenc*
*e:
k ss(k)
0 ! MSp* .wm!MSp* ! MSp*=(wkm) ! 0 (15)
where (wkm) is the principal ideal generated by wkmand ss(k)is the natural proj*
*ec
tion. Since D(wm ) = 0, the exact sequence (15) induces the exact sequence
of complexes:
k ss(k)
0 ! M .wm!M! M =(wkm) ! 0 : (16)
We paste the sequences (16) together to obtain the following commutative dia
gram: (1)
0 ! Mfi.wm!Mss!M=(wm )! 0
fi ffii fifi
Idfifiy .wmfifiy .wmfifiy
2 ss(2)
0 ! Mfi.wm!M! M =(w2m)! 0
fi ffii fifi
Idfifiy .wmfifiy .wmfifiy
.. . .
.fi .. .. (17)
fi ffii fifi
Idfifiy .wmfifiy .wmfifiy
k ss(k)
0 ! Mfi.wm!M! M =(wkm)! 0
fi ffii fifi
Idfifiy .wmfifiy .wmfifiy
.. . .
. .. ..
Taking the direct limit of the rows of (17), we obtain the following short sequ*
*ence
of complexes:
0 ! M ! w1mM ss!M=(w1m) ! 0 (18)
where i j
w1mM = lim!M .wm!M.wm!. .a.nd
i .w .w j
M =(w1m) = lim!M=(wm ) m!M =(w2m) m!. .:.
The short exact sequence of complexes (18) induces the following long exact
sequence in homology:
0 ! H0(M ) ! H0(w1mM) ! H0(M =w1m) ffin!H1(M ) !
H1(w1mM) ! H1(M =w1m) ffin!H2(M ) ! . . .
(19)
The key point about (19) is that the complex w1mM is acyclic.
16
Lemma 4.2 For n 1, n m 0 and s 1:
Hs(w1mM) = 0: (20)
It follows that we have the exact sequence
0 ! H0(M ) ! H0(w1mM) ! H0(M =w1m) ffim!H1(M ) ! 0;
(21)
and for s 1 we have group isomorphisms
Hs(M =w1m) ~=Hs+1(M ):
Proof. Consider the following subalgebra of w1mM:
L(i1; . .;.ik) = w1mMSp*[ui1; . .;.uik];
where n < i1 < . .<.ik. This subalgebra is closed under the differential D,
so it is also a subcomplex of w1mM. To prove (20) it is enough to show that
Hs(L(i1; . .;.ik)) ~=0 for allk 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, fiixm;i= wm . In terms of the algebra L(i) this means that
1
D wm xm;i = ui:
Let xuai2 L(i); a 1, be any element such that D (xuai)= (fiix)ua+1i= 0.
Then fiix = 0 and
1 a1 1 a1 a
D wm xm;ixui = fii wm xm;ix ui = xui:
The induction step: Hs(L(i1; . .;.ik1)) ~=0 =) Hs(L(i1; . .;.ik)) ~=0.
We have the following exact sequences of complexes:
0 ! L(i1; . .;.ik1) ! L(i1; . .;.ik) ! L(ik) ! 0 :(23)
The long exact sequence determined by (23) implies that Hs(L(i1; . .;.ik)) ~=0
for s 1. 
Now let n 3. We describe the structure of the subring
H0(w1n1M) w1n1MSp*:
Let wi, xi;j, xi1;:::;isbe the polynomial generators of the ring MSp* described
in Theorem 2.4. Define the following polynomial generators of w1n1MSp*:
Zj = xj;n1; j = 1; 2; . .;.n  2; Zn = xn1;n; (24)
17
Xi= 2xn1;i_w wi; Yi= xn1;i_2(xn1;i wiwn1); i n + 1:(25)
n1 wn1
Note that
X2i= 4Yi+ w2i:
Let 1 i < j; i; j 6= n  1. Then define
Xi;j= xi;j wixn1;j_wjxn1;i_w+ 2xn1;ixn1;j_2: (26)
n1 wn1
Note that if 1 i < n  1, j 6= n  1, then we could have chosen the polynomial
generators
X0i;j= xi;j wixn1;j_w: (27)
n1
We can also choose polynomial generators Xi1;:::;isof w1n1MSp* for s 3
as xi1;:::;is+ . ... We only need their existence. Their exact definition, is *
*not
necessary for our computations. However, for completeness, we define them as
follows.
