Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.
We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.
Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homolog…
We study the moduli and determine a homotopy type of the space of all generalized Morse functions on d-manifolds for given d. This moduli space is closely connected to the moduli space of all Morse functions studied in the paper math.AT/0212321, and the classifying space of the corresponding cobordism category.
Let $(X,ω)$ be a closed symplectic manifold. A loop $ϕ: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $π: P_ϕ \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $π$, Lalonde, McDuff and Polterovich proved that if $ϕ$ lifts to the Hamiltonian group $\mathrm{Ham}(X,ω)$, then the rational cohomology of $P_ϕ$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_ϕ$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. …
We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_θ$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $θ: X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_θ$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_θ$ having classifying space homotopy equivalent to that of $C_θ$, the smaller such $D$ the better. We prove that in most …
We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be …
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.
We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalising a conjecture of Newstead.
We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(π_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute the (rational) equivariant cohomology ring of Hom(π_1(X),SU(2)) and use this to compute the ordinary cohomology groups of the quotient Hom(π_1(X),SU(2))/SU(2). A key property is that the conjugation action is equivariantly formal.
The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of the holonomy around a certain loop determines a Bott-Morse function on the moduli space which is perfect, meaning that the Morse inequalities are equalities. This leads to an attractive recursion for the Betti numbers of the moduli space, which agrees with the Harder-Narasimhan formula in the case of one puncture with holonomy -1.
For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -- namely a certain cohomology class in H^3(G^2;Z) -- that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G.
We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in parti…
We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.
This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the sufficient conditions for the pre-quantization of $M_G(Σ)$, the moduli space of flat $G$-bundles over a closed surface $Σ$. For a simply connected, compact, simple Lie group $G$, $M_G(Σ)$ is known to be pre-quantizable at integer levels. For non-simply connected $G$, however, integrality of the level is not sufficient for pre-quantization, and this thesis…
If $K$ is a compact Lie group and $g\geq 2$ an integer, the space $K^{2g}$ is endowed with the structure of a Hamiltonian space with a Lie group valued moment map $Φ$. Let $β$ be in the centre of $K$. The reduction $Φ^{-1}(β)/K$ is homeomorphic to a moduli space of flat connections. When $K$ is simply connected, a direct consequence of a recent paper of Bott, Tolman and Weitsman is to give a set of generators for the $K$-equivariant cohomology of $Φ^{-1}(β)$. Another method to construct classes in $H^*_K(Φ^{-1}(β))$ is by using the so called universal bundle. When the group is $\Sun$ and $β$ i…
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surf…
