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The homotopy theory of fusion systems
"... The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group con ..."
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Cited by 35 (10 self)
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The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group consists, roughly speaking, of a finite pgroup S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on pcompleted classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig [Pu], [Pu2] of systems of fusion among subgroups of a given pgroup. The pcompletion of a space X is a space X ∧ p which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −− → Y induces a homotopy equivalence
A New Finite Loop Space at the Prime Two
 Journal of the A.M.S
, 1993
"... . We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We conjecture that it is also the last one. x1. Introduction From the point of view of homotopy theory a comp ..."
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Cited by 28 (5 self)
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. We construct a space BDI(4) whose mod 2 cohomology ring is the ring of rank 4 mod 2 Dickson invariants. The loop space on BDI(4) is the first example of an exotic finite loop space at 2. We conjecture that it is also the last one. x1. Introduction From the point of view of homotopy theory a compact Lie group G has the following remarkable combination of properties: (1) G can be given the structure of a finite CWcomplex, and (2) there is a pointed space BG and a homotopy equivalence from G to the loop space\Omega BG. Of course the space BG in (2) is the ordinary classifying space of G. In general, a finite complex X together with a chosen equivalence X ! \Omega BX for some BX is called a finite loop space. If p is a prime number and the geometric finiteness condition on X is replaced by the requirement that X be F p complete in the sense of [3] and have finite mod p cohomology, then X is called a padic finite loop space or a finite loop space at the prime p. A (padic) fi...
THE CLASSIFICATION OF pCOMPACT GROUPS FOR p ODD
, 2003
"... A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we fi ..."
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Cited by 24 (13 self)
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A pcompact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined plocal analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a onetoone correspondence between connected pcompact groups and finite reflection groups over the padic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as pcompact groups by their Weyl groups seen as finite reflection groups over the padic integers. Our approach in fact gives a largely selfcontained proof of the entire
The Center Of A pCompact Group
"... this paper we continue the study by looking at the idea of the "center" of a pcompact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of sel ..."
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Cited by 20 (1 self)
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this paper we continue the study by looking at the idea of the "center" of a pcompact group and showing that two very different ways of defining the center are equivalent. This leads for instance to a reproof and generalization of a theorem from [15] about the identity component of the space of self homotopy equivalences of BG (G compact Lie). Along the way we find various familiarlooking elements of internal structure in a pcompact group X , enumerate the X 's which are abelian in the appropriate sense, and construct what might be called the "adjoint form" of X . Before describing in more detail the main results we are aiming at, we have to introduce some ideas from [12]. A loop space X is by definition a triple (X ; BX ; e) in which X is a space, BX is a connected pointed space (called the classifying space of X ), and e : X !
Deterministic pCompact Groups
, 1997
"... We investigate the class of Ndetermined pcompact groups and the class of pcompact groups with Ndetermined automorphisms. A pcompact group is said to be Ndetermined if it is determined up to isomorphism by the normalizer of a maximal torus. The automorphisms of a pcompact group are said to b ..."
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Cited by 1 (1 self)
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We investigate the class of Ndetermined pcompact groups and the class of pcompact groups with Ndetermined automorphisms. A pcompact group is said to be Ndetermined if it is determined up to isomorphism by the normalizer of a maximal torus. The automorphisms of a pcompact group are said to be Ndetermined if they are determined by their restrictions to this maximal torus normalizer.
Author address: Homotopy theory of diagrams
, 2001
"... Chapter I. Model approximations and bounded diagrams 5 ..."