Minimal atomic complexes

Baker, Andrew; May, J. P.

doi:10.1016/j.top.2003.09.004

Cited by 17 publications

(38 citation statements)

References 16 publications

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“…However, this co-H -structure is not necessarily co-associative and neither are the inclusion and projection maps necessarily co-H -maps. Furthermore, each Y i is minimal atomic in the sense of [5], and cannot be equivalent to a suspension except in the case i = p − 1 when it does desuspend [14].…”

Section: A P-local Splittingmentioning

confidence: 99%

Quasisymmetric functions from a topological point of view

Baker¹,

Richter²

2008

MATH. SCAND.

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It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space CP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on CP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU . The canonical Thom spectrum over CP ∞ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

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Section: A P-local Splittingmentioning

confidence: 99%

Quasisymmetric functions from a topological point of view

Baker¹,

Richter²

2008

MATH. SCAND.

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“…In [2,Theorem 3.3] it was shown that every p-local CW complex of finite-type is equivalent to a minimal one, so such minimal complexes exist in abundance.…”

Section: Minimal Atomic P-local Commutative S-algebrasmentioning

confidence: 99%

“…Proof The details follow from the proof of [2,Theorem 3.3], replacing ordinary homology with HAQ * (−), mutatis mutandis.…”

Section: Theorem 32 Let R Be P-local Cw Commutative S-algebra With mentioning

confidence: 99%

“…This allows us to revisit the results of [2] in the context of simply connected p-local CW commutative S-algebras. The notion of a nuclear algebra appears in [11,Definition 2.7], while that of an atomic algebra appears in [11,Definition 2.8].…”

Section: Theorem 32 Let R Be P-local Cw Commutative S-algebra With mentioning

confidence: 99%

“…The main goal is to develop arguments based on skeletal filtrations with a view to continuing the work begun in [11] by extending results of [2] to the case of CW commutative S-algebras. Some of this work originally appeared in the second author's Ph.D. Thesis [9], but we go further and use it to investigate some examples.…”

Section: Introductionmentioning

confidence: 99%

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Topological André-Quillen homology for cellular commutative S-algebras

Baker

Gilmour

Reinhard

2008

Abh. Math. Semin. Univ. Hambg.

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Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic commutative S-algebras. We would like to thank M. Basterra, P. Kropholler, M. Mandell, P. May, B. Richter, J. Rognes and S. Sagave for numerous helpful comments. We are also very grateful to the referee for encouraging us to rethink significantly issues of notation and structure, thus improving the structure of the paper. Present address:A. Baker ( )

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Adams filtration and generalized Hurewicz maps for infinite loopspaces

Kuhn

2018

Invent. math.

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We study the Hurewicz map h * : π * (X) → R * (Ω ∞ X)where Ω ∞ X is the 0th space of a spectrum X, and R * is the generalized homology theory associated to a connective commutative S-algebra R.We prove that the decreasing filtration of the domain associated to an R-based Adams resolution is compatible with a filtration of the range associated to the augmentation ideal filtration of the augmented commutative S-algebra Σ ∞ (Ω ∞ X)+. The proof of our main theorem makes much use of composition properties of this filtration and its interaction with Topological André-Quillen homology.An application is a Connectivity Theorem: Localize away from (p−1)! and suppose X is (c − 1)-connected with c > 0. If α ∈ π * (X) has Adams filtration s and |α| < cp s , then h * (α) = 0 ∈ R * (Ω ∞ X). When specialized to mod p homology, this implies a Finiteness Theorem: if H * (X; Z/p) is finitely presented as a module over the Steenrod algebra, then the image of the Hurewicz map in H * (Ω ∞ X; Z/p) is finite. We illustrate these theorems with calculations of the mod 2 Hurewicz image of BO, its connected covers, and Ω ∞ tmf , and the mod p Hurewicz image of all the spaces in the BP and BP n spectra. En route, we get new proofs of theorems of Milnor and Wilson.In the special case when X is the suspension spectrum of a space Z and R = HZ/2, we recover results announced by Lannes and Zarati in the 1980s, relating the Adams filtration of π S * (Z) to Dyer-Lashof length in H * (QZ; Z/2), and generalize them to all primes p. For any X, we also get parallel results for the Hurewicz map for Morava E-theory, where π * (X) is now given the Adams-Novikov filtration.

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Minimal atomic complexes

Cited by 17 publications

References 16 publications

Quasisymmetric functions from a topological point of view

Quasisymmetric functions from a topological point of view

Topological André-Quillen homology for cellular commutative S-algebras

Adams filtration and generalized Hurewicz maps for infinite loopspaces

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