Realising formal groups

We show that a large class of formal groups can be realised functorially by even periodic ring spectra. The main advance is in the construction of morphisms, not of objects.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-8.abs.htm

Gross-Hopkins duality

We give a new and simpler proof of a result of Hopkins and Gross relating Brown-Comenetz duality to Spanier-Whitehead duality in the K(n)-local stable homotopy category

K(n)-local duality for finite groups and groupoids

We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the usual inner product on the representation ring when n=1, and to the Hopkins-Kuhn-Ravenel generalised character theory. We show that LG is a Frobenius algebra object in the K(n)-local stable category, and we recall the connection between Frobenius algebras and topological quantum field theories to help analyse this structure. In many places we find it convenient to use groupoids rather than groups, and to assist with this we include a detailed treatment of the homotopy theory of groupoids. We also explain some striking formal similarities between our duality and Atiyah-Poincare duality for manifolds.Comment: 37 pages; one included postscript figur

Morava E-theory of symmetric groups

We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups.Comment: To appear in Topolog

Common subbundles and intersections of divisors

Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that V_0\cap V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-42.abs.htm

Complex cobordism of involutions

We give a simple and explicit presentation of the Z/2-equivariant complex cobordism ring.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper11.abs.htm