Reducible connections and cup products 7.1 The maps D 1 , D 2 7.2 Manifolds with b + = 0, 1 1 6 7.2.1 The case b + = 1 7.2.2 The case b + = 0 7.3 The cup product 7.3.1 Algebro-topological interpretation 7.3.2 An alternative description 7.3.3 The reducible connection 7.3.4 Equivariant theory 7.3.5 Limitations of existing theory 7.4 Connected sums 7.4.1 Surgery and instanton invariants 201 7.4.2 The Hom F-complex and connected sums 8 Further directions 8.1 Floer homology for other 3-manifolds Contents vii 8.2 The blow-up formula Bibliography Index

Abstract. We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle, and are related to symplectic geometry and Seiberg-Witten theory.We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.

We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products.

No abstract

For any closed oriented surface g of genus g ¿ 3, we prove the existence of foliated g -bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ! on the ÿber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism Flux : Symp g → H 1 ( g ; R) which is an extension of the ux homomorphism Flux : Symp 0 g → H 1 ( g ; R) from the identity component Symp 0 g to the whole group Symp g of symplectomorphisms of g with respect to the symplectic form !. ?