The algebra of secondary homotopy operations in ring spectra

Baues, Hans–Joachim; Muro, Fernando

doi:10.1112/plms/pdq034

Proceedings of the London Mathematical Society

2010

DOI: 10.1112/plms/pdq034

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The algebra of secondary homotopy operations in ring spectra

Hans–Joachim Baues

Fernando Muro

Abstract: Abstract. The primary algebraic model of a ring spectrum R is the ring π * R of homotopy groups. We introduce the secondary model π * , * R which has the structure of a secondary analogue of a ring. The homology of π * , * R is π * R and triple Massey products in π * , * R coincide with Toda brackets in π * R. We also describe the secondary model of a commutative ring spectrum Q from which we derive the cup-one square operation in π * Q. As an application we obtain for each ring spectrum R new derivations of t… Show more

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Cited by 2 publications

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“…This subject is developed in work of Baues, Jibladze, Pirashvili, Muro and the second author, see e.g. [1][2][3][4][5].…”

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confidence: 99%

Quadratic maps between modules

Gaudier

Hartl

2009

Journal of Algebra

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Proposition 1.2. Suppose that R contains an element r such that r and r − 1 are invertible. Then any Rquadratic map f : M → N decomposes uniquely as a sum f = f 1 + f 2 of an R-linear map f 1 and a homoge-This criterion is improved in Example 4.5(2) below. Proof.We can takeby Remark 1.3 below. Uniqueness of f 1 and f 2 follows from the fact that under the hypothesis any map which is R-linear and homogenous R-quadratic is trivial; in fact,Note that the proposition applies whenever R is a field different from F 2 . If R = F 2 any Rquadratic map is homogenous.Finally, we discuss the case R = Z. Passi [15] defines a map f : G → A from a group G to an abelian group A to be (normalized) polynomial of degree. An inductive characterization of this property [9] shows that f is polynomial of degree 2 iff its cross-effect d f is homomorphic in each variable. If G is abelian, this is equivalent towhich is homogenous Z-quadratic; this suffices by Remark 1.3 below.Let us exhibit some elementary properties of R-quadratic maps. First note that if f is R-quadratic,Next for x, y, z in M and r, s in R the first condition in 1.1 can be written as(1.2) Additivity of f [r] then follows from (1.2) with s = r, and its R-linearity can be written as:(1.3) Remark 1.3. Relation (1.3) can be written as f s (rx) = r 2 f s (x), that is f s is a homogeneous R-quadratic map. Thus we see that f is R-quadratic iff its cross-effect is R-bilinear and its cross-actions are homogeneous R-quadratic.Clearly the set R-Quad(M, N) (resp. R-HQuad(M, N)) of the R-quadratic maps (resp. homogeneous R-quadratic maps) from M to N is an R-module, and pre-or postcomposition of an R-quadratic map (resp. homogeneous R-quadratic map) by an R-linear map is an R-quadratic (resp. homogeneous Rquadratic) map.Throughout this paper the tensor product of R-modules M and N is denoted by M ⊗ N instead of M ⊗ R N. Lemma 1.4. For any R-modules M, M , N one has a natural isomorphism R-Quad(M ⊕ M , N) ∼ = R-Quad(M, N) ⊕ R-Quad(M , N) ⊕ R-Hom(M ⊗ M , N).

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“…This subject is developed in work of Baues, Jibladze, Pirashvili, Muro and the second author, see e.g. [1][2][3][4][5].…”

mentioning

confidence: 99%

Quadratic maps between modules

Gaudier

Hartl

2009

Journal of Algebra

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“…Namely, quadratic pair algebras are natural objects representing classes in the third dimensional Mac Lane cohomology. Moreover the secondary homotopy groups of each ring spectrum form a quadratic pair algebra [8]. In particular, the sphere spectrum yields a quadratic pair algebra encoding all its secondary homotopy structure like triple Toda brackets.…”

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confidence: 99%

Quadratic algebra of square groups

Baues

Jibladze²,

Pirashvili

2008

Advances in Mathematics

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Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.There is a long-standing problem of algebra to extend the symmetric monoidal structure of abelian groups, given by the tensor product, to a non-abelian setting, see for example [13]. In this paper we show the somewhat surprising fact that such an extension is possible. Moreover our non-abelian tensor product remains even right exact and balanced. We describe the new nonabelian tensor product in the context of quadratic algebra which extends linear algebra."Linear algebra" is the algebra of rings and modules. A ring is a monoid in the symmetric monoidal category of abelian groups (Ab, ⊗, Z). ✩ The second and third authors are grateful to the

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The algebra of secondary homotopy operations in ring spectra

Cited by 2 publications

References 43 publications

Quadratic maps between modules

Quadratic maps between modules

Quadratic algebra of square groups

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