Parametrized scissors congruence K-theory of manifolds and cobordism categories

2025/04/02 by Mona Merling, Merling, Mona, George Raptis +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2504.01810

openalex publication_date 2025/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a parametrized version of scissors congruence K-theory of manifolds with tangential structure, which includes a topologized version of the scissors congruence K-theory of oriented manifolds as a special case. We examine the relation of this K-theory spectrum with cut-and-paste invariants, the (parametrized) cobordism category and with (bivariant) algebraic K-theory of spaces. We show that the scissors congruence K-theory of oriented manifolds agrees on π₀ with a version of the oriented cobordism category where we allow cobordisms to have free boundaries. Lastly, we show that the spectrum level refinement of the Euler characteristic from the scissors congruence K-theory to K(ℤ) detects on π₁ the Kervaire semicharacteristic.

Cited by

• A genuine G-spectrum for the cut-and-paste K-theory of G-manifolds

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