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Cut and paste invariants of manifolds via algebraic \(K\)-theory. (English) Zbl 1490.19003

Summary: Recent work of J. A. Campbell and I. Zakharevich [“Devissage and localization for the Grothendieck spectrum of varieties”, Preprint, arXiv:1811.08014] has focused on building machinery for studying scissors congruence problems via algebraic \(K\)-theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a \(K\)-theory space that recovers the classical SK (“schneiden und kleben,” German for “cut and paste”) groups for manifolds on \(\pi_0\), and we construct a derived version of the Euler characteristic.

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
19D99 Higher algebraic \(K\)-theory
19A49 \(K_0\) of other rings
57K99 Low-dimensional topology in specific dimensions
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References:

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