# Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles

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*O. García-Prada; P. B. Gothen; V. Muñoz*

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 the authors 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 the authors carry out a careful analysis of them by studying their variation with this parameter. Thus the authors 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 their Morse theoretic analysis, the authors 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.

#### Table of Contents

# Table of Contents

## Betti Numbers of the Moduli Space of Rank 3 Parabolic Higgs Bundles

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Parabolic Higgs bundles 514 free
- Chapter 3. Morse theory on the moduli space 1120
- Chapter 4. Parabolic triples 1928
- Chapter 5. Critical values and flips 2534
- Chapter 6. Parabolic triples with r[sub(1)] = 2 and r[sub(2)] = 1 3342
- Chapter 7. Critical submanifolds of type (1,1,1) 3948
- Chapter 8. Critical submanifolds of type (1,2) 4756
- Chapter 9. Critical submanifolds of type (2,1) 5160
- Chapter 10. Betti numbers of the moduli space of rank three parabolic bundles 5766
- Chapter 11. Betti numbers of the moduli space of rank three parabolic Higgs bundles 6372
- Chapter 12. The fixed determinant case 6776
- Bibliography 7988