There are two natural candidates for the group of relative CheegerSimons differential characters. The first directly extends the work of Cheeger and Simons and the second extends the description given by Hopkins and Singer of the Cheeger-Simons group as the homology of a certain cochain complex. We discuss both approaches and relate the two relative groups.

We discuss the homotopy surface category of a space which generalizes the 1+1dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Probenius algebras with additional structure. 855J. Knot Theory Ramifications 2000.09:855-864. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/05/15. For personal use only. Remark 4. In other words, morphisms of TX are morphisms of S with a labelling of each component from ^X. In light of this it is easy to see that if X is 2-connected then there is an equivalence of categories SX <-> S. Lemma 4. Let E and E' be surfaces. Then where fa is the product of all g^s and h^s forming the i'th connected component of E'E. Proof. Gluing of two connected components corresponds to multiplication in ^X, and for the i'th component of E'E we multiply all components of E' and E forming that component. • *We thank R. Levi and S. Willerton for correcting earlier sloppiness in this proof J. Knot Theory Ramifications 2000.09:855-864. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 02/05/15. For personal use only.

We study a variation of Turaev's homotopy quantum field theories using 2-categories of surfaces. We define the homotopy surface 2-category of a space X and define an S X -structure to be a monoidal 2-functor from this to the 2-category of idempotent-complete additive k-linear categories. We initiate the study of the algebraic structure arising from these functors. In particular we show that, under certain conditions, an S X -structure gives rise to a lax tortile π-category when the background space is an Eilenberg-Maclane space X = K(π, 1), and to a tortile category with lax π 2 X-action when the background space is simply-connected.

Abstract. In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk given by the second author at the Xth Oporto Meeting on Geometry, Topology and Physics. Homotopy Quantum Field TheoriesHomotopy quantum field theories were invented by Turaev [10], though the idea goes back to Segal's discussion of the possible geometry underlying elliptic cohomology [7]. Segal's construction is a generalisation of his definition of conformal field theory to the situation where one has a target or background space X. He assigns a topological vector space E(γ) to each collection of loops γ in a space X and a trace-class map E(σ) : E(γ) → E(γ ′ ) to each Riemann surface Σ equipped with a map σ : Σ → X agreeing with γ op ⊔ γ ′ on the boundary. The assignment is multiplicative in the sense that E(γ 1 ⊔ γ 2 ) is isomorphic to E(γ 1 ) ⊗ E(γ 2 ). The result can be thought of as a kind of infinite dimensional bundle on the free loop space of X, together with a generalised connection which describes "parallel transport" along surfaces.

The paper presents examples of complete singular toric varieties associated to the Niemeier lattices. The singularities and automorphisms of these varieties are seen to be closely related to the Golay codes.