Pith. Sign in

RECORD

Homotopically non-trivial maps with small k-dilation

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Larry Guth

Differential Geometry · Submitted 9 Sep 2007

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

The authors' main claim We construct homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n).

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID0709.1241 (version 1)
Pith IDUUZDESTR, Pith's permanent record of this paper and its review
CitationLarry Guth. "Homotopically non-trivial maps with small k-dilation." arXiv:0709.1241 (2007). https://pith.science/paper/UUZDESTR
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, 3,546 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.38 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.52 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We construct homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is m=4, n=3. Other examples include arbitrarily large values of m and n. We show that a homotopy class in pi_7(S^4) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion.

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 homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n). contribution 0.57 0.95 0.74 asserted 1.00
s1 The simplest example is m=4, n=3. result 0.84 0.72 0.06 asserted 1.00
s2 Other examples include arbitrarily large values of m and n. result 0.86 0.80 0.07 asserted 0.98
s3 We show that a homotopy class in pi_7(S^4) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion. result 0.98 0.90 0.36 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 construct homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is m=4, n=3. Other examples include arbitrarily large values of m and n. We show that a homotopy class in pi_7(S^4) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

1 arXiv paper cite this in 1 sentence

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

Papers citing it

Cites it as background 1 paper
  • The most widely discussed alternative is the direct collapse scenario, in which primordial gas in atomic-cooling haloes ($T_vir≳10^4$ K) avoids fragmentation into stars when H$_2$ cooling is suppressed by a strong Lyman–Werner radiation field, and instead collapses to form a direct collapse black hole (DCBH) seed with $M_seed∼10^4$–$10^6 M_⊙$ [ref] .

    Revisiting the Claim for a Direct-Collapse Black Hole in UHZ1 at $z=10.05$ · Introduction · 0.99

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.