We determine the Fukaya-Floer (co)homology groups of the three-manifold Y = Σ×S 1 , where Σ is a Riemann surface of genus g ≥ 1. These are of two kinds. For the 1-cycle S 1 ⊂ Y , we compute the Fukaya-Floer cohomology HF F * (Y, S 1 ) and its ring structure, which is a sort of deformation of the Floer cohomology HF * (Y ). On the other hand, for 1-cycles δ ⊂ Σ ⊂ Y , we determine the Fukaya-Floer homology HF F * (Y, δ) and its HF * (Y )-module structure.We give the following applications:• We show that every four-manifold with b + > 1 is of finite type.• Four-manifolds which arise as connected sums along surfaces of four-manifolds with b 1 = 0 are of simple type and we give constraints on their basic classes.• We find the invariants of the product of two Riemann surfaces both of genus greater or equal than one. * Supported by a grant from Ministerio de Educación y Cultura of Spain A(X) with that of Kronheimer and Mrowka [16], as we do not consider 3-homology classes (this is done in this way since the techniques here contained are not well suited to deal with these classes).We say that X is of w-simple type when D w X ((x 2 − 4)z) = 0, for any z ∈ A(X). If X has b 1 = 0 and it is of w-simple type, then it is of w ′ -simple type, for any other w ′ and it is said to be of simple type for brevity. Analogously, we say that X is of w-finite type when there is some n ≥ 0 such that D w X ((x 2 − 4) n z) = 0, for any z ∈ A(X). The order is the minimum such n, so order 1 means simple type and order 0 means that the Donaldson invariants are identically zero. X is of finite type if it is of w-finite type for any w. For completeness we introduce the notion of X being of w-strong simple type when D w X ((x 2 − 4)z) = 0, for any z ∈ A(X) and D w X (γz) = 0, for any γ ∈ H 1 (X) and any z ∈ A(X). This condition is the right one for extending the concept of simple type in the case b 1 = 0 to the case b 1 > 0. It gives the same structure theorem for the invariants presented in [16]. Introducing 3-homology classes, we would have (potentially) different definitions, but we shall not deal here with that issue. Also the order of w-finite type of X does not depend on w (see [25]).Let Σ = Σ g be a Riemann surface of genus g ≥ 1 and consider the three-manifold Y = Σ × S 1 .In [23] we computed the ring structure of the (instanton) Floer (co)homology of Y , together with the SO(3)-bundle with w 2 = P.D.[S 1 ]. This gadget encondes all the relations R ∈ A(Σ) satisfied by all 4-manifolds X containing an embedded surface Σ, representing an odd homology class and with Σ 2 = 0. More accurately, for such X, D w X (Rz) = 0, for any z ∈ A(Σ ⊥ ) and w ∈ H 2 (X; Z) with w · Σ ≡ 1 (mod 2). This is so since we have a decomposition X = X 1 ∪ Y A, where A is a tubular neighbourhood of Σ, and we can consider R ∈ A(A) and z ∈ A(X 1 ). Then the (relative) Donaldson invariants for A corresponding to R are already vanishing.In order to drop the condition z ∈ A(Σ ⊥ ), the useful space to work in is no longer the Floer homology, but the extension developed by F...