Dedicated to Michael Boardman on the occasion of his sixtieth birthday.Vorone gde-to Bog poslal kusoqek syra. I. A. KrylovAbstract. We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swisscheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the openclosed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad.

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We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n − 1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie the Mahowald inverse limit lim. We also give a multiplicative splitting of tE(n) after a suitable base extension.

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Abstract. Let E be a circle-equivariant complex-orientable cohomology theory. We show that the fixedpoint formula applied to the free loop space of a manifold X can be understood as a Riemann-Roch formula for the quotient of the formal group of E by a free cyclic subgroup. The quotient is not representable, but (locally at p) its p-torsion subgroup is, by a p-divisible group of height one greater than the formal group of E.