Spaces over BO are thickened manifolds

Tanaka, Hiro Lee

doi:10.2140/agt.2025.25.4287

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Spaces over $\mathrm{BO}$ are thickened manifolds

Hiro Lee Tanaka

Algebraic & Geometric Topology 25 (2025) 4287–4319

DOI: 10.2140/agt.2025.25.4287

Abstract

Consider the topologically enriched category of compact smooth manifolds (possibly with corners), with morphisms given by codimension-zero smooth embeddings. Now formally identify any object $X$ with its thickening $X \times [- 1, 1]$ . We prove that the resulting $\infty$ -category of thickened smooth manifolds is equivalent to the $\infty$ -category of finite spaces over $BO$ . (This is one formalization of the philosophy that embedding questions become homotopy-theoretic upon passage to higher dimensions.) The central tool is a geometric construction of pushouts in this $\infty$ -category, carried out with an eye toward proving analogous results in exact symplectic geometry. Notably, the proof never invokes smooth approximation nor any $h$ -principle.

Keywords

manifolds, embeddings

Mathematical Subject Classification

Primary: 57N65, 57R40

References

Publication

Received: 24 March 2024

Revised: 18 August 2024

Accepted: 16 September 2024

Published: 29 October 2025

Authors

Hiro Lee Tanaka
Department of Mathematics Texas State University San Marcos, TX United States

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Volume 25, issue 7 (2025)