Tate Cohomology of Circle Actions as a Heisenberg Group

We study the Madsen-Tillmann spectrum CP∞ −1 as a quotient of the Mahowald pro-object CP∞ −∞ , which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten. 1. Tate cohomology of circle actions 1.1 If E is a geometric bordism theory (such as integral homology), its Tate cohomology t∗TE can be constructed by tom Dieck stabilization from the geometric theory of E-manifolds with T-action, the action required to be free on the boundary [8]. If E is multiplicative, so is t∗TE; there is a cofibration sequence · · · → EBT+ → t∗TE → E−∗−2BT+ → · · · in which the boundary map sends a T-manifold with boundary to the quotient of its boundary by the (free) T-action. When E is complex-oriented [eg MU or HZ] this sequence reduces to a short exact sequence which identifies t∗TE with the Laurent series ring E((e)) obtained by inverting the Euler class in E(BT) = E[[e]], and the boundary map can be calculated as a formal residue; more precisely, the formal Laurent series f maps to the residue of fd logE at e = 0, where d logE is the invariant differential of the formal group law of E. When E is not complexorientable, tTE can behave very differently [4], as the Segal conjecture shows: up to a profinite completion, tTS 0 ∼ S ∨ S ∏ BT/C where C runs through proper subgroups of T. There is a related but simpler theory τ TE defined by manifolds with free T-action on the boundary alone, which fits in an exact sequence · · · → E → τ TE → E−∗−2BT+ → · · · ; ignoring the interior T-action defines a truncation t∗TE → τ TE . Date: 15 September 2001. 1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx. The author was supported in part by the NSF.