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On the K-theory of matroids with Tutte coverings

Luigi Caputi, Sabino Di Trani

Abstract

The aim of this work is to explicitly compute the K-theory of the category of matroids with respect to the covering family of Tutte coverings. In particular, we show that this is equivalent to the K-theory spectrum of the category of graphic matroids on looped forests, with the covering family generated by isomorphisms. Further, we show that this yields an equivalence of $C_2$-spectra.

On the K-theory of matroids with Tutte coverings

Abstract

The aim of this work is to explicitly compute the K-theory of the category of matroids with respect to the covering family of Tutte coverings. In particular, we show that this is equivalent to the K-theory spectrum of the category of graphic matroids on looped forests, with the covering family generated by isomorphisms. Further, we show that this yields an equivalence of -spectra.
Paper Structure (5 sections, 15 theorems, 28 equations)

This paper contains 5 sections, 15 theorems, 28 equations.

Table of Contents

  1. Matroids and their Algebraic Invariants
  2. $K$-theory of categories with covering families
  3. Matroids with Tutte coverings
  4. Indecomposable Coverings
  5. The $K$-theory of (indecomposable) matroids

Key Result

Theorem 1

There is an equivalence of spectra where $S_p$ denotes the symmetric group on $p$ elements.

Theorems & Definitions (41)

  • Theorem 1: cf. Corollary \ref{['cor:splitting']}
  • Example 1.1: Uniform Matroids
  • Example 1.2: Graphic Matroids
  • Definition 1.3: Direct Sum Matroid
  • Definition 1.4: Dual Matroid
  • Definition 1.5
  • Definition 1.6: Deletion Matroid
  • Definition 1.7: Contraction Matroid
  • Remark 1.8
  • Proposition 1.9
  • ...and 31 more