The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 241-244).

Thesis (Ph.D.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/08). Includes bibliographical references.

In this thesis an extension of the classical intersection cohomology of Goresky and MacPherson, which we call multiperverse cohomology, is defined for a certain class of depth 1 controlled stratified spaces, which we call multicontrolled stratified spaces. These spaces are spaces with singularities -- this being their controlled structure -- with additional multicontrol data. Multiperverse cohomology is constructed using a cochain complex of tau-multiperverse forms, defined for each case tau of a parameter called a multiperversity. For the spaces that we consider these multiperversities, forming a lattice M, extend the general perversities of intersection cohomology. Multicontrolled stratified spaces generalise the structure of (the compactifications of) Q-rank 1 locally symmetric spaces. In this setting multiperverse cohomology generalises some of the aspects of the weighted cohomology of Harder, Goresky and MacPherson. We define two special cases of multicontrolled stratified spaces: the product-type case, and the flat-type case. In these cases we can calculate the multiperverse cohomology directly for cones and cylinders, this yielding the local calculation at a singular stratum of a multicontrolled space. Further, we obtain extensions of the usual Mayer-Vietoris sequences, as well as a partial Kunneth Theorem. Using the concept a dual multiperversity we are able to obtain a version of Poincare duality for multiperverse cohomology for both the flat-type and the product-type case. For this Poincare duality there exist self-dual multiperversities in certain cases, such as for non-Witt spaces, where there are no self-dual perversities. For certain cusps, called double-product cusps, which are naturally compactified to multicontrolled spaces, the multiperverse cohomology of the compactification of the double-product cusp for a certain multiperversity is equal to the L2-cohomology, analytically defined, for certain doubly-warped metrics.

The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.

The affine Grassmannian associated to a reductive group G is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[MO2] to write an explicit formula for this connection. In order to do this, we construct a recursive relation for the stable envelopes in the G = PSL_2 case and compute the first-order correction in the general case. The computation of the purely quantum part of the multiplication is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[BMO]. For the case of simply-laced G, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group G^\vee.

The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an R-module. The construction of the R-module is similar to the Alexander Wallace Spanier construction of the abelian group.

We propose a coherent analogue of the non-archimedean case of Venkatesh's conjecture on the cohomology of locally symmetric spaces for Shimura varieties coming from unitary similitude groups. Let G be a unitary similitude group with an indefinite signature at at least one archimedean place. Let Π be an automorphic cuspidal representation of G whose archimedean component Π∞ is a non-degenerate limit of discrete series and let 𝑊 be an automorphic vector bundle such that Π contributes to the coherent cohomology of its canonical extension. We produce a natural action of the derived Hecke algebra of Venketesh with torsion coefficients via cup product coming from étale covers and show that under some standard assumptions this action coincides with the conjectured action of a certain motivic cohomology group associated to the adjoint representation Ad𝜌π of the Galois representation attached to Π. We also prove that if the rank of G is greater than two, then the classes arising from the \'etale covers do not admit characteristic zero lifts, thereby showing that previous work of Harris-Venkatesh and Darmon-Harris-Rotger-Venkatesh is exceptional.

No description available.