Pattern equivariant functions and cohomology

Pattern-Equivariant Cohomology with Integer Coefficients

We relate Kellendonk and Putnam's pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with coefficients in any Abelian group. It also provides an easy proof of Kellendonk and Putnam's original theorem relating PE cohomology to the Cech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.

Twisted noncommutative equivariant cohomology: Weil and Cartan models

We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel'd twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken; we show that one can implement a Drinfel'd twist of their models in order to take into account the noncommutativity of the spaces we are acting on.

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula.

Holomorphic Equivariant Cohomology via a Transversal Holomorphic Vector Field

In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when $v$ is a transversal holomorphic vector field on a compact complex manifold $X$ with a zero point set $Y$, the embedding $j:Y\to X$ induces a natural isomorphism between the holomorphic equivariant cohomology of $X$ via $v$ with coefficients in $ξ$ and the Dolbeault cohomology of $Y$ with coefficients in $ξ|_Y$, where $ξ\to X$ is a holomorphic vector bundle over $X$.

等变上同调中的局部化

Localisation in Equivariant Cohomology

本文探讨等变上同调的理论基础与应用，重点阐述Atiyah-Bott局部化定理如何简化带李群作用的辛流形上的积分，并展示其在理论物理和几何学等领域的广泛影响。