Johannes Huebschmann

Cohomology of nilpotent groups of class 2

Abstract We compute the cohomology rings of a number of nilpotent groups of class 2 for appropriate coefficients, and we do some more sample calculations of various cohomology groups. The basic tool is a suitable small free resolution for an arbitrary nilpotent group of class 2 constructed in a previous paper.

The mod-p cohomology rings of metacyclic groups

Abstract Let e : 1→N→G→K→1 be an extension of a finite cyclic group N by a finite cyclic group K, and let R be a commutative ring with 1. Using homological perturbation theory, we construct a free resolution of R over the group ring RG of G in such a way that the resolution reflects the structure of G as an extension of N by K, and we use this resolution to compute the mod p cohomology ring of G for a prime p via the spectral sequence of the extension e.

Perturbation theory and free resolutions for nilpotent groups of class 2

Abstract Using homological perturbation theory, we construct a small free resolution for any nilpotent group G of class 2 which reflects the structure of G as a central extension of an abelian group by an abelian group.

Cohomology of metacyclic groups

group N by a finite cyclic group K . Using homological perturbation theory, we introduce the beginning of a free resolution of the integers Z over the group ring ZG of G in such a way that the resolution reflects the structure of G as an extension of N by K, and we use this resolution to compute the additive structure of the integral cohomology of G in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension e. The calculation is then completed by comparing the two bounds.

Kähler quantization and reduction

Abstract Exploiting a notion of Kähler structure on a stratified space introduced elsewhere we show that, in the Kähler case, reduction after quantization is equivalent to quantization after reduction in a certain weak sense. Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kähler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization is weakly equivalent to quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond as complex vector spaces (whether or not the natural inner products correspond is unknown whence the qualifier “weakly”) but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kähler spaces arise from representations of compact Lie groups and from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces. For illustration, we carry out Kähler quantization on various spaces of that kind including singular Fock spaces.

Origins and Breadth of the Theory of Higher Homotopies

Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940s. Prompted by the failure of the Alexander–Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of A n -spaces and A ∞ -spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through.Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy structures behave much better relative to homotopy than strict structures, and HPT enables one to exploit this observation in various concrete situations which, in particular, leads to the effective calculation of various invariants which are otherwise intractable.Higher homotopies abound but they are rarely recognized explicitly and their significance is hardly understood; at times, their appearance might at first glance even come as a surprise, for example in the Kodaira–Spencer approach to deformations of complex manifolds or in the theory of foliations.

Singularities and Poisson geometry of certain representation spaces

Certain representation spaces have been investigated by algebraic geometers as moduli spaces of holomorphic bundles over a Riemann surface. Such moduli spaces exhibit symplectic and Kahler structures as well as gauge theory interpretations. The purpose of this article is to elucidate the local structure of such a space, and the focus will be on the singularities. Among the tools will be the interconnection between the theory of algebraic and symplectic quotients and, furthermore, Poisson structures, a concept which has been known in mathematical physics for long and is currently of much interest in mathematics as well.

EXTENDED MODULI SPACES, THE KAN CONSTRUCTION, AND LATTICE GAUGE THEORY

Let Y be a CW-complex with asingle 0-cell, let K be its Kan group, a free simplicial group whose geometric realization is a model for the space ΩY of based loops on Y, and let G be a Lie group. By means of simplicial and cosimplicial techniques involving fundamental results of Kan’s and the standard W- and bar constructions, we obtain a weak G-equivariant homotopy equivalence from the geometric realization |Hom(K,G)| of the cosimplicial manifold Hom(K,G) of homomorphisms from K to G to the space Mapo(Y,BG) of based maps from Y to the classifying space BG of G where G acts on BG by conjugation. Thereafter we carry out an explicit purely finite dimensional construction of generators of the equivariant cohomology of the geometric realization of Hom(K,G) and hence of the space Mapo(Y,BG) of based maps from Y to the classifying space BG of G. For a smooth manifold Y, this may be viewed as a rigorous approach to lattice gauge theory, and we show that it then yields, (i) when dim(Y)=2, equivariant de Rham representatives of generators of the equivariant cohomology of twisted representation spaces of the fundamental group of a closed surface including generators for moduli spaces of semi-stable holomorphic vector bundles on complex curves so that, in particular, when G is compact, the known structure of stratified symplectic space on the twisted representation spaces results and (ii) when dim(Y)=3, equivariant cohomology generators including a rigorous combinatorial description of the Chern–Simons function for a closed 3-manifold. The latter is illustrated by a calculation of the Chern–Simons invariants for flat SU(2)-connections over 3-dimensional lens spaces.

The singularities of Yang-Mills connections for bundles on a surface

Let Σ be a closed surface, G a compact Lie group, not necessarily connected, with Lie algebra g, endowed with an adjoint action invariant scalar product, let ξ: P → Σ be a principal G-bundle, and pick a Riemannian metric and orientation on Σ, so that the corresponding Yang-Mills equations

Higher homotopies and Maurer-Cartan algebras: Quasi-Lie-Rinehart, Gerstenhaber, and Batalin-Vilkovisky algebras

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the Maurer-Cartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the “space of leaves” and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including the Hodge-de Rham spectral sequence.