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The homotopy type of the topological cobordism category

Mauricio Gomez Lopez, Alexander Kupers

TL;DR

This work extends the GMTW framework from smooth to topological manifolds, establishing a stable description of the topological cobordism category in terms of a Thom spectrum: for $d\neq 4$, $B\mathsf{Cob}^{\rm Top}(d)\simeq \Omega^{\infty-1}MT^{\rm Top}(d)$. The authors develop topological analogues of the smooth machinery—spaces of topological submanifolds, universal tangential structures, and spectra—and prove an excision principle for tangential data, together with smoothing theory to compare topological and smooth spectra. They then prove the topological scanning equivalence and show that the classifying spaces are infinite loop spaces modeled by the corresponding Thom spectra, including variants with tangential structures and boundary. The results bridge topological and smooth cobordism categories and lay groundwork for PL analogues, with implications for related index and surgery-theory frameworks. Overall, the paper provides a comprehensive, technically robust topological analogue of the GMTW theorem and its parametrized tangential-generalizations.

Abstract

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $Ω^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle over $BTop(d)$. We also give versions with tangential structures and boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.

The homotopy type of the topological cobordism category

TL;DR

This work extends the GMTW framework from smooth to topological manifolds, establishing a stable description of the topological cobordism category in terms of a Thom spectrum: for , . The authors develop topological analogues of the smooth machinery—spaces of topological submanifolds, universal tangential structures, and spectra—and prove an excision principle for tangential data, together with smoothing theory to compare topological and smooth spectra. They then prove the topological scanning equivalence and show that the classifying spaces are infinite loop spaces modeled by the corresponding Thom spectra, including variants with tangential structures and boundary. The results bridge topological and smooth cobordism categories and lay groundwork for PL analogues, with implications for related index and surgery-theory frameworks. Overall, the paper provides a comprehensive, technically robust topological analogue of the GMTW theorem and its parametrized tangential-generalizations.

Abstract

We define a cobordism category of topological manifolds and prove that if its classifying space is weakly equivalent to , where is the Thom spectrum of the inverse of the canonical bundle over . We also give versions with tangential structures and boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.

Paper Structure

This paper contains 52 sections, 62 theorems, 67 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Conventions
  3. Acknowledgements
  4. Recollection on smooth cobordism categories
  5. Spaces of smooth manifolds
  6. Spaces of smooth submanifolds
  7. Spaces of smooth submanifolds with tangential structure
  8. Simplicial sets of smooth submanifolds
  9. The universal tangential structure
  10. An aside on spaces of bundle maps
  11. Spectra of smooth submanifolds
  12. Smooth cobordism categories
  13. The definition of the smooth cobordism category
  14. The classifying space of the smooth cobordism category
  15. The scanning argument
  16. ...and 37 more sections

Key Result

Theorem 1

Let $d \neq 4$, then $B\mathsf{Cob}^{\rm Top}(d) \simeq \Omega^{\infty-1}MT{\rm Top}(d)$.

Figures (9)

  • Figure 1: A path in $\Psi_1^{\rm Diff}(\mathbb{R}^2)$. Note that parts of manifolds may "disappear at infinity."
  • Figure 2: Two morphisms in ${\rm Cob}^{\rm Diff}(1,2)$ and their composition.
  • Figure 3: We need to extend $X$ rightwards compatibly with $Y_i$. This will be done using isotopy extension.
  • Figure 4: The starting and end point of the family obtained by applying the argument of Lemma \ref{['lem.sourcetargetfib']} to the Figure \ref{['fig:kanfibrationinput']}.
  • Figure 5: A $0$-simplex in $\psi^{{\rm Top},\xi}(1,2,1)((t_0,\overline{Y}_0),(t_1,\overline{Y}_1))$.
  • ...and 4 more figures

Theorems & Definitions (167)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 157 more