The cohomology theory known as Tmf, for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to re...

Given a closed complex manifold X of even dimension, we develop a systematic (vertex) algebraic approach to study the rational orbifold cohomology rings H∗ orb(X /Sn) of the symmetric products. We present constructions and establish results on the rings H∗ orb(X /Sn) including two sets of ring generators, universality and stability, as well as connections with vertex operators and W algebras. T...

A hyperplane arrangement A is a finite collection of hyperplanes in some fixed (typically real or complex) vector space V. For simplicity, in this overview we work over the complex numbers C. There is a host of beautiful mathematics associated to the complement X = V A. Perhaps the first interesting result in the area was Arnol’d’s computation [2] of the cohomology ring of the complement of the...

Abstract. Given an n-tuple of positive real numbers (α1, . . . , αn), Konno [K2] defines the hyperpolygon space X(α), a hyperkähler analogue of the Kähler variety M(α) parametrizing polygons in R3 with edge lengths (α1, . . . , αn). The polygon space M(α) can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, X(...

We define the quantum exterior product ∧h and quantum exterior differential dh on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of dh. We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is ...

Partial quotients of moment-angle complexes are topological analogues smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring such partial quotient in terms torsion product involving corresponding Stanley-Reisner ring. We show that their gives correct cup if 2 is invertible chosen coefficient ring, but general. rectify this by de...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

0. Introduction. The main goal of this paper is to give a unified description for the structure of the small quantum cohomology rings for all projective homogeneous spaces SLn(C)/P , where P is a parabolic subgroup. The quantum cohomology ring of a smooth projective variety, or more generally of a symplectic manifoldX, has been introduced by string theorists (see [Va] and [W1]). Roughly speakin...

Let G be a compact connected Lie group with maximal torus T , and H a closed subgroup containing T . We compute the equivariant cohomology ring and the equivariant characteristic numbers of the homogeneous space G/H under the natural action of the maximal torus T . The computation is based on the localization theorems of Borel and of Atiyah-Bott-Berline-Vergne. Let G be a compact connected Lie ...

Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC∞ ∗ (Y ) of the cohomology ring of a closed 3manifold Y whose behavior mimics that of the Heegaard Floer homology HF∞(Y, s) for s a torsion spin structure. We derive from this a numerical invariant h(Y ) ∈ Z, and obtain upper and lower bounds on h(Y ). We describe the behavior of h(Y ) under connected sum, and dedu...