If 1 < i1 < . .<.is; i1; . .;.is 6= n  1 then define
Xi1;:::;is= xi1;:::;is
0 1
s2X w X
+ @ (1)k___1_k xn1;it1. .x.n1;itkxi1;...;bit;...;bit;...;isA
k=1 wn1 1t1<...* = Z(2)[w1; . .;.wn2]:
18
We will also need the following polynomial subrings of w1n1MSp*:
R1 = Z(2)[Zj; Xi; Yi  j = 1; . .;.n  2; n; i; n + 1]
R2 = Z(2)[Xi;j 1 < i < j; i; j 6=;n  1]
R3 = Z(2)[Xi1;:::;is s 3; 1 < i1 < . .<.is];
R = R1 R2 R3:
Lemma 4.3 There is a ring isomorphism:
1
H0(w1n1M) ~=Z(2) wn1; wn1 W* P* R
Proof. There is a ring isomorphism:
1
w1n1MSp* ~=Z(2) wn1; wn1 W* P* T R2 R3;
where
T = Z(2)[wn1; . .;.wn+k; . .;.x1;n1; . .;.xn2;n1; xn1;n; . .;.xn1;n+k; .*
* .].
Since the subring T of MSp* is closed under the action of the Bockstein operato*
*rs
fij for j n + 1, T generates the subcomplex
T = T [un+1; . .;.un+k; . .]. (28)
of M. To prove this lemma, it suffices to establish the isomorphism:
H0(T ) ~=w1n1R1: (29)
For k 1, define the following subrings of w1n1MSp*:
T (k)= Z(2)[wn1; . .;.wn+k; x1;n1; . .;.xn2;n1; xn1;n; . .;.xn1;n+k];
T0(k)= Z(2)[wn1; wn+k; xn1;n+k];
R(k) = Z(2)[wn1; Zj; Xi; Yi  j = 1; . .;.n  2; n; i = n + 1;;. .;.n + k]
R(k)0= Z(2)[wn1; Xn+k; Yn+k]:
Since the rings T (k), T0(k)are also closed under the action of the Bockstein
operators fij for j n + 1, we can define the subcomplexes T (k)and T0(k)of
M as in (28). Direct computation shows that
H0(w1nT0(k)) ~=w1n1R(k)0: (30)
19
In particular, we have the isomorphisms:
H0(w1n1T (1))~= H0(w1n1T0(1)) Z(2)[x1;n1; . .;.xn2;n1; xn1;n]
~= w1n1R(1)0 Z(2)[x1;n1; . .;.xn2;n1; xn1;n]= w1n1R(1):
By induction on k 1, we have the a homomorphism of short exact sequences
of complexes
0 ____w1n1T (k)________w1n1T (k+1)_______w1n1T0(k+1)__0
6 6 6
  
  
  
0 _____w1n1R(k)_________w1n1R(k+1)________w1n1R(k+1)0_0
where the complexes on the bottom line have zero differential. The left and rig*
*ht
vertical maps induce isomorphisms in homology. Thus, the Five Lemma com
pletes the induction proof of the isomorphisms (30). Taking the direct limit ov*
*er
k of the isomorphisms (30) establishes the isomorphism (29). 
Define
tn(i1; . .;.i2s) = ffin xn1;i1._.x.n1;i2s_w2 H1(M()31)
n1
where 3 n < i1 < . .i.s, ffin is the boundary homomorphism of (21) and
H1(M ) ~=Ext1;*ABP(BP *(MSpn ); BP *):
These are the required elements of higher torsion in the first line of the ANSS.
Proposition 4.4 The element
tn(i1; . .;.is) 2 Ext1;4*+1ABP(BP *(MSpn ); 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 proof
for the case when s odd is a slight modification of the even case. The following
technical lemma will be used to show that 2s1tn(i1; . .;.i2s) 6= 0.
Lemma 4.5 There does not exist an element Y 2 MSp*, such that the element
Z = 2s1xn1;i1. .x.n1;i2s wn1Y; (32)
belongs to the ring H0(M ).
