Spaces over BO are thickened manifolds

TL;DR

This work shows that the ∞‑category of thickened compact manifolds, with morphisms given by codimension‑zero embeddings, is equivalent to the ∞‑category of finite spaces over BO. It achieves this by constructing a concrete pushout framework for thickened manifolds, reducing gluing to homotopy pullbacks of spaces, and proving that the core functor to spaces over BO is fully faithful and essentially surjective. The analysis leverages a smoothing construction and a thickened mapping‑space model (H^(∞)), avoiding the h‑principle and smooth approximation. The result provides a robust, presentable framework with potential applications to Liouville sectors in exact symplectic geometry, where thickened gluing behaves well in the homotopy category.

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 x [-1,1]. We prove that the resulting infinity-category of thickened smooth manifolds is equivalent to the infinity-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 infinity-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.

Figures (3)

• Figure 1: Illustrations of Choice \ref{['choice. smoothing corner']}. In all illustrations is a neighborhood of $\partial W \times \{-1\}_u \times \{0\}_v$ inside $P'$, projected onto the square $(-1-\epsilon,-1+\epsilon)_u \times [-\epsilon,\epsilon)_v$. As in Figure \ref{["figure. P and P'"]}, the $u$ direction is drawn horizontally. The $v$ direction is drawn vertically. In $(i)$, $\partial W \times \{-1\}_u \times \{0\}_v$ is the fiber over the point where the solid line intersects the dashed line. The fiber above the dashed line is the locus $\partial W \times (-1-\epsilon,-1+\epsilon)_u \times \{0\}_v$. The region below the dashed locus represents the region $\partial W \times (-1-\epsilon,-1+\epsilon)_u \times [-\epsilon,0)_v$. In $(ii)$ is now a thin dotted line, indicating the locus $\partial W \times (-1-\epsilon,-1+\epsilon)_u \times \{ -\epsilon\}_v$. The new curve in $(iii)$ indicates $\gamma$. The region consisting of points to the left of or below $\gamma$ is $R_X$. Note that $R_X$ intersects some portion of the thick dashed line, but does not contain all of the thick dashed line.
• Figure 2: A cartoon of $P'$ on the left (with regions indicated), and of $P$ on the right. The intervals runs horizontally, while the vertical directions are meant to convey movement in the $W, X, Y$ directions. We have drawn not only $W \times [-1,1]$, but the whole $W \times [-1-\epsilon,1+\epsilon]$ to emphasize that $W$ admits a collar inside $X \times [-2,-1]$ and inside $Y \times [1,2]$. The drawing of $P$ (note that the cornered are rounded) is meant to indicate that $P$ is obtained by removing the not-obviously-smooth regions where the rectangles in the $P'$ drawing intersect.
• Figure 3: An isotopy of the map \ref{['eqn. the two variable map u and s']} to the identity. The horizontal direction is the $[-2,2]_u$ direction, while the vertical direction is the $I_s$ direction, with orientations indicated (in the directions of increasing values of $u$ and of $s$). The dotted arrows indicate isotopies, with the images of each stage of the isotopy indicated as the shaded regions. Note also that the dashed lines indicate the lines $u = \pm 1$ and their images under various stages of the isotopy.

Theorems & Definitions (69)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4: Precedents
• Remark 1.5
• Corollary 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Example 1.10: A model for the terminal object of $\mathrm{Ind}(\mathcal{M}\!\operatorname{fld}^{\diamond})$