Hopf Algebras of Cooperations for Real and Complex <i>K</i>
-Theory

Adams, J. F.; Harris, A.; Switzer, Robert M.

doi:10.1112/plms/s3-23.3.385

“…(a) The group Ext)t\{K)(K*(S0), K*(S0)) is isomorphic to ker ¿},2/im ¿?'2. By Proposition 3 and Theorem (2.3) of [3], ker dl'2 = {r(v -u)\r is a suitable rational}. The integrality condition (2.4) of [3] requires that r = |(mod 1).…”

Section: Generalized Steenrod-hopf Invariants For Stable Homotopy Thementioning

confidence: 99%

Generalized Steenrod-Hopf invariants for stable homotopy theory

Krueger¹

1973

Proc. Amer. Math. Soc.

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Abstract.In his paper On the groups J(X). IV, Adams suggested that one might try to continue his d and e invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his d and e invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams' e invariant for this theory.

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“…This gives a simplified proof of the unitary case of the main result of [5]. We refer to elements of ,4[u;~'] as stably numerical polynomials.…”

Section: The A-theory Of the Classifying Spaces Of Torimentioning

confidence: 99%

Untitled

1989

Trans. Amer. Math. Soc.

0

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Abstract.We study the torsion-free part of the stable homotopy groups of the space BU, by considering upper and lower bounds. The upper bound is furnished by the ring PK,(BU) of coaction primitives into which n^(BU) is mapped by the complex /^-theoretic Hurewicz homomorphism nst(BV)^ PKt(BU).We characterize PK,(BU) in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in BU . From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain e-invariant calculations of J. F. Adams.

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“…That this class is nonzero in Ext2 can be shown by calculating with d\A and d\A and the integrality condition (2.4) of [3].…”

Section: Generalized Steenrod-hopf Invariants For Stable Homotopy Thementioning

confidence: 99%

“…It is not hard to show that a is represented by t2ß2 e Kt(BU) with BU regarded as the second term, K2, of the spectrum K. When BU is taken to be K0, t2ß2 projects to p'2=\v(v-u) by (6.13) of [3], so when BU is taken to be A"2, t2ß2 projects to |(i>-u) according to (6.1) of [3]. Thus Ii(r¡)=j0*c(?…”

Section: Generalized Steenrod-hopf Invariants For Stable Homotopy Thementioning

confidence: 99%

Generalized Steenrod-Hopf Invariants for Stable Homotopy Theory

Krueger¹

1973

Proceedings of the American Mathematical Society

2

0

View full text Add to dashboard Cite

Abstract.In his paper On the groups J(X). IV, Adams suggested that one might try to continue his d and e invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his d and e invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams' e invariant for this theory.

show abstract

Hopf Algebras of Cooperations for Real and Complex K -Theory

Cited by 43 publications

References 3 publications

Generalized Steenrod-Hopf invariants for stable homotopy theory

Generalized Steenrod-Hopf invariants for stable homotopy theory

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Generalized Steenrod-Hopf Invariants for Stable Homotopy Theory

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