Tate Cohomology of Framed Circle Actions as a Heisenberg Group

The complex Mahowald pro-spectrum CP∞ −∞ is not, as might seem at first sight, self-dual; rather, its natural dual is its own double suspension. This assertion makes better sense as a claim about the Tate cohomology spectrum tTS defined by circle actions on framed manifolds. A subtle twist in some duality properties of infinite-dimensional projective space results, which has consequences [via work of Madsen and Tillmann] for the Virasoro symmetries [discovered by Witten and Kontsevich] of the stable cohomology of the Riemann moduli space. An action of a compact Lie group G on a space is free if every point has trivial isotropy group. The class of such free G-spaces is mapped to the class of unrestricted G-spaces, by a forgetful functor which defines a homomorphism from the (geometric) bordism theory of free G-manifolds, perhaps with extra structure, to the geometric bordism theory of unrestricted G-manifolds. This note is written in terms of actions of the circle group T on framed manifolds, and is hence concerned with T-equivariant stable homotopy theory, but I have tried to use the more accessible geometric language as much as possible. [The stabilization construction of tom Dieck allows us to pass nicely from the language of geometry to the language of spectra.] However, a reader interested in equivariant homotopy theory itself may be disappointed to learn that the real focus of this note is a certain bilinear form on tTS, nondegenerate (and hence symplectic) modulo torsion, and its relevance to work of Madsen and Tillmann on the moduli space of Riemann surfaces. My hope is to understand the geometry behind this pairing, so although the final results concern the homology tTS ∧HZ = tTHZ (which might also be described as periodic cyclic cohomology) the discussion begins with Tate cohomology of complex (as opposed to framed) cobordism. The first section below is concerned with the geometric construction of the bilinear form. The second summarizes basic properties of CP −∞ and discusses some of the work of Madsen and Tillmann. The third attempts to explain how this is related to Kontsevich-Witten theory. I owe Tillmann and Madsen thanks for many helpful and patient conversations about their thinking, but the mistakes and excesses in this note are my own responsibility. Date: 20 May 2001. 1991 Mathematics Subject Classification. 19Dxx, 57Rxx, 83Cxx. The author was supported in part by the NSF. 1