## SECONDARY ALGEBRAS ASSOCIATED TO RING SPECTRA (2006)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Baues06secondaryalgebras,

author = {Hans-joachim Baues and Fernando Muro},

title = {SECONDARY ALGEBRAS ASSOCIATED TO RING SPECTRA},

year = {2006}

}

### OpenURL

### Abstract

Abstract. Homotopy groups of a connective ring spectrum R form an-graded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R represents a cohomology class in the third Mac Lane cohomology of the algebra π∗R. If R is commutative then π∗,∗R has an E∞-structure and encodes the cup-one squares in π∗R. Contents

### Citations

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Citation Context .... 1. Ring spectra and module spectra In this paper the framework for stable homotopy theory will be the stable model category of symmetric spectra of compactly generated topological spaces defined in =-=[MMSS01, 9]-=-. The smash product of symmetric spectra X ∧Y defines a symmetric monoidal structure in this category. The unit of this monoidal structure is the sphere spectrum S. Monoids in the category of symmetri... |

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Citation Context ...mmutative connective ring spectrum) yields the E∞-quadratic pair algebra π∗,∗S which enriches the structure of the commutative algebra π∗S considerably. Many of the stable results on Toda brackets in =-=[Tod62]-=- are, in fact, local computations in this global structure of an E∞quadratic pair algebra and therefore hold for commutative ring spectra in general. It is likely that the E∞-quadratic pair algebra π∗... |

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Citation Context ...Moreover, this operadic cohomology group should map to the third topological Andre-Quillen cohomology group, where the first Postnikov invariant of a connective commutative ring spectrum R lives, see =-=[Bas99]-=-. In fact, if R neglects the Hopf map the class R should be mapped this way to the first Postnikov invariant of R as in the non-commutative case, see Remark 2.9. of the ungraded E∞-pair algebra π adc ... |

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Citation Context ...ociated to a connective ring spectrum R 〈modf(R)〉 ∈ HML ∗ (π∗R, Σ −1 π∗R). This universal Toda bracket determines all Toda brackets in modf(R), see [BD89], i.e. all matric Toda brackets in R, compare =-=[May69]-=-. The goal of this section is the construction of a small algebraic model for this universal Toda bracket. For this we will consider universal Toda brackets associated to quadratic pair algebras. Give... |

32 | HZ–algebra spectra are differential graded algebras, preprint
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Citation Context ...g-Mac Lane spectrum HZ also neglect the Hopf map. Any connective HZ-algebra is weakly equivalent to the HZ-algebra HC∗ of a differential graded algebra C∗ concentrated in non-negative dimensions, see =-=[Shi06]-=-. The theory developed in this paper allows one to compute a small model of πadd ∗,∗ HC∗. Indeed this pair algebra is quasi-isomorphic to ¯d: Σ −1 (C∗/d(C∗)) −→ Z∗. Here Z∗ ⊂ C∗ is the subring of cycl... |

30 |
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Citation Context ... x1 ⌣1 x2): x1 ⊗ x2 −→ x2 ⊗ x1. 9. Universal Toda brackets of ring spectra Given a ring spectrum R the category mod(R) of (right) R-modules is a Quillen model category, see [MMSS01, Theorem 12.1]. By =-=[Bau89]-=- the full subcategory mod fc (R) of fibrant-cofibrant R-modules is a groupoid-enriched category whose morphisms are homotopy classes of homotopies, also termed tracks. A free R-module is an R-module o... |

28 |
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Citation Context ...)(τ⊗ ⊗ 1) µ(1 ⊗ µ) µ((µτ � ⊗ ) ⊗ 1) ��� �� �� �� �� �� �� �� �� �� �� �� �� µ(⌣1⊗1) � �������������� �������������� µ(µ ⊗ 1) commutes. These are the usual axioms in a symmetric monoidal category, see =-=[Bor94]-=-. Indeed there is an additional diagram, called the pentagon, which should commute. In this case it commutes automatically since the product in B is strictly associative. Pair algebras carry a notion ... |

28 |
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Citation Context ...y SH 3 (A, M) is naturally included in Mac Lane cohomology, see [JP86, BP], which is isomorphic to the topological Hochschild cohomology of the corresponding Eilenberg-Mac Lane ring spectrum, compare =-=[PW92]-=-, (2.10) SH 3 (A, M) ֒→ HML 3 (A, M) ∼ = THH 3 (HA, HM). If R is a connective ring spectrum neglecting η we can consider the ungraded pair algebra πadd 0,∗ R in the bottom degree of πadd ∗,∗ R in Theo... |

28 |
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(Show Context)
Citation Context ...c pair algebra π∗,∗R in (4.4), k: π∗R ⊗ Z/2 −→ Σ −1 π∗R, coincides with the multiplication by η where η is the stable Hopf map. We prove this theorem in Section 14. Example 5.5. Waldhausen defined in =-=[Wal78]-=- the K-theory spectrum KW of a category W with cofibrations and weak equivalences. This spectrum is a symmetric spectrum, see [GH99]. Moreover, if W is a strict monoidal category with biexact tensor p... |