20
Proof. By the definition of Xi;jin (26),
(i; j) = 2xn1;ixn1;j wn1(wixn1;j wjxn1;i) + w2n1xi;j
= w2n1Xi;j2 MSp*:
Thus, (i; j) 2 H0(M ), and there is an element a 2 MSp* such that
(i1; i2) . .(.i2s1; i2s) = 2sxn1;i1. .x.n1;i2s wn1a 2 H0(M ):
Suppose that the element Z from (32) does exist. Then we have:
2Z = 2sxn1;i1. .x.n1;i2s 2wn1Y or
(i1; i2) . .(.i2s1; i2s)  2Z = wn1(2Y  a):
In particular, in the polynomial ring H0(w1n1M) Z=2 we have
(i1; i2) . .(.i2s1; i2s) = w2sn1Xi1;i2.X.i.2s1;i2s= wn1(2Y  a):
Thus,
2Y  a = w2s1n1Xi1;i2.X.i.2s1;i2s:
It remains to observe that the element w2s1n1Xi1;i2.X.i.2s1;i2sdoes not belo*
*ng
to the ring H0(M ) 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
w1n1xn1;i1. .x.n1;i2s2 H0(M =w1n1)
is a Dcycle. Suppose that
2s1tn(i1; . .;.i2s) = 0
in H1(M ). By the exact sequence (21), there is an element Y 2 MSp* such
that s1
Y + 2___xn1;i1._.x.n1;i2s_w
n1
is a cycle in H0(w1n1M). Then the element
wn1Y + 2s1xn1;i1. .x.n1;i2s2 MSp*
is a cycle in M , which contradicts Lemma 4.5. 
Note 4.3 Proposition 4.4 fails for n = 2 since the generators Xi1;...;isfor the
ring H0(w11M<2>) are essentially different from the case n > 2 because the
elements x1;j1;...;jtare not defined in the ring MSp*.
21
5 Existence of Higher Torsion Elements
This section is devoted to the proof of Theorem A. In particular, we construct
elements on(i1; . .;.is) 2 MSpn* which project to the elements tn(i1; . .;.is) *
*of
higher 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 exist
indecomposable elements on(i) 2 MSpn4*+1with the following properties:
(i) on(i1) = OE2i12;
(ii) on(i1; . .;.is) 2 ;
(iii)on(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 deduce
the existence of higher torsion in MSp* of all orders.
Theorem B. For each k 1 there exist elements of order 2k in the symplectic
cobordism ring MSp*.
The motivation for defining the on(i1; . .;.is) by induction on s 1 so that th*
*ey
satisfy conditions (i) and (ii) of Theorem A is that their projections tn(i1; .*
* .;.is)
into the one line of E1;4*+12(MSpn ) of the ANSS satisfy the corresponding
algebraic conditions:
(i) tn(i1) = ui1;
(ii) tn(i1; . .;.is) 2 .
Note that the complex M has a nice product structure which enables us to
define Massey products in E2 = H*(M) in the usual way.
Our construction begins with the following result. We let MSp0* denote
MSp and OE1 denote OE0 = j.
Theorem 5.1 (V.Gorbunov, [2, Theorem 4.3.5]) For j > 2n2 and n 1 the
Toda bracket contains zero in the ring MSpn1*.
Recall from Section 2 that MSpn* 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 = j and Pk is an Spmanifold which represents
the basic Ray element uk = OE2k2for k 2. We also chose Spmanifolds Wi
such that @Wi = 2Pi. Let j n 2. By Theorem 5.1, the Toda bracket
contains zero in the ring MSpn2*. In other words, there ex*
*ist
n2manifolds eXn1;j, Wj(n1)and Wn(j)1such that
ffiWn(j)1= Pn1 x 2; ffiWj(n1)= 2 x Pj and
22
ffiXen1;j= Wn(j)1x Pj [ Pn1 x Wj(n1):
As nmanifolds we have
ffiXen1;j= Wn(j)1x Pj: (33)
Note that cobordism classes of the manifolds Wn(j)1depend, in general, on j.
Lemma 5.2 For j n 2, there exist nmanifolds Wn1 and Xn1;j, such
that
ffiXn1;j= Wn1 x Pj (34)
where Wn1 does not depend on j.
