Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userClassifying spaces for 1-truncated compact Lie groups
82
0
0.0
(
0
)
Authors
Charles Rezk
Ask ChatGPT about the research
No Arabic abstract
A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.
rate research
Read More
The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $langle -rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $langle L_{0}rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}oplus L_{1}$, concentrated in degrees 0 and 1. We establish that this category is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $mathcal{C}(L)$ associated to $L$. We prove that $mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(langle Lrangle, langle L_{0}rangle)$. Our main result is the identification of $langle Lrangle$ with the classifying space of $mathcal{C}(L)$.
Among the generalizations of Serres theorem on the homotopy groups of a finite complex we isolate the one proposed by Dwyer and Wilkerson. Even though the spaces they consider must be 2-connected, we show that it can be used to both recover known results and obtain new theorems about p-completed classifying spaces.
In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${rm Comm}(G)$ and ${rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.
For n>2, we prove the mod 2 cohomology of the finite Chevalley group Spin_n(F_q) is isomorphic to that of the classifying space of the loop group of the spin group Spin(n).
In this article we study the homology of spaces ${rm Hom}(mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${rm Hom}(F_n/Gamma_n^m,G)$, where the subgroups $Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