25 |
Stable homotopy categories
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(Show Context)
Citation Context ...ious cobordism spectra. The homotopy group πnR coincides with the group of morphisms Σ n R → R in the stable homotopy category of right R-modules. Therefore Toda brackets in π∗R are defined following =-=[Hel68]-=- by using the triangulated structure in the homotopy category of right R-modules. More precisely, given elements a, b, c ∈ π∗R of degree p, q, r, respectively, with ab = 0 and bc = 0 a generic element... |

22 |
Homotopy theory of A∞ ring spectra and applications to MU - modules, K-theory 24
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(Show Context)
Citation Context ...nsider the ungraded pair algebra πadd 0,∗ R in the bottom degree of πadd ∗,∗ R in Theorem 2.7 yielding the associated Shukla cohomology class 〈π add 0,∗ R〉 ∈ SH3 (π0R, π1R). On the other hand Lazarev =-=[Laz01]-=- introduced the “first k-invariant” k 1 R ∈ THH3 (H(π0R), H(π1R)) of the ring spectrum R. We claim that the image under (2.10) of the class 〈π add 0,∗ R〉 in topological Hochschild cohomology can be id... |

21 |
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Citation Context ...d(C∗)) −→ Z∗ in Example 2.8 is an E∞-pair algebra with cup-one product induced by ⌣1 in (3.6). Remark 3.7. The coherence conditions in Remark 3.2 are related to the BarrattEccles operad introduced in =-=[BE74]-=-. Indeed, one can check that E∞-pair algebras are algebras over the following operad in pm. Let BE be the simplicial BarrattEccles operad and let C∗BE be the operad of chain complexes obtained by taki... |

20 | Morita theory in abelian, derived and stable model categories. Structured Ring Spectra
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(Show Context)
Citation Context ... X ⊗ Y . The smash product X ∧ Y comes equipped with natural maps, p, q ≥ 0, (10.1) jp,q : Xp ∧ Yq −→ (X ∧ Y )p+q satisfying a universal property which characterizes X ∧Y as a symmetric spectrum, see =-=[Sch04]-=-. The category Sp of spectra is a stable model category, see [MMSS01, 4.1 and 9], while Sp Σ is in addition a symmetric monoidal model category, see [MMSS01, 4.2 and 9]. We wish to emphasize that amon... |

14 |
The cohomology of homotopy categories and the general linear group, K-Theory
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(Show Context)
Citation Context ...Σ p+q �� c where the upper row is an exact triangle. The existence of such a commutative diagram follows from the axioms of a triangulated category. One can also define Toda brackets in π∗R following =-=[BD89]-=- by using tracks in the category of right R-modules (i.e. homotopy classes of homotopies between maps). Let us sketch this alternative construction. Since the homotopy groups of any fibrant replacemen... |

14 |
Quadratic algebra of square groups
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(Show Context)
Citation Context ... functors h0, h1: qpm −→ Ab defined by h0C = Coker∂ and h1C = Ker∂ as in (2.1). A quasi-isomorphism of quadratic pair modules is a morphism inducing isomorphisms on h0 and h1. Remark 4.3. As shown in =-=[BJP05]-=-, the category SG is a symmetric monoidal category with the tensor product X ⊙ Y of square groups and with unit object Znil. Moreover, there is a symmetric monoidal structure ⊙ on qpm defined by using... |

13 |
modules, and algebras in infinite loop space theory
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(Show Context)
Citation Context ...r algebras. For this we introduce “quadratic pair algebras” and “E∞-quadratic pair algebras”. These algebraic structures yield examples of ring categories and bipermutative categories in the sense of =-=[EM05]-=-. For any connective ring spectrum R we describe a quadratic pair algebra C = π∗,∗R satisfying (C) such that Toda brackets in R coincide with Massey products in π∗,∗R. The quadratic pair algebra π∗,∗R... |

8 | Crossed extensions of algebras and Hochschild cohomology, Homology Homotopy Appl - Baues, Minian |

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8 | Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili - Springer-Verlag, edition - 1998 |

7 |
Cobordism of Massey products
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(Show Context)
Citation Context ...otopy groups of a ring spectrum π∗R carry Toda bracket operations which enrich the ring structure of π∗R. Toda brackets have been considered, for instance, in [Tod62] for the sphere spectrum S and in =-=[Ale72]-=-, under the name of Massey products, for various cobordism spectra. The homotopy group πnR coincides with the group of morphisms Σ n R → R in the stable homotopy category of right R-modules. Therefore... |

7 |
Topological cyclic homology of schemes, in Algebraic K-theory
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- 1997
(Show Context)
Citation Context ...ove this theorem in Section 14. Example 5.5. Waldhausen defined in [Wal78] the K-theory spectrum KW of a category W with cofibrations and weak equivalences. This spectrum is a symmetric spectrum, see =-=[GH99]-=-. Moreover, if W is a strict monoidal category with biexact tensor product then KW is a ring spectrum. In [MT06] we give a small algebraic model for the quadratic pair algebra π0,∗KW which is generate... |