Proof. We prove this lemma by induction on n 2. Let n = 2. For each j 2
wehhaveithat fi1W1(j)= 2 by construction. For j 2, all the 1bordism classes
W1(j) equal the same element w1 2 MSp12 since w1 is the unique cobordism
class such that fi1w1 = 2.
Now assume that this lemma is true for n  1. Let Wn1 be any Spmanifold
such that @Wn1 = 2Pn1. Let j n with eXn1;jand Wn(j)1n2manifolds as
above. We will define an nmanifold Xn1;jthat satisfies (34). Since MSpn*
is a polynomial ring in degrees less than or equal to 2n+2  4, we have that
h i
fl = Wn(j)1 [Wn1];
is a polynomial in the generators w1; . .;.wn1, xr, r 2n3, r 6= 2l1, degxr =
4r, asPin the statement of Theorem 2.1. Let fl be the sum of k momomials:
fl = ki=1fli. Since dim Wn1 = 2(2n1  1), each monomial fli contains at
least one factor wmi, mi n  2; write fli = efliwmi. By induction, there is a
mimanifold Xmi;j, mi < n  1, such that ffiXmi;j= Wmi x Pj. Let ei be a
nmanifold which represents efli. Define a nmanifold Xn1;jas the disjoint
union of the following nmanifolds:
[k i j
Xn1;j= eXn1;j mn ei; Xmi;jx Pj
i=1
where the nmanifolds eXn1;jare as in (33). Clearly ffiXn1;j=Wn1xPj. 
Lemma 5.3 There exist elements on (i1; . .;.is), in the ring MSpn4*+1for s 1
and n < i1 < . .<.is such that:
(i) on(i) = ui;
(ii) on (i1; . .;.is)2 for s 2;
(ii) wn1on (i1; . .;.is)= 0.
23
Proof. We construct the elements on (i1; . .;.is)by induction on s 1. For
s = 1, Theorem 5.1 gives that wn1uj = 0. Assume that this lemma is true for
s  1. Select any element on (i1; . .;.is)of the Toda bracket
:
We must show that wn1on (i1; . .;.is)= 0. Let Wn1 be a nmanifold as in
Lemma 5.2 which represents wn1 so that there exists a nmanifold Xn1;is
with ffiXn1;is= Wn1 x Pis. By Lemma 3.4 we have that
wn1on (i1; . .;.is)2wn1
on (i1; . .;.is1):
Note that the nmanifold (Wn1) of (11) has dimension 2(2n1  1)  1 =
dimun. The element un is the unique nontrivial element of this degree in
MSpn1*. Thus,
[(Wn1)] n1 = un; where = 0 or1: (35)
Therefore, in MSpn*, [(Wn1)] = 0. Thus in MSpn*, Lemma 3.3 and the
induction hypothesis give
on (i1; . .;.is1)=(unuis+wn1a) on (i1; . .;.is1)=0: 
Next we determine the projection of on (i1; . .;.is)into the E2term of the
ANSS for MSpn to be tn(i1; . .;.is) which we defined in (31) and studied in
Proposition 4.4. We do this by constructing a nmanifold which represents
on (i1; . .;.is)and a nmanifold whose boundary equals wn1on (i1; . .;.is)
modulo the AdamsNovikov 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 on(i1; . .;.is) of the Toda bracket
such that the projection of on(i1; . .;.is) into E1;4*+12(MSpn ) of the ANSS
equals
Xs
tn(i1; . .;.is) = uirxn1;i1. .b.xn1;ir.x.n.1;is:
r=1
Proof. Lemma 5.3 gives us a nmanifold T (i1; . .;.is)which represents the
element o (i1; . .;.is)and a nmanifold H (i1; . .;.is)with
ffiH (i1; . .;.is)= m (Wn1; T (i1; . .;.is)):
We use induction on s 1 to define specific nmanifolds Ts and Hs such that:
24
(i) ffiHs = m (Wn1; Ts)[ Ls;
(ii) Ts projects in the one line of the ANSS to tn(i1; . .;.is) 2 H1(M) ;
(iii)Hs projects in the zero line of the ANSS to xn1;i1. .x.n1;is2 H0(M) ;
(iv) Ls has AdamsNovikov filtration degree two.
If s = 1 let T1 = Pi1, H1 = Xn1;i1and L1 = ;. By induction suppose that
we have nmanifolds Ts1 and Hs1 which satisfy the above four conditions.
