Two Torsion and the Loop Space Conjecture

Lin, James P.

doi:10.2307/1971339

Cited by 28 publications

(14 citation statements)

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“…We also remark that the classification, together with results of Bott for compact Lie groups, gives that H * (ΩX; Z p ) is p-torsion free and concentrated in even degrees for all p-compact groups. This result was first proved by Lin and Kane, in fact in the more general setting of finite mod p H-spaces, in a series of celebrated, but highly technical, papers [39,40,41,36], using completely different arguments. Theorem 1.2 also implies a classification for non-connected p-compact groups, though, just as for compact Lie groups, the classification is less calculationally explicit: Any disconnected p-compact group X fits into a fibration sequence BX 1 → BX → Bπ with X 1 connected, and since our main theorem also includes an identification of the classifying space of such a fibration B Aut(BX 1 ) with the algebraically defined space (B 2 Z(D X 1 )) h Out(D X 1 ) , this allows for a description of the moduli space of p-compact groups with component group π and whose identity component has Z proot datum D. More precisely we have the following theorem, which in the case where π is the trivial group recovers our classification theorem in the connected case.…”

Section: Introductionmentioning

confidence: 91%

The classification of 2-compact groups

Andersen

Grodal

2008

J. Amer. Math. Soc.

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We prove that any connected 2 2 –compact group is classified by its 2 2 –adic root datum, and in particular the exotic 2 2 –compact group DI ⁡ ( 4 ) \operatorname {DI}(4) , constructed by Dwyer–Wilkerson, is the only simple 2 2 –compact group not arising as the 2 2 –completion of a compact connected Lie group. Combined with our earlier work with Møller and Viruel for p p odd, this establishes the full classification of p p –compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p p –compact groups and root data over the p p –adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen–Grodal–Møller–Viruel methods by incorporating the theory of root data over the p p –adic integers, as developed by Dwyer–Wilkerson and the authors. Furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by Jackowski–McClure–Oliver in the early 1990s.

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Section: Introductionmentioning

confidence: 91%

The classification of 2-compact groups

Andersen

Grodal

2008

J. Amer. Math. Soc.

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“…Different choices of m give us a set of obstructions {c2(/)} c [G A G; Í2G']. The sets (c2(/)} have proved to be useful in the study of //-spaces, see for example [7,1]. We shall deal exclusively with the case G = Q,X, G' = QX', where X and X' are //-spaces and q, q' are the usual commuting homotopies for the loop multiplications.…”

Section: I2xg2^g'mentioning

confidence: 99%

Primitive Obstructions in the Cohomology of Loopspaces

Williams¹

1984

Proceedings of the American Mathematical Society

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“…The author also thanks the Institute for Advanced Study in Jerusalem for their warm hospitality while work was in progress. Throughout the paper it will be assumed that the reader is familiar with the notation of [2]. §2.…”

Section: + •••+2 Slmentioning

confidence: 99%

“…On the Hopf Algebra Structure of the mod 2 Cohomology of a Finite ff-Space By James LIN* § 1. Introduction…”

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confidence: 99%

On the Hopf Algebra Structure of the mod 2 Cohomology of a Finite $H$-Space

Lin¹

1984

Publ. Res. Inst. Math. Sci.

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The action of the Steenrod algebra on the cohomology of a finite //-space has placed several restrictions on the possible spaces which can be finite //"-spaces. It is now known, for example, that the loop space of a finite //-space has no homology torsion. We continue this study here by showing that in the mod 2 cohomology of a finite //-space, every generator of degree 4/e + l is in the image of Sq 2k .This result has several consequences. For example, if an integer n has dyadic expansion n=2where O^s^St+i we say n has length I. If 4& + 1 has length / and there is a generator of the mod 2 cohomology of a finite //-space in degree 4& + 1, then there is a generator in degree 2 Z -1. Hence if a finite //-space is highly connected this result implies the non-existence of generators of small length.Another implication is that there are no 4/e dimensional primitive indecomposables in the mod 2 cohomology of the loop space. Also, all generators of degree 8/2+5 in the cohomology of a finite //-space are in the image of Sq 2 . Throughout this paper it is assumed that all finite //-spaces used in this paper are simply connected and have associative mod 2 homology rings. Under this assumption it is easily shown that finite //-spaces that have no two torsion have primitively generated mod 2 cohomology rings. Hence their structure over the Steenrod algebra is restricted by several results of Thomas [4]. It is therefore more interesting to study finite //-spaces that have two torsion in their homology.Under these circumstances it was shown that the two torsion is of order at most two and arises from odd degree mod 2 cohomology generators whose cup product is non trivial. In chapter 3 of this paper a structure theorem is given for the action of the Steenrod algebra on these generators. In particular if y is Communicated by N.

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Two Torsion and the Loop Space Conjecture

Cited by 28 publications

References 25 publications

The classification of 2-compact groups

The classification of 2-compact groups

Primitive Obstructions in the Cohomology of Loopspaces

On the Hopf Algebra Structure of the mod 2 Cohomology of a Finite $H$-Space

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