Homotopy Theory of Classifying Spaces of Compact Lie Groups

Jackowski, Stefan; McClure, James E.; Oliver, Bob

doi:10.1007/978-1-4613-9526-3_4

Cited by 23 publications

(23 citation statements)

References 37 publications

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“…Hence when G is connected, the exact sequence of Proposition 7.2 gives a new way to describe Out.BGp /, which is different from but closely related to the descriptions in [19; 20] and [22].…”

Section: Proposition 72mentioning

confidence: 66%

Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

2007

Self Cite

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Section: Proposition 72mentioning

confidence: 66%

Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

2007

Self Cite

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“…In [38] the second author showed that p -completed classifying spaces of compact connected Lie groups admit unstable Adams operations q of exponent a p -adic unit q 2 ‫ޚ‬ p . This is extended to p -compact groups for odd primes p in [53].…”

Section: Introductionmentioning

confidence: 99%

Chevalleyp–local finite groups

Broto

Møller

2007

Algebr. Geom. Topol.

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“…Proposition [17]. For any compact Lie group G there exist ( 1) a category A~ie whose objects are the (conjugacy classes of) nontrivial elementary abelian 2-subgroups of G, Unless G has a central element of order 2, this proposition provides a decomposition of BG up to F 2 cohomology as a generalized pushout of classifying spaces of proper subgroups of G. There is also an algebraic version of the above.…”

Section: The Basic Technique Our Present Methods Of Building B Di( 4)mentioning

confidence: 99%

“…One might even conjecture on the basis of the tables in [5] that the classifying space of any connected 2-adic finite loop space is the product of the F 2 -completion of the classifying space of a connected compact Lie group with a number of copies of B DI( 4) . [20,17,13]. The starting point is again the theory of Lie groups, but this time on the homotopy-theoretic side.…”

mentioning

confidence: 99%

A new finite loop space at the prime two

Dwyer¹,

Wilkerson²

1993

J. Amer. Math. Soc.

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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 CW-complex, and (2) there is a pointed space BG and a homotopy equivalence from G to the loop space QB G .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 -+ nBX 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 Fp-complete in the sense of [3] and have finite mod p cohomology, then X is called a p-adic finite loop space or a finite loop space at the prime p. A (p-adic) finite loop space is a strong homotopy-theoretic analogue of a compact Lie group. The study of these spaces is related to many classical questions in topology (for instance, to the problem of determining all spaces with polynomial cohomology rings).Call a p-adic finite loop space X exotic if it is not the Fp-completion of G for a compact Lie group G. There are many known examples of exotic p-adic finite loop spaces at odd primes p [5] and the classification of these spa~es is partially understood [8,9]. However, until now there have been no known exotic 2-adic finite loop spaces.Recall [32] that the ring of rank 4 mod 2 Dickson invariants is the ring of invariants of the natural action of GL( 4, F 2) on the rank 4 polynomial algebra H*((BZ/2)4, F 2 ); this ring of invariants is a polynomial algebra on classes c s ' C 12 ' C 14 ' and C l5 with Sq 4 C s = c 12 ' Sq 2 c l2 = c 14 ' and Sql C l4 = C 15 • Our main theorem is the following one.1.1. Theorem. There exists an F 2-complete space B DI( 4) such that H*(BDI(4) , F 2 ) is isomorphic as an algebra over the Steenrod algebra to the ring of rank 4 mod 2 Dickson invariants.Let DI(4) = QBDI(4). Standard methods show that H*(DI(4) , F 2 ) is multiplicatively generated by elements x 7 , y 11' and z I3' with S q 4 X = y, Sq2y = z, Sql Z = x 2 =1= 0, and X4 = i = z2 = O. This space DI(4) is an exotic 2-adic finite loop space.

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Homotopy Theory of Classifying Spaces of Compact Lie Groups

Cited by 23 publications

References 37 publications

Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

Chevalleyp–local finite groups

A new finite loop space at the prime two

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