Consider the nmanifold
H(0)= m (Hs1; Xn1;is)
with
ffi(H(0)) = m(m(Wn1; Ts1); Xn1;is)[m(Hs1; Wn1xPis)[m(Ls1; Xn1;is):
Glue the nmanifolds H(0)and A(Wn1; Ts1; Xn1;is) together along their
common boundary m(m(Wn1; Ts1); Xn1;is) to obtain the nmanifold
H(1)= H(0)[ A (Wn1; Ts1; Xn1;is)
with
ffi(H(1))= m(Wn1; m(Ts1; Xn1;is)) [ m(Hs1; Wn1 x Pis)[
A(Wn1; Ts1; Wn1 x Pis) [ m(Ls1; Xn1;is)
= m (Wn1; m (Ts1; Xn1;is))[ m(Hs1; Wn1) x Pis[
An(Wn1; Ts1; Wn1) x Pis[ m(Ls1; Xn1;is):
Next glue the nmanifolds H(1)and K(Hs1; Wn1)xPistogether along their
common boundary mn(Hs1; Wn1) x Pisto obtain the nmanifold
Hs = H(1)[ K(Hs1; Wn1) x Pis
with
ffi(Hs)= m(Wn1; m(Ts1; Xn1;is)) [ m(Wn1; Hs1) x Pis[
K (m(Wn1; Ts1) [ Ls1; Wn1)x Pis[
A(Wn1; Ts1; Wn1) x Pis[ m(Ls1; Xn1;is)
= m(Wn1; m(Ts1; Xn1;is) [ Hs1 x Pis)[
{K(m(Wn1; Ts1) [ Ls1; Wn1)[ A (Wn1; Ts1; Wn1)}x Pis[
m(Ls1; Xn1;is)
= m(Wn1; Ts) [ Ls
25
where
Ts = m(Ts1; Xn1;is) [ Hs1 x Pis:
and
Ls = {K (m(Wn1; Ts1) [ Ls1; Wn1) [ A(Wn1; Ts1; Wn1)}x Pis[
m(Ls1; Xn1;is):
Since Ts1, Ls1 has AdamsNovikov filtration degree one, two, respectively,
the projection of the nmanifold Ls to the one line of the ANSS are trivial.
Therefore, Ls has AdamsNovikov filtration degree two. By construction and
the induction hypothesis, the projection of Ts to the one line of the ANSS equa*
*ls
tn (i1; . .;.is1)xn1;is+ xn1;i1. .x.n1;is1uis= tn (i1; . .;.is):
Since Ts1 and Pishave AdamsNovikov filtration degree one, the projections of
A (Wn1; Ts1; Xn1;is)and K(Hs1; Wn1) x Pisto the zero line of the ANSS
are trivial. Thus, the projection of Hs to the zero line of the ANSS equals the
projection of H(0)which is xn1;i1. .x.n1;is1. xn1;is. 
We complete the proof of Theorem A. By the previous lemma, the element
on (i1; . .;.is)2 MSpn* can be defined as required so that it projects to
tn(i1; . .;.is) 2 Ext1;*ABP(BP *(MSpn ); BP *)
which has order 2[(s+1)=2]by Lemma 4.4. Therefore, on (i1; . .;.is)has order
grater or equal to 2[(s+1)=2]in MSpn*. Finally, note that on (i1; . .;.is)is i*
*n
decomposable in MSpn* because its projection tn(i1; . .;.is) into the algebra
E*;*2(MSpn ) is indecomposable.
Proof of Theorem B. By Theorem A, there are torsion elements of order
greater than or equal to 2k for all k 1 in the ring MSp3 . Consider the
BocksteinSullivan exact sequences:
. ..!MSp2* .OE2!MSp2* ss2!MSp3* fi3!MSp2* ! . . .
. ..!MSp1* .OE1!MSp1* ss1!MSp2* fi2!MSp1* ! . . .
. ..!MSp* .j!MSp* ss0!MSp1* fi1!MSp* ! . . .
