Applications of Nonconnective Im(J)-theory

Crabb, M. C.; Knapp, Karlheinz

doi:10.1016/b978-044481779-2/50013-4

Cited by 3 publications

(5 citation statements)

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“…Let p be an odd prime, k be an integer which generates .‫=ޚ‬p 2 / and k be the (stable) Adams operation in p -local complex K -theory. Then the spectrum of nonconnected Im.J /-theory Ad may be defined by the cofiber sequence: Knapp [13] and Crabb and Knapp [5]. Connective Im.J /-theory A is then defined as the ( 1)-connected cover of Ad.…”

Section: Introductionmentioning

confidence: 99%

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Connective Im(J)–theory for cyclic groups

Knapp

2007

Algebr. Geom. Topol.

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797Connective Im.J /-theory for cyclic groups KARLHEINZ KNAPPWe study connective Im.J /-theory for the classifying space B‫=ޚ‬p a of a finite cyclic p -group and compute the Im.J /-cohomology groups completely. We also compute the Im.J /-homology groups, with the exception of a finite range of dimensions. 55N35, 19L64; 19D99, 19L20 IntroductionIm.J /-theory A is a generalized homology theory appearing both in homotopy theory and algebraic K -theory. It is (sometimes) a fairly good first approximation to stable homotopy capturing v 1 -periodic phenomena, and is closely related to the classical J -homomorphism and the e -invariant. Let p be an odd prime, k be an integer which generates .‫=ޚ‬p 2 / and k be the (stable) Adams operation in p -local complex K -theory. Then the spectrum of nonconnected Im.J /-theory Ad may be defined by the cofiber sequence: Knapp [13] and Crabb and Knapp [5]. Connective Im.J /-theory A is then defined as the ( 1)-connected cover of Ad. More importantly, A appears as the p -localization of the K -theory of a finite field. If k is chosen as a prime power, thenwhere K‫ކ‬ k is the algebraic K -theory spectrum of the finite field ‫ކ‬ k .

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Section: Introductionmentioning

confidence: 99%

“….B/ (see Crabb and Knapp [5] for the relation between the e -invariant and Im.J /-theory). But, of course, We now solve the group extension (18) for n with p .n/ a 1.…”

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confidence: 99%

Connective Im(J)–theory for cyclic groups

Knapp

2007

Algebr. Geom. Topol.

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“…Alternatively, this shows that the vector bundle mH * over P m−k is stably fibre homotopy trivial at the prime 2 (see [4], the proof of Proposition 2.8 and [18], the proof of Theorem 2.2) and leads to the equivalent condition that m(F−H * ) ∈ KO 0 (P m−k ) (2) lies in the image of ψ 3 − 1. (See, for example, [10], Theorem 5.1 modified to KO. )…”

Section: A Choice Of Homotopy Induces a Diagram Of Fibrationsmentioning

confidence: 99%

“…and mapping to the generator of Z/8 = KO −6 (C 8 ). The reason is that the map J −6 (C 8 ) → KO −6 (C 8 ) is onto (since the Adams operator ψ 3 acts as multiplication by 3 4 on (KO 8 ) (2) , and so ψ 3 − 1 as 80 on KO −6 (C 8 ) (2) ) and that the Hurewicz homomorphism h J : ω 7 → J 7 is also onto, which follows from the computation of the J-homomorphism and the e-invariant in [2], Theorem 1.6, and the relation between the e-invariant and h J (see [10], Section 1).…”

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confidence: 94%

$G$-Structures on Spheres

Čadek

Crabb

2006

Proc. Lond. Math. Soc.

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This paper deals with the problem of determining those groups G and the homomorphisms ρ to which the structure group G n in the fibrations above can be reduced. The problem has been solved in many interesting special cases. Considering standard inclusions ρ : G = G k → G n we get the famous problem on sections of Stiefel manifolds over spheres resolved in [1, 3, 5, 23]. The other standard inclusions SU(k) → SO(n), Sp(k) → SO(n) and Sp(k) → SU(n) are dealt with in [12], [18] and [19], respectively. In these cases the question was to find a minimal standard subgroup to which G n can be reduced.In [15] Leonard asked an opposite question: find all maximal proper subgroups to which G n can be reduced. He solved it in the cases when G is a reducible maximal subgroup of G n . Moreover, he proved that G n cannot be reduced to any proper subgroup (G, ρ) if (1) n is even and G n = SO(n) or SU(n), unless G n = SO(6) and G = SU(3);(2) n ≡ 11 mod 12 and G n = Sp(n);(3) G is a non-simple irreducible maximal proper subgroup of G n . 793 can be reduced to G via a homomorphism ρ : G → G n if and only if one of the following cases occurs: (A) G n = SO(n), G = SO(k), n = m − 1, m ≡ 0 mod a(m − k) and, up to conjugation, ρ is the standard inclusion SO(k) → SO(n); (B) G n = SO(n), G = SU(k), n = 2m − 1, m ≡ 0 mod 2 ν2(b(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions SU(k) → SO(2k) → SO(n) or the composition SU(4) → SO(8) × SO(6) → SO(15) where the first homomorphism is given on the first factor by the standard inclusion and on the second factor by the double covering SU(4) ∼ = Spin(6) → SO(6); (C) G n = SO(n), G = Sp(k), n = 4m − 1, m ≡ 0 mod 2 ν2(c(m−k)) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SO(4k) → SO(n) or the exterior square Sp(3) → SO(15); (D) G n = SU(n), G = SU(k), n = m − 1, m ≡ 0 mod b(m − k) and, up to conjugation, ρ is the standard inclusion SU(k) → SU(n); (E) G n = SU(n), G = Sp(k), n = 2m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the composition of the standard inclusions Sp(k) → SU(2k) → SU(n); (F) G n = Sp(n), G = Sp(k), n = m − 1, m ≡ 0 mod c(m − k) and, up to conjugation, ρ is the standard inclusion Sp(k) → Sp(n).As a consequence of Theorem 2.

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“…In particular, this means that no such sum can desuspend to [S 2j−1 L, S 2j−1 ] (p) . On the other hand, it is known that the generator of π 2n−1 (J p ) does desuspend to π (2n−1)+(2ν+1) (S 2ν+1 ) (for example, see [26]), and if we combine this with Proposition 6. Finally, we shall use Proposition 7.2 to verify the conjecture on exceptional dimensions when n = j(p − 1) for j = 1, .…”

Section: Proofs Of Theorems 4-7mentioning

confidence: 99%

Tangential thickness of manifolds

Kwasik

Schultz

2015

Proceedings of the London Mathematical Society

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Abstract. A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to a classification of such manifolds with thickness 1, 3 or 2k, for k ≥ 1. On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.

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Applications of Nonconnective Im(J)-theory

Cited by 3 publications

References 22 publications

Connective Im(J)–theory for cyclic groups

Connective Im(J)–theory for cyclic groups

$G$-Structures on Spheres

Tangential thickness of manifolds

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