Let \Omega^3(SU(n)) be the Lie group of based mappings from S^3 to SU(n). We construct a Lie group extension of \Omega^3(SU(n)) for n\geq 3 by the abelian group \exp 2\pi i {\cal A}_3^{\ast}, where ...{\cal A}_3^{\ast} is the affine dual of the space of SU(n)-connections on S^3. J. Mickelsson in 1987 constructed a similar Lie group extension. In this article we give several improvement of his results, especially we give a precise description of the extension of those components that are not the identity component. We also correct several argument about the extension of \Omega^3(SU(2)) which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of \Omega^3(SU(2)) reduces to the extension by Z_2 is correct. Then we shall investigate the adjoint representation of the Lie group extension of \Omega^3(SU(n)) for n\geq 3.
We prove that expanding endomorphisms on arbitrary tori are 1-sided Bernoulli with respect to their corresponding measure of maximal entropy and are thus, measurably, as far from invertible as ...possible. This applies in particular to expanding linear toral endomorphisms and their smooth perturbations. Then we study toral extensions of expanding toral endomorphisms, in particular probabilistic systems on skew products, and prove that under certain not too restrictive conditions on the extension cocycle, these skew products are 1-sided Bernoulli too. We also give a large class of examples of group extensions of expanding maps in higher dimensions, for which we check the conditions on the extension cocycle.
A central extension of the form E:0→V→G→W→0, where V and W are elementary abelian 2-groups, is called Bockstein closed if the components qi∈H⁎(W,F2) of the extension class of E generate an ideal ...which is closed under the Bockstein operator. In this paper, we study the cohomology ring of G when E is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of G has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg–Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps Q:W→V associated to the extensions E of the above form.
We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology
HS
2
(
G
,
A
). We also give conditions for the map
HS
n
(
G
,
A
) →
H
n
(
G
,
A
) to ...be injective.
In cite{Demp2} Dempwolff proved the existence of a group of theform $2^{5}{^{cdot}}GL(5,2)$ (a non split extension of theelementary abelian group $2^{5}$ by the general linear group$GL(5,2)$). This ...group is the second largest maximal subgroup of thesporadic Thompson simple group $mathrm{Th}.$ In this paper wecalculate the Fischer matrices of Dempwolff group $overline{G} =2^{5}{^{cdot}}GL(5,2).$ The theory of projective characters isinvolved and we have computed the Schur multiplier together with aprojective character table of an inertia factor group. The fullcharacter table of $overline{G}$ is then can be calculated easily.
PDF