We show that exponents of the groups TorsMSp2*, TorsMSp1* and TorsMSp*
must be infinite since the exponent of Tors MSp3* is infinite. Assume, to
the contrary, that all torsion of MSp2* has exponent 2k. We take an element
a 2 MSp3* of order 22k+1. Then the element a1 = fi2(a) has order no more
than 2k. From the above BocksteinSullivan exact sequence,
n ss o
2ka 2 Im MSp2* !1MSp3* MSp3*:
26
Let ss2(a2) = 2ka. Then 2k+1a2 2 Ker ss2 = Im (.OE2), so 2k+1a2 = OE2x. Conse
quently 2k+2a2 = 0, and a2 has finite order. Since
ss2(2ka2) = 22ka 6= 0 ;
the element a2 2 MSp2* has order grater than or equal to 2k+1, contradicting
the assumption that Tors MSp2* has exponent 2k. Thus, the exponent of
Tors MSp2* is infinite. 
Note 5.1 Theorem B is just an existence theorem, and its proof does not give
a specific way to construct torsion elements of higher order in MSp*. However,
for each n 3 the family on(i) 2 MSpn* determines a different family of higher
order torsion elements in MSp*. In the next two sections and in [3] we study
the family that is determined by o3(i) 2 MSp3*. The analysis of the cases n 4
requires more topological information about MSp* than we have at present.
6 Construction of Elements in MSp2* and MSp1*
In Lemma 5.3 we constructed elements of higher torsion
o (i1; . .;.is)= o3(i1; . .;.is)2 MSp34*+1:
In this section we study the elements:
fl (i1; . .;.is)=fi3o (i1; . .;.is)2 MSp24*+3;
ff0(i1; . .;.is)=fi2fl (i1; . .;.is)2 MSp14*+1:
In particular we compute their projection to the three line of the ANSS. Throug*
*h
out this section, let m and K denote the canonical constructions m2 and K2 of
Section 3.
We begin by interpreting fi3(o3(i1; . .;.is))in terms of manifolds with sin
gularities. Recall that by Lemma 5.3 there is a representative 3manifold
T (i1; . .;.is)of o3(i1; . .;.is)and a 3manifold H (i1; . .;.is)such that
ffiH (i1; . .;.is)= m(W2; T (i1; . .;.is)):
We can consider the manifold T (i1; . .;.is), H (i1; . .;.is)as a 2manifold
eT(i1; . .;.is), eH(i1; . .;.is), respectively, with
ffiHe(i1; . .;.is)=m(W2; eT(i1; . .;.is)) [ P3 x E (i1; . .;.is);
(36)
ffiTe(i1; . .;.is)=P3 x G (i1; . .;.is)
where G (i1; . .;.is)= fi3T (i1; . .;.is)represents the 2cobordism class
fl (i1; . .;.is). Note that
ffiE (i1; . .;.is)= m (W2; G (i1; . .;.is)):
27
To determine the projection of E (i1; . .;.is)to the ANSS we need to identify t*
*he
quadratic construction (W2) which was defined in Section 3. The following
lemma is an easy computation in the ASS for MSp2*. We defer its proof to
Lemma 4.3 of [3].
Lemma 6.1 The cobordism class of the 2manifold (W2) equals P3 in MSp2*.

We are now ready to compute the projection of E (i1; . .;.is)into the two line
of the ANSS for MSp2 . This will lead directly to the identification of the
projection of the fl (i1; . .;.is)into the three line of the ANSS for MSp2 .
Lemma 6.2 (a) E (i1)= ;.
(b) For s 2, E (i1; . .;.is)projects in E2;4*+21of the ANSS for MSp2 to
X
e (i1; . .;.is)= uit1uit2x2;i1. .b.x2;it1.b.x.2;it2.x.2.;is:
1t1;
(d) For s 3, fl (i1; :::;pis)rojects in E3;*1(MSp2 ) of the ANSS to
X
g (i1; . .;.is)= uit1uit2uit3x2;i1.b.x.2;it1.b.x.2;it2.b.x.2;it3.*
*x.2.;is:
1t1.
