Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

Mona Merling, George Raptis, Julia Semikina

TL;DR

This work constructs a parametrized, topologized scissors-congruence $K$-theory spectrum $K^{\square}(\mathrm{Mfd}^{\partial}_d)$ that sits between the cobordism category and algebraic $K$-theory of spaces, with $\pi_0$ recovering a cut-and-paste cobordism group. It develops a bivariant framework for manifolds with tangential structures, defines topologized squares $K$-theory $K^{\square}(\mathsf{Mfd}^{\theta}_{\Delta})$, and builds a cobordism-to-$K^{\square}$ map via a parametrized cobordism category with free boundary. A compatible map to bivariant $A$-theory is constructed, aligning with the Bökstedt–Madsen map and showing factorization through the thick model $A_{\Delta}(p)$; this yields a precise comparison between $K^{\square}(\mathrm{Mfd}^{\partial}_d)$ and $A(p)$. The paper proves that $\pi_0(\Omega B\mathsf{Cobf}_n)$ matches the oriented cut-and-paste group $\mathrm{SK}^{\partial}_n$, and identifies the image of the mapping-torus subgroup in $\mathrm{SKK}_n$, linking cobordism data to scissors-congruence invariants. Open questions remain about whether the larger cobordism category with free boundary fully recovers the $K^{\square}$-theory after passing to suitable parametrizations.

Abstract

We construct a parametrized version of scissors congruence $K$-theory of manifolds, which in particular gives a topologized version of the scissors congruence $K$-theory of oriented manifolds, and we describe this spectrum as mediating between the cobordism category and usual algebraic $K$-theory of spaces. We show that on $π_0$, the scissors congruence $K$-theory of oriented manifolds agrees with a version of the cobordism category where we allow free boundaries.

Parametrized scissors congruence $K$-theory of manifolds and cobordism categories

TL;DR

This work constructs a parametrized, topologized scissors-congruence -theory spectrum that sits between the cobordism category and algebraic -theory of spaces, with recovering a cut-and-paste cobordism group. It develops a bivariant framework for manifolds with tangential structures, defines topologized squares -theory , and builds a cobordism-to- map via a parametrized cobordism category with free boundary. A compatible map to bivariant -theory is constructed, aligning with the Bökstedt–Madsen map and showing factorization through the thick model ; this yields a precise comparison between and . The paper proves that matches the oriented cut-and-paste group , and identifies the image of the mapping-torus subgroup in , linking cobordism data to scissors-congruence invariants. Open questions remain about whether the larger cobordism category with free boundary fully recovers the -theory after passing to suitable parametrizations.

Abstract

We construct a parametrized version of scissors congruence -theory of manifolds, which in particular gives a topologized version of the scissors congruence -theory of oriented manifolds, and we describe this spectrum as mediating between the cobordism category and usual algebraic -theory of spaces. We show that on , the scissors congruence -theory of oriented manifolds agrees with a version of the cobordism category where we allow free boundaries.

Paper Structure

This paper contains 23 sections, 9 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline
  4. Acknowledgments
  5. Parametrized $K$-theory of manifolds
  6. Bivariant theories
  7. The squares $K$-theory of $\theta$-manifolds
  8. Bivariant $K$-theory of manifolds
  9. Comparison with cut-and-paste invariants
  10. Recollections on classical cut-and-paste groups of manifolds
  11. The cut-and-paste groups $SK(\theta)$
  12. The map from the cobordism category
  13. The parametrized cobordism category
  14. The idea of the bivariant transformation from the cobordism category
  15. The definition of the bivariant transformation from the cobordism category
  16. ...and 8 more sections

Key Result

Theorem 3.1

Given a $\theta$-structure $\theta=(B, p, \xi)$, there is an isomorphism of groups

Figures (4)

  • Figure 1: Example of a cut-and-paste operation illustrating that $T\sqcup T$ is $SK$ equivalent to $T \# T \sqcup S^2$ (here $T$ denotes the torus)
  • Figure 2: Example of a cobordism with free boundary
  • Figure 3: Idea of the map $\mathrm{sd} N_{\,\space} \mathsf{Cobf}(\theta) \to N_{\,\space}^{\square}(\textsf{Mfd}^{\theta})$ on the level of $1$-simplices
  • Figure 4: Example of an SK-embedding L(i)

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1: WIT
  • Definition 3.2
  • Definition 3.3
  • ...and 28 more