اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe third cohomology group classifies crossed module extensions
134
0
0.0
(
0
)
تأليف
Sebastian Thomas
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We give an elementary proof of the well-known fact that the third cohomology group H^3(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext^2(G, M) of equivalence classes of crossed module extensions of G with M.
قيم البحث
اقرأ أيضاً
For a number ring $mathcal{O}$, Borel and Serre proved that $text{SL}_n(mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $text{GL}_n(mathcal{O})$ is a virtual duality group. In contrast to
$text{SL}_n(mathcal{O})$, we prove that the dualizing module of $text{GL}_n(mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $text{GL}_n(mathcal{O})$ in its virtual cohomological dimension.
We construct a finitely generated group that does not satisfy the generalized Burghelea conjecture.
The main aim of this paper is the construction of a smooth (sometimes called differential) extension hat{MU} of the cohomology theory complex cobordism MU, using cycles for hat{MU}(M) which are essentially proper maps Wto M with a fixed U(n)-structur
e and U(n)-connection on the (stable) normal bundle of Wto M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that hat{R}(M):=hat{MU}(M)otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.
For a central perfect extension of groups $A rightarrowtail G twoheadrightarrow Q$, we study the maps $H_3(A,mathbb{Z}) to H_3(G, mathbb{Z})$ and $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$ provided that $Asubseteq G$. First we show that the image of $H
_3(A, mathbb{Z})to H_3(G, mathbb{Z})/rho_ast(Aotimes_mathbb{Z} H_2(G, mathbb{Z}))$ is $2$-torsion where $rho: A times G to G$ is the usual product map. When $BQ^+$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G, mathbb{Z}) to H_3(Q, mathbb{Z})$.
Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of monoidal functors.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
