Cohomology of U(2,1) representation varieties of surface groups

In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surjectivity theorem in the non-fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles.Comment: 34 page

Morse theory and stable pairs

We study the Morse theory of the Yang-Mills-Higgs functional on the space of pairs (A; \Phi), where A is a unitary connection on a rank 2 hermitian vector bundle over a compact Riemann surface, and \Phi is a holomorphic section of (E; d_A). We prove that a certain explicitly defined substratification of the Morse stratification is perfect in the sense of G-equivariant cohomology, where G denotes the unitary gauge group. As a consequence, Kirwan surjectivity holds for pairs. It also follows that the twist embedding into higher degree induces a surjection on equivariant cohomology. This may be interpreted as a rank 2 version of the analogous statement for symmetric products of Riemann surfaces. Finally, we compute the G-equivariant Poincare polynomial of the space of semistable pairs. In particular, we recover an earlier result of Thaddeus. The analysis provides an interpretation of the Thaddeus flips in terms of a variation of Morse functions

Cohomology of U(2,1) representation varieties of surface groups

In this paper, we use the Morse theory of the Yang–Mills–Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2, 1)‐ and SU(2, 1)‐Higgs bundles with fixed Toledo invariant. In the non‐coprime case, this gives new results about the topology of the U(2, 1) and SU(2, 1) character varieties of surface groups. The main results are a calculation of the equivariant Poincaré polynomials, a Kirwan surjectivity theorem in the non‐fixed determinant case, and a description of the action of the Torelli group on the equivariant cohomology of the character variety. This builds on earlier work for stable pairs and rank 2 Higgs bundles

Microbial sulfate reduction and metal attenuation in pH 4 acid mine water

Sediments recovered from the flooded mine workings of the Penn Mine, a Cu-Zn mine abandoned since the early 1960s, were cultured for anaerobic bacteria over a range of pH (4.0 to 7.5). The molecular biology of sediments and cultures was studied to determine whether sulfate-reducing bacteria (SRB) were active in moderately acidic conditions present in the underground mine workings. Here we document multiple, independent analyses and show evidence that sulfate reduction and associated metal attenuation are occurring in the pH-4 mine environment. Waterchemistry analyses of the mine water reveal: (1) preferential complexation and precipitation by H2S of Cu and Cd, relative to Zn; (2) stable isotope ratios of 34S/32S and 18O/16O in dissolved SO4 that are 2–3 ‰ heavier in the mine water, relative to those in surface waters; (3) reduction/oxidation conditions and dissolved gas concentrations consistent with conditions to support anaerobic processes such as sulfate reduction. Scanning electron microscope (SEM) analyses of sediment show 1.5-micrometer, spherical ZnS precipitates. Phospholipid fatty acid (PLFA) and denaturing gradient gel electrophoresis (DGGE) analyses of Penn Mine sediment show a high biomass level with a moderately diverse community structure composed primarily of iron- and sulfate-reducing bacteria. Cultures of sediment from the mine produced dissolved sulfide at pH values near 7 and near 4, forming precipitates of either iron sulfide or elemental sulfur. DGGE coupled with sequence and phylogenetic analysis of 16S rDNA gene segments showed populations of Desulfosporosinus and Desulfitobacterium in Penn Mine sediment and laboratory cultures

Cohomology of SL(2,C) Character Varieties of Surface Groups and the Action of the Torelli Group

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists of the even alternating products of degree two Prym representations, so that the kernel of the action is precisely the Prym-Torelli group. We compute the Betti numbers of the ordinary cohomology of the moduli space of flat SL(2,C) connections. Using results of Cappell-Lee-Miller we show that the Prym-Torelli group, which acts trivially on equivariant cohomology, acts non-trivially on ordinary cohomology