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Rational toral rank of a map

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Toshihiro Yamaguchi

Algebraic Topology · Submitted 1 Oct 2013

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

The authors' main claim As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$.

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arXiv ID1310.0105 (version 1)
Pith ID6VIZWUUZ, Pith's permanent record of this paper and its review
CitationToshihiro Yamaguchi. "Rational toral rank of a map." arXiv:1310.0105 (2013). https://pith.science/paper/6VIZWUUZ
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,813 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.10 · mentions code or data availability 0.02.

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

Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over $\Q$ \cite{FHT}.

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 $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. background 1.00 0.06 0.02 asserted 0.69
s1 The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. background 1.00 0.05 0.04 asserted 1.00
s2 As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. contribution 1.00 0.84 0.12 asserted 1.00
s3 We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over $\Q$ \cite{FHT}. method 0.94 0.51 0.04 asserted 0.92

What kind of paper the classifier reports this is: theoretical result 0.72 · evidence from proof or derivation 0.98.

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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

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Original abstract

Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over $\Q$ \cite{FHT}.

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