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A K-theory spectrum for cobordism cut and paste groups

Renee S. Hoekzema, Carmen Rovi, Julia Semikina

TL;DR

This work extends cut-and-paste cobordism theory to manifolds with boundary by defining the cobordism cut-and-paste group $\overline{\mathrm{SK}}^{\partial}_n$ and relating it to a cobordism theory with trivial boundary via exact sequences. It introduces a three-direction K-theory-with-cubes construction $K^{\mancube}(\overline{\mathrm{Mfd}}^{\partial}_n)$, whose $\pi_0$ recovers $\overline{\mathrm{SK}}^{\partial}_n$, and a quotient-lifting map from the classical $K^{\square}$-theory that lifts the map $\mathrm{SK}^{\partial}_n \to \overline{\mathrm{SK}}^{\partial}_n$. The core technical development is the quadrisimplicial object $X^{*}_{\bullet,\bullet,\bullet}$, encoding manifolds with boundary, SK-embeddings, and trivial-boundary cobordisms, from which an infinite loop space spectrum is obtained via Segal-Gamma machinery. The framework provides a refined, spectrum-level understanding of cut-and-paste cobordism invariants with boundary and connects to cobordism-category fibrations, offering tools for further computations and structural insights in bordism theory.

Abstract

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $π_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$.

A K-theory spectrum for cobordism cut and paste groups

TL;DR

This work extends cut-and-paste cobordism theory to manifolds with boundary by defining the cobordism cut-and-paste group and relating it to a cobordism theory with trivial boundary via exact sequences. It introduces a three-direction K-theory-with-cubes construction , whose recovers , and a quotient-lifting map from the classical -theory that lifts the map . The core technical development is the quadrisimplicial object , encoding manifolds with boundary, SK-embeddings, and trivial-boundary cobordisms, from which an infinite loop space spectrum is obtained via Segal-Gamma machinery. The framework provides a refined, spectrum-level understanding of cut-and-paste cobordism invariants with boundary and connects to cobordism-category fibrations, offering tools for further computations and structural insights in bordism theory.

Abstract

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group is defined. In this paper, we extend this definition to manifolds with boundary obtaining and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on the cobordism cut and paste groups of manifolds with boundary, , and a map of spectra that lifts the canonical quotient map .
Paper Structure (17 sections, 12 theorems, 74 equations, 4 figures)

This paper contains 17 sections, 12 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cobordism cut and paste groups of manifolds with boundary
  3. Classical $\overline{\mathrm{SK}}$ groups
  4. Cobordism with trivial boundary
  5. $\overline{\mathrm{SK}}$ groups of manifolds with boundary
  6. Fibrations of cobordism categories
  7. Cubes of manifolds with boundary
  8. The simplicial set $X^*_{0,0,0}$
  9. Simplicial sets $X^*_{1,0,0}$ and $X^*_{0,1,0}$
  10. The simplicial set $X^{*}_{0,0,1}$
  11. The simplicial set $X^*_{1,1,0}$
  12. Simplicial sets $X^*_{1,0,1}$ and $X^*_{0,1,1}$
  13. The simplicial set $X^*_{1,1,1}$
  14. General construction of $X^*_{\bullet, \bullet,\bullet}$
  15. Recovering cobordism cut and paste groups as $K^\text{\mancube}_0$
  16. ...and 2 more sections

Key Result

Theorem A

There is an isomorphism $K_0^\text{\mancube}(\overline{\mathrm{Mfd}}^\partial_n) \cong \overline{\mathrm{SK}}^{\partial}_n$.

Figures (4)

  • Figure 1: Relation in $\Omega^{\partial}$
  • Figure 2: The $\epsilon_c$ and $\epsilon_{\partial}$ collars on a cobordism with trivial boundary.
  • Figure 3: SK embedding of cobordisms
  • Figure 4: Composing squares (or cubes) that coincide in only one vertex is not always possible

Theorems & Definitions (33)

  • Theorem A
  • Theorem B
  • Definition 2.2.1
  • Definition 2.2.2
  • Remark 2.2.3
  • Lemma 2.2.4
  • proof
  • Definition 2.3.1
  • Lemma 2.3.2
  • proof
  • ...and 23 more