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Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of space-time can be naturally…

Let $\pi: P\to B$ be a locally trivial fiber bundle over a connected CW complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then $\pi$ is said to be a symplectic fiber bundle if its structural group is the group of…

Let f: M -> N be an even codimensional immersion between smooth manifolds. We derive an explicit formula for the Pontrjagin numbers and signature of the multiple point manifolds in terms of singular cohomology of M and N, the maps induced…

We introduce extended toric Deligne-Mumford stacks. We use an extended toric Deligne-Mumford stack to get the toric stack bundle and compute its orbifold Chow ring. Finally we generalize one result of Borisov, Chen and Smith so that the…

We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the…

This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. Cohomologies of wrap groups…

We conjecture and prove closed-form index expressions for the cohomology dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for all compact toric surfaces we provide a simple algorithm which allows expression of any…

In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a complete description of this Batalin-Vilkovisky algebra for…

We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial…

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the homology of the unit tangent bundle of the surface. This stable twisted cohomology…

We determine the integral cohomology ring of the homogeneous space E_8/T^1E_7 by the Borel presentation and a method due to Toda. Then using the Gysin exact sequence associated with the circle bundle S^1 -> E_8/E_7 -> E_8/T^1E_7, we also…

In this review, novel non-standard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational…

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For…

We introduce Deligne cohomology that classifies U(1) fibre bundles over 3-manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (non-perturbative) computations in U(1)…

This article uses basic homological methods for evaluating examples of compactly supported cohomology groups of line bundles over projective curve.

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…

We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line. We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their…

We study moduli stacks of principal $\Bbb C^*$-bundles over nodal complex algebraic curves and determine their rational cohomology algebras in terms of Chern classes.

The join construction produces a third Sasaki manifold from two others, and we investigate the algebraic topology of the joins of circle bundles over surfaces of positive genus with weighted three-spheres. Topologically, such a join has the…

We present a finite-dimensional and smooth formulation of string structures on spin bundles. It uses trivializations of the Chern-Simons 2-gerbe associated to this bundle. Our formulation is particularly suitable to deal with string…