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A Geometric Vietoris-Begle Theorem, with an Application to Convex Subsets of Topological Vector Lattices

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Andrew McLennan

General Topology · Submitted 25 Sep 2019

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

The authors' main claim We show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and thus an AR.

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arXiv ID1909.11347 (version 3)
Pith ID4MFJGIKB, Pith's permanent record of this paper and its review
CitationAndrew McLennan. "A Geometric Vietoris-Begle Theorem, with an Application to Convex Subsets of Topological Vector Lattices." arXiv:1909.11347 (2019). https://pith.science/paper/4MFJGIKB
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,304 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.51 · mentions code or data availability 0.01.

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

We show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and thus an AR. This is proved by verifying the hypotheses of a second result: if $X$ is a connected space that is homotopy equivalent to an ANR, $Y$ is an ANR, and $f \colon X \to Y$ is a continuous surjection such that for each $y \in Y$ and each neighborhood $V \subset Y$ of $y$, there is a neighborhood $V' \subset V$ of $y$ such that $f^{-1}(V')$ can be contracted in $f^{-1}(V)$, then $f$ is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.

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 show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and thus an AR. result 0.97 0.95 0.91 asserted 1.00
s1 This is proved by verifying the hypotheses of a second result: if $X$ is a connected space that is homotopy equivalent to an ANR, $Y$ is an ANR, and $f \colon X \to Y$ is a continuous surjection such that for each $y \in Y$ and each neighborhood $V \subset Y$ of $y$, there is a neighborhood $V' \subset V$ of $y$ such that $f^{-1}(V')$ can be contracted in $f^{-1}(V)$, then $f$ is a homotopy equivalence. method 0.81 0.77 0.07 asserted 1.00
s2 The latter result is a geometric analogue of the Vietoris-Begle theorem. interpretation 0.87 0.52 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · 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 show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and thus an AR. This is proved by verifying the hypotheses of a second result: if $X$ is a connected space that is homotopy equivalent to an ANR, $Y$ is an ANR, and $f \colon X \to Y$ is a continuous surjection such that for each $y \in Y$ and each neighborhood $V \subset Y$ of $y$, there is a neighborhood $V' \subset V$ of $y$ such that $f^{-1}(V')$ can be contracted in $f^{-1}(V)$, then $f$ is a homotopy equivalence. The latter result is a geometric analogue of the Vietoris-Begle theorem.

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