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Universal Toda brackets of ring spectra

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Steffen Sagave

Algebraic Topology · Submitted 27 Nov 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra.

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arXiv IDmath/0611808 (version 3)
Pith IDFUSRXW25, Pith's permanent record of this paper and its review
CitationSteffen Sagave. "Universal Toda brackets of ring spectra." arXiv:math/0611808 (2006). https://pith.science/paper/FUSRXW25
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,152 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We construct and examine the universal Toda bracket of a highly structured ring spectrum R. contribution 1.00 0.96 0.20 asserted 1.00
s1 This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. contribution 0.52 0.70 0.29 asserted 1.00
s2 It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. interpretation 0.64 0.73 0.25 asserted 0.98
s3 For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. contribution 1.00 0.86 0.06 asserted 0.95
s4 The real and complex K-theory spectra serve as our main examples. contribution 0.49 0.64 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples.

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