We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to normal crossing divisor can be described in terms delta-admissible chains. Roughly speaking, chain is simplicial semi-algebraic meeting faces properly. As an application, we Abel-Jacobi map for higher Chow cycles via example, describe Hodge realization polylog constructed by Bloch-Kriz map.

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Qu...

An equivariant bundle gerbe à la Meinrenken over a G-manifold M is known to be a special type of S-gerbe over the differentiable stack [M/G]. We prove that the natural morphism relating the Cartan and simplicial models of equivariant cohomology in degree 3 maps the Dixmier-Douady class of an equivariant bundle gerbe à la Meinrenken to the Behrend-Xu-Dixmier-Douady class of the corresponding S-g...

We show that the boundary map in periodic cyclic cohomology introduced by Cuntz and Quillen satisfies properties similar to the properties of the boundary map in simplicial homology. We also prove that the boundary map is compatible with the boundary map in algebraic and topological K-Theory, which leads to index theorems. As an application we obtain a new proof of the Connes-Moscovici index th...

We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization holds in equivariant Chow cohomology with integer coefficients. We also compute the equivariant Chow cohomology of toric prevarieties and general complex hyper...

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families Seiberg-Witten equations. This is formulated as a cohomology class on certain abstract simplicial complex consisting embedded surfaces 4-manifold. also give examples which admit positive scalar curvature metrics and for this does not vanish. non-vanishing result our provides new adjunction-type genus constraints confi...

The notion of geometric nerve of a 2-category (Street, [18]) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding sim-plicial maps. These facts allow us to prove a representation theorem of the general non abelian cohomolo...

We show that a finite regular cell complex with the intersection property is a Cohen-Macaulay space iff the top enriched cohomology module is the only nonvanishing one. We prove a comprehensive generalization of Balinski’s theorem on convex polytopes. Also we show that for any Cohen-Macaulay cell complex as above, although there is now generalization of the Stanley-Reisner ring of simplicial co...