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The homotopy type of the Poincaré cobordism category for surfaces

Azélie Picot

TL;DR

The paper develops a 2D oriented Poincaré cobordism category Cob_2^{SG}(X) and analyzes its classifying functor B Cob_2^{SG}(-) via Goodwillie calculus. It shows B Cob_2^{SG}(-) is not 1-excisive and identifies its first Goodwillie derivative with the Thom spectrum Th(ν_{S^2}^{haut}) over Bhaut_*(S^2), using a parametrized Pontryagin–Thom framework and a pushout with the smooth cobordism datum. The authors construct a delooping PH(2,-) that encodes the best excisive approximation and prove two main theorems: (A) a natural transformation from B Cob_2^{SG}(-) to the infinite-looped Thom data, and (C) a nontrivial obstruction in cohomology implying non-excisiveness. The work combines Poincaré duality theory, enriched category theory, and Goodwillie calculus to connect cobordism categories with Thom spectra and spectral pushouts, providing a robust framework for understanding cobordism categories beyond smooth manifolds. Overall, it advances the homotopy-theoretic understanding of cobordism categories and their excisive approximations with explicit spectral models.

Abstract

We define a version of the surface cobordism category $\mathrm{Cob}_2^{\mathrm{SG}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B\mathrm{Cob}_2^{\mathrm{SG}}(-): \mathcal{S} \rightarrow \mathcal{S}$ is not $1$-excisive. We show its first derivative $\partial_1 B\mathrm{Cob}_2^{\mathrm{SG}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B\mathrm{haut}_*^+(S^2)$.

The homotopy type of the Poincaré cobordism category for surfaces

TL;DR

The paper develops a 2D oriented Poincaré cobordism category Cob_2^{SG}(X) and analyzes its classifying functor B Cob_2^{SG}(-) via Goodwillie calculus. It shows B Cob_2^{SG}(-) is not 1-excisive and identifies its first Goodwillie derivative with the Thom spectrum Th(ν_{S^2}^{haut}) over Bhaut_*(S^2), using a parametrized Pontryagin–Thom framework and a pushout with the smooth cobordism datum. The authors construct a delooping PH(2,-) that encodes the best excisive approximation and prove two main theorems: (A) a natural transformation from B Cob_2^{SG}(-) to the infinite-looped Thom data, and (C) a nontrivial obstruction in cohomology implying non-excisiveness. The work combines Poincaré duality theory, enriched category theory, and Goodwillie calculus to connect cobordism categories with Thom spectra and spectral pushouts, providing a robust framework for understanding cobordism categories beyond smooth manifolds. Overall, it advances the homotopy-theoretic understanding of cobordism categories and their excisive approximations with explicit spectral models.

Abstract

We define a version of the surface cobordism category over a base space where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor is not -excisive. We show its first derivative in the Goodwillie sense is equivalent to a Thom spectrum over .

Paper Structure

This paper contains 22 sections, 63 theorems, 273 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The main character
  3. Main Results
  4. Outline of the paper
  5. Constructing a Poincaré Cobordism Category
  6. Recollections on Poincaré Complexes
  7. A pointset model for B haut partial(Q,P)
  8. The Poincaré Cobordism Category as a category internal to simplicial sets
  9. Delooping the functor B PCobdG(-)
  10. A formula for B sCobtwo(X)
  11. Diffeomorphisms versus self-homotopy equivalences of surfaces
  12. Deleting the spheres: Proof of Theorem \ref{['thmB']}
  13. Quillen's Theorem B for the reduction functor
  14. Parametrized Pontryagin-Thom Construction for Fibrations
  15. Dualizing objects and Spherical Fibrations
  16. ...and 7 more sections

Key Result

Theorem A

The first approximation map is equivalent to the natural transformation where $P$ is the pullback of the cospan in the following diagram and $\gamma$ is induced by $\alpha$: \begin{tikzcd} {} & \textcolor{mypurple}{{\mathrm{BCob}_2^{\mathrm{SG}}(-)}} \\ && \textcolor{mypurple}{P} & {\Omega^{\infty-1} (\mathrm{Th}(\nu_{S^2}^{\mathrm{haut}}) \wedge \Sigma^{\infty}_+-)}

Figures (7)

  • Figure 1: The relative thickening $(U_1,U_0)$
  • Figure 2: On the left: examples of $0$-simplices of $\psi_0^{\mathrm{SG}}(2)$. On the right: a $1$-simplex of $\psi_0^{\mathrm{SG}}(1)$.
  • Figure 3: Three $0$-simplices $U_1,U_2$ and $U_3$ of $\psi_{1}^{\mathrm{SG}}(2)$
  • Figure 4: A $0$-simplex in $\psi_1^{\mathrm{(S)G}}(2,1)$, or a $0$-simplex of the simplicial set of morphisms of $\mathrm{Cob}_1^{\mathrm{(S)G}}(2)$
  • Figure 5: Action of the zigzag of functors $\mathrm{Cob}_{\mathrm{d}}^{\mathrm{(S)O}} \leftarrow \mathrm{Cob}_d^{\mathrm{(S)O,tub}} \rightarrow \mathrm{Cob}_{\mathrm{d}}^{\mathrm{(S)G}}$ on $0$-simplices of objects
  • ...and 2 more figures

Theorems & Definitions (147)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary 1
  • Definition 1
  • Definition 2
  • Theorem 2.1: Spivak Normal Fibration
  • Definition 3
  • Theorem 2.2: Theorem $4.2$ in Wall1967PoincareCI, Corollary $3$ and Theorem $2$ in Eckmann1980
  • Proposition 1
  • ...and 137 more