2 - Quaternionic elliptic objects and K3-cohomology

Summary

Abstract. We shall outline a research program which aims to provide a geometric approach to K3-cohomology. We define for this a quaternionic analogue of Segal's elliptic objects.

INTRODUCTION

The relation between one dimensional formal group laws and complex oriented generalised cohomology theories, which is a consequence of Quillen's theorem which relates Lazard's ring – the ring over which is defined the universal formal group law – and the coefficient ring of complex cobordism, has been a very fruitful source of research in algebraic topology. Among one dimensional formal group laws three have a direct geometric interpretation. They are obtained from the additive group, the multiplicative group and elliptic curves respectively by taking completion at the origin. The corresponding cohomology theories, Ordinary cohomology, K-theory and Elliptic cohomology also have geometric interpretations. These interpretations have extended their use, and helped to find applications of them, to other fields of mathematics and physics besides algebraic topology.

No other one dimensional commutative formal group law is associated to a group in this direct way. However M. Artin and B. Mazur showed that one can associate formal groups to certain algebraic varieties – among them K3 surfaces – in a way which generalises the case of elliptic curves. M. Hopkins used their work to define an associated generalised cohomology theory: K3-cohomology. The geometric origin of the formal group laws associated to K3 surfaces gives a reason to hope that there exists a geometric definition of K3 cohomology.