(d) We use induction on s 3. The case s = 3 follows from (b). Assume
the case s  1. By the description of the 2manifold G (i1; . .;.is)in the proof
of (c), fl (i1; . .;.is)projects in the three line of the ANSS to
g (i1; . .;.is)= xis;2g (i1; . .;.is1)[ uisx e (i1; . .;.is1):
By the induction hypothesis and the previous lemma,
X
g (i1; . .;.is)= xis;2 uit1uit2uit3x2;i1.b.x.2;it1.b.x.2;it2.b.x.2;it3.*
*x.2.;is1
1t1 is. The element x depends on the generators xj1;...;jk; wj and
uj. In particular, the formula for the first differential is invariant under t*
*he
transposition (3; n)in all entries of the elements x and g(3; i2; . .;.is). App*
*lying
this permutation we obtain an element x0, such that d1(x0) = 2tg(i2; . .;.is; n*
*), a
contradiction. 
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 will
denote i1; . .;.i2s for the remainder of this section. We prove Proposition 7.1*
*(i)
by showing that g(i) has order at least 2s1. Let MSp2 ss!MSp3 denote the
canonical map which induces the homomorphism
E*;*2(MSp2 ) ss*!E*;*2(MSp3 ):
Let eg(i) = ss*(g(i)) 2 E3;*2(MSp3 ). By Proposition 6.3(d),
X
eg(i) = uit1uit2uit3x2;i1. .b.x2;it1.b.x.2;it2.b.x.2;it3.x.2.;i2*
*s:
1t1 ! w12M<3> ! M<3> =(w12) ! 0;
0 ! M<3> =(w12) ! w11M<3>=(w12) ! M<3> =(w12; w11) ! 0;
0 ! M<3> =(w12; w11) ! w13M<3>=(w12; w11) ! M<3> =(w13; w12; w11) ! 0:
Recall that by Theorem 5.1 there exist elements x1;k2 MSp1*, x2;k2 MSp2*,
and x3;k2 MSp3* such that
fik (x1;k)= w1; fik (x2;k)= w2; fik (x3;k)= w3:
The arguments used to prove Lemma 4.2 can be used to prove the following
lemma.
Lemma 7.3 The complexes
w12M<3>; w11w13M<3>; w11M<3>=(w12);
w13M<3>=(w12); w13M<3>=(w12; w11)
are acyclic, i.e. their nth homology groups are zero for n 1. 
34
Consider the following composition of boundary homomorphisms:
H0(M<3> =(w12; w13; w11))ffi(0)!H1(M<3> =(w12; w11))ffi(1)!
H2(M<3> =(w12))ffi(2)!H3(M<3> ):
By Lemma 7.3, ffi(0) is an epimorphism and ffi(1), ffi(2) are isomorphisms.
Define the following elements:
eg2(i) 2 H2(M<3> =(w12)); eg1(i) 2 H1(M<3> =(w12; w11));
eg0(i) 2 H0(M<3> =(w12; w13; w11));
where
X
eg2(i) = w12 uit1uit2x2;i1. .b.x2;it1.b.x.2;it2.x.2.;i2s;
1t1 =(w12; w11))has order at
least 2s1.
Proof. Assume that 2s2 eg1(i) = 0. Then ffi(0) 2s2eg0(i)= 2s2eg1(i) = 0.
Therefore, there is an element Z1 2 H0 w13M<3>=(w11; w12) such that
ss1(Z1) = 2s2 eg0(i)
where ss1 is the homomorphism in the diagram below. In the following diagram
the row and both columns are exact.
35
0




?
0 H0(M<3>=(w11; w12))
 
6 
 
 
 ?
1 1 1 ss2 1
H0 w3 M<3>=w12 __H0 w1 w3 M<3>=w12 __H0 w3 M<3>=(w11; w12) __0
 
6 
ss3 ss1
 
 ?
1 1 1
H0 w2 w1 w3 M<3> H0(M<3>=(w11; w12w13))
 
6 
 ffi(0)
 
 ?
1 1
H0 w1 w3 M<3> H1(M<3>=(w11; w12))
 
6 
 
 
 ?
0 0
From the diagram above, we see that there exist
1 1 1 1 1 1
Z2 2 H0 w1 w3 M<3>=w2 and Z3 2 H0 w2 w1 w3 M<3>
such that ss3(Z3) = Z2 and ss2(Z2) = Z1. Let
0 1
X
S = 2s2@ x1;it1x3;it2x2;i1. .b.x2;it1.b.x.2;it2.x.2.;i2sA:
1t1) , 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 2 MSp*
such that
C = wp11wq12S + wp1B1 + wq2B2 + w3B3
belongs to the ring H0(M<3>) .