7 |
The 1-type of a Waldhausen K-theory spectrum
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(Show Context)
Citation Context ...y W with cofibrations and weak equivalences. This spectrum is a symmetric spectrum, see [GH99]. Moreover, if W is a strict monoidal category with biexact tensor product then KW is a ring spectrum. In =-=[MT06]-=- we give a small algebraic model for the quadratic pair algebra π0,∗KW which is generated just by the objects, weak equivalences, and cofiber sequences in W. The next result is an illustrating example... |

4 | Computation of the E3-term of the Adams spectral sequence
- Baues, Jibladze
- 2004
(Show Context)
Citation Context ...operations B, for p a fixed prime, computed in [Bau06], is a pair algebra. This pair algebra has proved to be useful for computations of d2 differentials in the classical Adams spectral sequence, see =-=[BJ04]-=-. The pair algebra B corresponds to the coconnective version of π∗,∗R for R = End(HZ/p) the endomorphism spectrum of the mod p Eilenberg-Mac Lane spectrum. The coconnective theory, however, is not con... |

4 |
products for secondary homotopy groups
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- 2006
(Show Context)
Citation Context ...category with the tensor product X ⊙ Y of square groups and with unit object Znil. Moreover, there is a symmetric monoidal structure ⊙ on qpm defined by using the tensor product of square groups, see =-=[BM05c]-=-. The unit object for the monoidal structure in qpm is ⎛ Z P Znil = ⎜ ⎝ ��������� ⎞ ��������� H ⎟ ⎠ Z/2 �� Z where P is the non-trivial homomorphism and H(n) = ( ) n 2 as in (4.1). The functors h0 and... |

3 |
Mac Lane cohomology
- Third
- 2006
(Show Context)
Citation Context ...specified isomorphisms h0B ∼ = A and h1B ∼ = M such that any element in A is the image of some x ∈ B0 with H(x) = 0. Morphisms are quadratic pair algebra morphisms over A and under M. It is proved in =-=[BJP06]-=- that the set of connected components of this category is in natural bijection with the 3-dimensional Mac Lane cohomology group HML 3 (A, M). As we mention in Remark 2.9 this cohomology group coincide... |

3 | Crossed modules over operads and operadic cohomology, K-Theory 31 - Baues, Minian, et al. |

3 |
and symmetric spectra in general model categories
- “Spectra
(Show Context)
Citation Context ...structure fibrant objects coincide in both cases with the Ω-spectra. Symmetric sequences and symmetric spectra defined in this way are available over monoidal categories more general than Top ∗ , see =-=[Hov01]-=-. They can also be defined by using right actions of symmetric groups instead of left actions. 11. Secondary homotopy groups of spaces A groupoid-enriched category, also termed track category, is a 2-... |

2 |
The homotopy category of pseudofunctors and translation cohomology
- Baues, Muro
- 2005
(Show Context)
Citation Context ...d only if there is a pseudofunctor ϕ: C → B such that for any morphism f : X → Y in C the equality pBϕ(f) = pC(f) holds, and for any x ∈ D(X, Y ) the equation σ ϕ(f)(x) = ϕσf(x) is satisfied, compare =-=[BM05a]-=-. We now prove Theorem 9.7 by constructing an appropriate pseudofunctor ϕ: modf(R) −→ wmodf(π∗,∗R).�� � � �� SECONDARY ALGEBRAS ASSOCIATED TO RING SPECTRA 27 The functor sending a fibrant replacement... |

2 |
homotopy groups, Preprint of the Max-Planck-Institut für Mathematik MPIM2006-36, http://arxiv.org/abs/math.AT/0604029
- Secondary
- 2006
(Show Context)
Citation Context ...is a relative homotopy class of homotopies between them. Similarly for the category of fibrantcofibrant spectra or symmetric spectra, compare [Bau89]. For the convenience of the reader we recall from =-=[BM05b]-=-, [BM05d] the following definition. Definition 11.1. Let n ≥ 3. For a pointed space X we define the additive secondary homotopy group, Πn,∗X which is a 0-free quadratic pair module with Πn,(0)X = Znil... |

2 |
Comparison of MacLane cohomology, Shukla and Hochschild
- Baues, Pirashvili
(Show Context)
Citation Context ...h1B ∼ = M. Morphisms are pair algebra morphisms over A and under M. Then the set of connected components of this category is in natural bijection with 3-dimensional Shukla cohomology SH 3 (A, M), see =-=[BP]-=-. Shukla cohomology is derived Hochschild cohomology. If the inclusions h1B ⊂ B1 and ∂(B1) ⊂ B0 split additively then the Shukla cohomology class associated to a pair algebra B is in the image of the ... |