Singularities and Higher Torsion in
Symplectic Cobordism I
Boris I. Botvinnik and Stanley O. Kochman
Abstract
In this paper we construct higher two-torsion 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 Adams-Novikov spectral sequence in termsof 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
M Sp and classifies up to cob ordism the ring of smooth manifolds with an Sp-
structure on their stable normal bundles. Although MSp only has two-torsion,
its ring structure is far more complicated than any of the other cobordism rings
M G for the classical Lie groups G = O, SO, Spin, U, SU which have been
completely computed. Over the past thirty years, these cobordism rings M G
have had a major impact on differential topology and homotopy theory. On the
other hand, if the complexity of the ring M Sp were understood,then symplectic
cob ordism theory M Sp () would have the potential to become a powerfultool
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 cantry 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 MS p which says that all of its torsionelements are
nilpotent.!Another basic structural question is:
!
!! Do there exist elements of order 2kin the ring MSp for all k 1?(1)
Note!that!the corresponding structural property is well-known for all otherclas*
*si-
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.