Homotopy groups of moduli spaces of representations

We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott-Morse theory on the corresponding moduli spaces.     

Homological Algebra of Twisted Quiver Bundles

Several important cases of vector bundles with extra structure(such as Higgs bundles and triples) may be regarded as examplesof twisted representations of a finite quiver in the categoryof sheaves of modules on a variety/manifold/ringed space. Itis shown that the category of such representations is an abeliancategory with enough injectives by the construction of an explicitinjective resolution. Using this explicit resolution, a longexact sequence is found that computes the Ext groups in thisnew category in terms of the Ext groups in the old category.The quiver formulation is directly reflected in the form ofthe long exact sequence. It is also shown that under suitablecircumstances, the Ext groups are isomorphic to certain hypercohomologygroups.     

Exotic components of SO(p,q) surface group representations,and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $\mathrm{SO}(p,q)$ with $2<p<q$. These groups lie outside formerly know classes of groups associated with exotic components.     

Rank two quadratic pairs and surface group representations

Let X be a compact Riemann surface. A quadratic pair on X consists of a holomorphic vector bundle with a quadratic form which takes values in a fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the SO₀(2, 3)-character variety of X.     

From singularity theory to finiteness of Walrasian equilibria

The main result of this paper states that there exists a residual subset of the set of critical economies whose associated equilibria are finite in number. We also show that this subset does not contain any open set and therefore the result is the best possible for our choice of topology (compact-open topology). The proof rests on results and concepts from singularity theory.     

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant Members of VBAC (Vector Bundles on Algebraic Curves). Second and Third authors partially supported by Ministerio de Educación y Ciencia and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP2002-0017 (Spain)/E–30/03 (Portugal). First and Second authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project MTM2004-07090-C03-01. Third author partially supported by the Centro de Matemática da Universidade do Porto and the project POCTI/MAT/58549/2004, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. The second author visited the IHES with the partial support of the European Commission through its 6th Framework Programme “Structuring the European Research Area” and the Contract No. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action