Abstract. To a topological group G, we assign a naive G-spectrum D G , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that D G detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if D G is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of finitely dominated spaces, the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H -space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H . In an appendix, we identify the homotopy type of D G for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Classification (1991): 55P91, 55N91, 55P42, 57P10, 55P25, 20J05, 18G15. Mathematics Subject

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [17], but our proofs are fundamentally different. Errata Minor errors were corrected on page 967 (18 February 2008).

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Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let A ζ denote the C * -algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA ζ , the group of unitaries of A ζ . The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

This article studies Markovian stochastic motion of a particle on a graph with finite number of nodes and periodically time-dependent transition rates that satisfy the detailed balance condition at any time. We show that under general conditions, the currents in the system on average become quantized or fractionally quantized for adiabatic driving at sufficiently low temperature. We develop the quantitative theory of this quantization and interpret it in terms of topological invariants. By implementing the celebrated Kirchhoff theorem we derive a general and explicit formula for the average generated current that plays a role of an efficient tool for treating the current quantization effects.