Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes

2-Gerbes bound by complexes of gr-stacks, and cohomology

We define 2-gerbes bound by complexes of braided group-like stacks. We prove a classification result in terms of hypercohomology groups with values in abelian crossed squares and cones of morphisms of complexes of length 3. We give an application to the geometric construction of certain elements in Hermitian Deligne cohomology groups.

Inertia and delocalized twisted cohomology

We show that the inertia stack of a topological stack is again a topological stack. We further observe that the inertia stack of an orbispace is again an orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a flat line bundle over its inertia stack. Via sheaf theory over topological stacks it gives rise to the twisted delocalized cohomology of the orbispace. With these results and constructions we generalize concepts, which are well-known in the smooth framework, to the topological case. In the smooth case we show, that our sheaf-theoretic definition of twisted delocalize…

Real bundle gerbes, orientifolds and twisted KR-homology

We consider Real bundle gerbes on manifolds equipped with an involution and prove that they are classified by their Real Dixmier-Douady class in Grothendieck's equivariant sheaf cohomology. We show that the Grothendieck group of Real bundle gerbe modules is isomorphic to twisted KR-theory for a torsion Real Dixmier-Douady class. Using these modules as building blocks, we introduce geometric cycles for twisted KR-homology and prove that they generate a real-oriented generalised homology theory dual to twisted KR-theory for Real closed manifolds, and more generally for Real finite CW-complexes, …

Periodic twisted cohomology and T-duality

The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in pe…

A1-invariants in Galois cohomology and a claim of Morel

We establish a variant of the splitting principle of Garibaldi-Merkurjev-Serre for invariants taking values in a strictly homotopy invariant sheaf. As an application, we prove the folklore result of Morel that pi_0 of the motivic localization of the group completion of the stack of finite étale schemes is given by the sheaf of unramified Grothendieck-Witt groups.

代数叠K-理论上的Lambda环结构

Lambda-ring structures on the K-theory of algebraic stacks

该研究构建光滑代数叠K-理论的lambda、gamma运算，证明其高阶K-理论是预lambda环，满足凝聚层为向量丛商条件时为lambda环，进而定义相关运算与上同调，还得到特定情形下绝对上同调与等变高阶Chow群的比较结果。