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arXiv · 2504.01810

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

Mona Merling·George Raptis·Julia Semikina
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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 $\pi_0$ 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 $K(\mathbb{Z})$ detects on $\pi_1$ the Kervaire semicharacteristic.

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Mona Merling, George Raptis, Julia Semikina. 2025-04-02. Parametrized scissors congruence $K$-theory of manifolds and cobordism categories. https://arxiv.org/abs/2504.01810

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