On the 2-compact group DI(4)

@article{Notbohm2003OnT2,
  title={On the 2-compact group DI(4)},
  author={Dietrich Notbohm},
  journal={Crelle's Journal},
  year={2003},
  volume={2003},
  pages={163-185}
}
  • D. Notbohm
  • Published 2003
  • Mathematics
  • Crelle's Journal
Besides the simple connected compact Lie groups there exists one further simple connected 2-compact group, constructed by Dwyer and Wilkerson, the group DI(4). The mod-2 cohomology of the associated classifying space BDI(4) realizes the rank 4 mod-2 Dickson invariants. We show that mod-2 cohomology determines the homotopy type of the space BDI(4) and that the maximal torus normalizer determines the isomorphism type of DI(4) as 2-compact group. We also calculate the set of homotopy classes of… Expand
N-determined 2-compact groups II
This is the first part of a paper that classifies 2-compact groups. In this first part we formulate a general classification scheme for 2-compact groups in terms of their maximal torus normalizerExpand
Mod 2 cohomology of 2-local finite groups of low rank
We determine the mod 2 cohomology over the Steenrod algebra of the classifying space of a free loop group LG for G=Spin(7), Spin(8), Spin(9), F_4, and DI(4). Then we show that it is isomorphic asExpand
The classification of 2-compact groups
We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compactExpand
The Cohomology of Exotic 2-local Finite Groups
There exist spaces BSol(q) which are the classifying spaces of a family of 2-local finite groups based on certain fusion system over the Sylow 2-subgroups of Spin7(q). In this paper we calculate theExpand
A faithful complex representation of the 2-compact group DI(4)
p-Compact groups, introduced by Dwyer and Wilkerson [DW2], are homotopy theoretic analougues of compact Lie groups (see also [N2] and [M]). The present paper contains a construction of a faithfulExpand
Loop space homology associated with the mod 2 Dickson invariants
Abstract. The spaces and have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces, one has the classifying spaces ofExpand
Lab Notes on the exceptional Lie group E 8 at the prime 2
This is an account of the author’s use of computer algebra tools to explore the structure of the maximal elementary abelian 2-subgroups of the exceptional Lie group E8. The principal result obtainedExpand
The classification of p-compact groups for p odd
A p-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined p-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objectsExpand
Symplectic groups are N-determined 2-compact groups
We show that for n � 3 the symplectic group Sp(n) is as a 2-compact group determined up to isomorphism by the isomorphism type of its maximal torus nor- malizer. This allows us to determine theExpand
Construction of 2-local finite groups of a type studied by Solomon and Benson
A p{local nite group is an algebraic structure with a classifying space which has many of the properties of p{completed classifying spaces of nite groups. In this paper, we construct a family ofExpand
...
1
2
...

References

SHOWING 1-10 OF 31 REFERENCES
Product splittings for p-compact groups
The purpose of this paper is to prove a theorem similar to 1.1 for p-compact groups, which are homotopy theoretic analogues of compact Lie groups. Suppose that X is a p-compact group (see §2) withExpand
Self homotopy equivalences of classifying spaces of compact connected Lie groups
We describe, for any compact connected Lie group G and any prime p, the monoid of self maps BGˆ ! BGwhich are rational equivalences. Here, BGdenotes the p-adic completion of the classifying space ofExpand
Unstable splittings of classifying spaces of P-compact groups
Dwyer and Wilkerson gave a definition of a p–compact group, which is a loop space with certain properties and a good generalisation of the notion of compact Lie groups in terms of classifying spacesExpand
Homotopy uniqueness of classifying spaces of compact connected lie groups at primes dividing the order of the weyl group
As A truism, Lie groups-in particular compact connected Lie groups-are very rigid objects. The perhaps best known instance of this rigidity was formulated in Hilbert’s fifth problem and proved byExpand
Homotopical uniqueness of classifying spaces
If G is a connected compact Lie group, then for almost all prime numbers p the mod p cohomology ring of the classifying space BG is a finitely generated polynomial algebra. In 1961, N. Steenrod [24]Expand
Cohomology of finite groups
The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fieldsExpand
Connected finite loop spaces with maximal tori
Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example having a maximal torus is a quite distinguishedExpand
Kernels of maps between classifying spaces
For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that theExpand
Rational isomorphisms of p-compact groups
A rational isomorphism is a p-compact group homomorphism inducing an isomorphism on rational cohomology. Finite covering homomorphisms and nontrivial endomorphisms of simple p-compact groups areExpand
A cohomology decomposition theorem
In [9] Jackowski and McClure gave a homotopy decomposition theorem for the classifying space of a compact Lie group G; their theorem states that for any prime p the space BG can be constructed at pExpand
...
1
2
3
4
...