Suplico a vuesa merced, señor don Quijote, que mire bien y especule con cien ojos lo que hay allá dentro: quizá habrá cosas que las ponga yo en el libro de mis Transformaciones (El ingenioso hidalgo don Quijote de la Mancha, Book 2, Chapter XXII) I beg you, don Quixote sir: look carefully, inspect with a hundred eyes what you see down there. Who knows, maybe you will find something that I can put in my book on Transformations.

Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex surface into SL(2, C), for the case of small genus g, and allowing the holonomy around a fixed point to be any matrix of SL(2, C), that is Id , − Id , diagonalisable, or of either of the two Jordan types.For this, we introduce a new geometric technique, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

We give a presentation for the Floer cohomology ring HF * (Σ × S 1), where Σ is a Riemann surface of genus g ≥ 1, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of flat SO(3)connections of odd degree over Σ. We study the spectrum of the action of H * (Σ) on HF * (Σ × S 1) and prove a physical assumption made in [1].

We answer in the affirmative the question posed by Babenko and Taimanov [3] on the existence of nonformal, simply connected, compact symplectic manifolds of dimension 8.