Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Given a compact connected Lie group G with an S 1 -module structure and a maximal compact torus T of G S 1 , we define twisted Weyl group W ( G , S 1 , T ) of G associated to S 1 -module and show that two elements of T are δ -conjugate if and only if they are in one W ( G , S 1 , T ) -orbit. Based on this, we prove that the natural map W ( G , S 1 , T ) \ H 1 ( S 1 , T ) → H 1 ( S 1 , G ) is bijective, which reduces the calculation for the nonabelian cohomology H 1 ( S 1 , G ) .

Second cohomotopy and nonabelian cohomology

The main difficulty in the theory of non-abelian cohomology is that for cosimplicial groups only zero-th and first dimensional cohomotopy are known. In this article we introduce a new class of cosimplicial groups, called centralised cosimplicial groups, for which we are able to define a second cohomotopy, with all expected properties. The main examples of such cosimplicial groups come from 2-categories.

On cocycle bitorsors and gerbes over a Grothendieck topos

The aim of this paper is to give a new description of Giraud's nonabelian cohomology setH2(L), [7], defined for a bandLof a Grothendieck toposEas the set ofL-equivalence classes ofL-gerbes overE. Our description is similar to that forH1by Čech cohomology, but with cocycles taken from the stack of bitorsors overE. We first continue to study the construction of gerbes by cocycle bitorsors or bouquets [13], originally given in [7] and [4]; Duskin [4, 5] showed that bouquetsBandB′ ofEgive rise to equivalent gerbes if and only if there exist essential equivalencesB″ →BandB″ →B′ for another bouquetB″.