Continuous cohomology

In mathematics , continuous cohomology is a variant of group cohomology , but only continuous co-cycles are allowed when defining it. It is often more amenable to computation than group cohomology and is therefore used in various areas of representation theory and global analysis .

definition

Let it be a topological group . The continuous cohomology is the cohomology of the complex with ${\displaystyle G}$ ${\displaystyle H_{c}^{*}(G)}$ ${\displaystyle (C_{c}^{n}(G),d^{n})}$

${\displaystyle C_{c}^{n}=\left\{f\colon G^{n+1}\to \mathbb {R} \ {\mbox{stetig}}\ \mid f(\sigma \sigma _{1},\ldots ,\sigma \sigma _{n+1})=\sigma \cdot f(\sigma _{1},\ldots ,\sigma _{n+1})\ \forall \ \sigma \in G\right\}}$

and

${\displaystyle (d^{n-1}f)(\sigma _{1},\ldots ,\sigma _{n+1})=\sum _{i=1}^{n+1}(-1)^{i}f(\sigma _{1},\ldots ,{\hat {\sigma }}_{i},\ldots ,\sigma _{n+1}).}$

The elements of this complex are called homogeneous continuous coquettes .

Examples

The continuous cohomology of semisimple Lie groups can be calculated using van Est's theorem . For example is

${\displaystyle H_{c}^{i}(SO(n,1))={\begin{cases}\mathbb {R} ,&{\text{für }}i=0\\0,&{\text{sonst}}\end{cases}}}$

and

${\displaystyle H_{c}^{*}(SL(n,\mathbb {C} ))=\Lambda _{\mathbb {Z} }(b_{3},b_{5},\ldots ,b_{2n-1}),}$

where denotes the i-th Borel class . ${\displaystyle b_{i}\in H_{c}^{i}(SL(n,\mathbb {C} ))}$

literature

• Armand Borel , Nolan Wallach : Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition. Mathematical Surveys and Monographs, 67. American Mathematical Society, Providence, RI, 2000. ISBN 0-8218-0851-6