Coarse cohomology with twisted coefficients

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

RIGIDIFIED TORSOR COCYCLES, HYPERCOVERINGS AND BUNDLE GERBES

We give a geometric interpretation of sheaf cohomology for higher degrees $n\geq 1$ in terms of torsors on the member of degree $d=n-1$ in hypercoverings of type $r=n-2$ , endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.

From Sheaf Cohomology to the Algebraic de Rham Theorem

This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, that there is an isomorphism H*(Xₐₙ,ℂ) ≃ H*X,Ω‎subscript alg superscript bullet) between the complex singular cohomology of Xan and the hypercohomology of the complex Ω‎subscript alg superscript bullet of sheaves of algebraic differential forms on X. The proof necessitates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology. The chapter then develops more machinery, mainly the Čech cohomology of a sheaf and the Čech cohomology of a complex of sheaves, as tools for computing hypercohomology. The chapter thus proves that the general case of Grothendieck's theorem is equivalent to the affine case.

The Hodge Theory of Maps

This chapter showcases two further lectures on the Hodge theory of maps, and they are mostly composed of exercises. The first lecture details a minimalist approach to sheaf cohomology, and then turns to the intersection cohomology complex, which is limited to the definition and calculation of the intersection complex Isubscript X of a variety of dimension d with one isolated singularity. Finally, this lecture discusses the Verdier duality. The second lecture sets out the Decomposition theorem, which is the deepest known fact concerning the homology of algebraic varieties. It then considers the relative hard Lefschetz and the hard Lefschetz for intersection cohomology groups.

Sheaf cohomology on network codings: maxflow-mincut theorem

<p>Surveying briefly a novel algebraic topological application sheaf theory into directed network coding<br />problems, we obtain the weak duality in multiple source scenario by<br />using the idea of modified graph. Furthermore,we establish the<br />maxflow-mincut theorem with network coding sheaves in the case of multiple source.</p>