1
Proof. Recall from Lemma 4.3 that H0 w2 M<3> is a polynomial ring and
that there is a ring monomorphism:
1
0 ! H0(M<3>) ss*!H0 w2 M<3> :
Assume that we have chosen Y1 and Y2 in (38) to make q so large that the cycle C
can be written as a polynomial in the polynomial generators of H0 w12M<3> .
Also, we can always choose integral polynomial generators for H0 w12M<3> ,
i.e. from the image ss*(H0(M<3> )), which include
1;j= w2x1;j w1x2;j= w2X01;j;
2;j= 2x2;j w2wj = w2Xj;
(39)
3;j= w2x3;j w3x2;j= w2X03;j;
i;j= 2x2;ix2;j w2(wix2;j wjx2;i) + w22xi;j= w22Xi;j
where 1 i < j; i; j =2{1; 2; 3}and X01;j; X03;j; Xi;j; Xj are the polynomial
generators of H0 w12M<3> defined in (25), (26), (27).
Define
X(j1; . .;.j2t) = Xj1;j2. .X.j2t1;j2t2 H0(w12M<3>);
(j1; . .;.j2t) = j1;j2. .j.2t1;j2t= w2t2X(j1; . .;.j2t) 2 H0(M<3> ):
It follows from (39) that
w22x1;it1x3;it2= 1;it13;it2+ w1x2;it13;it2+ w3x2;it21;it1+ w1w3x2;it1x2;it2
= 1;it13;it2+ w1a1 + w3a2
(40)
where a1; a2 2 MSp*. Consider the monomial
l = 2s1x1;it1x3;it2x2;i1. .b.x2;it1.b.x.2;it2.x.2.;i2s:
37
It follows from (39) and (40) that
2 s1
w22l = w2x1;it1x3;it22 x2;i1. .b.x2;it1.b.x.2;it2.x.2.;i2s
i j
= 1;it13;it2+ w1a1 + w3a2 (i1; . .;.bit1; . .;.bit2; . .;.i2s) + w2D
= 1;it13;it2(i1; . .;.bit1; . .;.bit2; . .;.i2s)
+ (w1a1(it1; it2) + w3a3(it1;(it2))i1; . .;.bit1; . .;.bit2; . .;.i2s)
+w32x1;it1x3;it2Dk(i1; . .;.bit1; . .;.bit2; . .;.i2s):
Assume that B1; B2; B3 exist which make C a cycle in M<3>. Then
p q
2C = wp11wq322w22S + 2 (w1B1 + w2B2 + w3B3)
X
= wp11wq32 1;it13;it2(i1; . .;.bit1; . .;.bit2; . .;.i2s)
1t1) :
2C  wp11wq32 = wp1K1 + wq2K2 + w3K3 = K: (42)
Let fil1;...;lkX denote the composition fil1(fil2(. .(.filkX).).).for an elemen*
*t X
of MSp*. Recall that fii(fijX)= fij(fiiX). Thus fil1;...;lkX does not depend on
the order of l1; . .;.lk. The proof of the following lemma is straightforward.
Lemma 7.6 Let X 2 MSp* with degX < 2(2n  1). Then
Xn X
2nX + (1)k2nk wl1. .w.lkfil1;...;lkX
k=1 4l1<...) MSp*. 
38
Proof of Lemma 7.5 Continued: Choose n so that 2(2n  1) > degK. Then
Xn X
Si = 2nKi+ (1)k2nk wl1. .w.lkfil1;...;lkKi
k=1 4l1<...) :
2n+1C  2nwp11wq32 = 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 w12M<3> . Then the equality (43) occurs in the
polynomial ring H0 w12M<3> . By the definition of in (41), wp11wq32 is
a linear combination of monomials in the canonical polynomial generators with
odd 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 in
the polynomial ring H0 w12M<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 to
the elements ff0(i) = fi2(fl(i))and prove Theorem 7.1(ii). Recall that we alrea*
*dy
know a projection of ff0(i) into the E3;*2(MSp1 ) from Proposition 6.4(d). In
E3;*2(MSp3 ), for 4 k 2s define
X
a(k4)(i) = wk42 p(it1; . .;.itk)x2;i1. .b.x2;it1.b.x.2;itk.x.2.;i2s
1t1<...