Réalisation topologique de certaines algèbres associées aux algèbres de Dickson

Jeanneret, Alain; Suter, Ulrich W.

doi:10.1007/bfb0087513

Cited by 5 publications

(3 citation statements)

References 4 publications

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“…Since G 1 := SO(3) = DI(2) and G 2 = DI(3) , it is tempting to put G 3 = DI(4) . No 2-compact group deserves the name DI(n) with n ≥ 5 [19,35].…”

Section: Theorem 24 [12]mentioning

confidence: 99%

Homotopy Lie groups

Møller¹

1995

Bull. Amer. Math. Soc.

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Abstract.Homotopy Lie groups, recently invented by W.G. Dwyer and C.W. Wilkerson [13], represent the culmination of a long evolution. The basic philosophy behind the process was formulated almost 25 years ago by Rector [32,33] in his vision of a homotopy theoretic incarnation of Lie group theory. What was then technically impossible has now become feasible thanks to modern advances such as Miller's proof of the Sullivan conjecture [25] and Lannes's division functors [22]. Today, with Dwyer and Wilkerson's implementation of Rector's vision, the tantalizing classification theorem seems to be within grasp.Supported by motivating examples and clarifying exercises, this guide quickly leads, without ignoring the context or the proof strategy, from classical finite loop spaces to the important definitions and striking results of this new theory.

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“…Since G 1 := SO(3) = DI(2) and G 2 = DI(3) , it is tempting to put G 3 = DI(4) . No 2-compact group deserves the name DI(n) with n ≥ 5 [19,35].…”

Section: Theorem 24 [12]mentioning

confidence: 99%

Homotopy Lie groups

Møller¹

1995

Bull. Amer. Math. Soc.

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“…It is known that no space of type BDI(n) can exist for n 2: 6 [31]. Lannes has recently used methods similar to ours to show the nonexistence of a space of type B DI (5) , while [24] proves the stronger statement that an H-space "of type nBDI(5)" does not exist (see also [18]). The construction of BDI (4) in this paper completes the determination of the set of integers n for which a space of type BDI(n) exists.…”

mentioning

confidence: 80%

A New Finite Loop Space at the Prime Two

Dwyer¹,

Wilkerson²

1993

Journal of the American Mathematical Society

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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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“…Lannes has recently used methods similar to ours to show the nonexistence of a space of type B DI(5) , while [24] proves the stronger statement that an H-space "of type nBDI(5)" does not exist (see also [18]). The construction of BDI (4) in this paper completes the determination of the set of integers n for which a space of type BDI(n) exists.…”

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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Réalisation topologique de certaines algèbres associées aux algèbres de Dickson

Cited by 5 publications

References 4 publications

Homotopy Lie groups

Homotopy Lie groups

A New Finite Loop Space at the Prime Two

A new finite loop space at the prime two

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