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Homological stability for coloured configuration spaces and symmetric complements

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TriThang Tran

Algebraic Topology · Submitted 22 Dec 2013

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

The authors' main claim We prove a homological stability theorem for certain complements of symmetric spaces.

how it is stated · asserted 1.00 read the record →

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arXiv ID1312.6327 (version 1)
Pith IDMHJQMSJ2, Pith's permanent record of this paper and its review
CitationTriThang Tran. "Homological stability for coloured configuration spaces and symmetric complements." arXiv:1312.6327 (2013). https://pith.science/paper/MHJQMSJ2
Claim paper

The compiled record

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.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,923 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.67 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We prove a homological stability theorem for certain complements of symmetric spaces. This is a variant of a conjecture by Vakil and Matchett Wood for subspaces of $\mathrm{Sym}^n(X)$ where $X$ is an open manifold admitting a boundary. To do this we prove a homological stability result for a type of "coloured" configuration space by adding points of the same colour.

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 prove a homological stability theorem for certain complements of symmetric spaces. result 0.52 0.96 0.94 asserted 1.00
s1 This is a variant of a conjecture by Vakil and Matchett Wood for subspaces of $\mathrm{Sym}^n(X)$ where $X$ is an open manifold admitting a boundary. background 0.89 0.07 0.07 asserted 0.93
s2 To do this we prove a homological stability result for a type of "coloured" configuration space by adding points of the same colour. method 0.92 0.94 0.12 asserted 1.00

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

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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 prove a homological stability theorem for certain complements of symmetric spaces. This is a variant of a conjecture by Vakil and Matchett Wood for subspaces of $\mathrm{Sym}^n(X)$ where $X$ is an open manifold admitting a boundary. To do this we prove a homological stability result for a type of "coloured" configuration space by adding points of the same colour.

Human review

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How the literature uses it

2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 8 sentences without a confident label

Uses the method or tool of 1 paper

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