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Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies

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Tahl Nowik

Geometric Topology · Submitted 3 Sep 1999

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

The authors' main claim We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

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arXiv IDmath/9909020 (version 1)
Pith IDQ4E6PW2Y, Pith's permanent record of this paper and its review
CitationTahl Nowik. "Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies." arXiv:math/9909020 (1999). https://pith.science/paper/Q4E6PW2Y
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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.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,808 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.15 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.86 · compares with prior results 0.49 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We denote this number by Q(i,i') \in Z/2. We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

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 Let F be a closed orientable surface. background 1.00 0.07 0.01 asserted 0.60
s1 If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. background 1.00 0.04 0.06 asserted 1.00
s2 We denote this number by Q(i,i') \in Z/2. background 0.66 0.39 0.04 asserted 0.80
s3 We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively. result 0.96 0.95 0.92 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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

Let F be a closed orientable surface. If i,i':F \to R^3 are two regularly homotopic generic immersions, then it has been shown in [N] that all generic regular homotopies between i and i' have the same number mod 2 of quadruple points. We denote this number by Q(i,i') \in Z/2. We show that for any generic immersion i:F\to R^3 and any diffeomorphism h:F\to F such that i and i\circ h are regularly homotopic, Q(i,i\circ h) = (rank(h_*-Id) + (n+1)e(h)) mod 2, where h_* is the map induced by h on H_1(F,Z/2), n is the genus of F and e(h) is 0 or 1 according to whether h is orientation preserving or reversing, respectively.

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