Relating cut and paste invariants and TQFTs

Rovi, Carmen; Schoenbauer, Matthew

doi:10.1093/qmath/haab044

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Relating cut and paste invariants and TQFTs

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Rovi, Carmen
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1093/qmath/haab044
(Publisher version)

https://doi.org/10.48550/arXiv.1803.02939
(Preprint)

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Citation

Rovi, C., & Schoenbauer, M. (2022). Relating cut and paste invariants and TQFTs. Quarterly Journal of Mathematics, 73(2), 579-607. doi:10.1093/qmath/haab044.

Cite as: https://hdl.handle.net/21.11116/0000-000A-C0A5-2

Abstract

In this paper, we shall be concerned with a relation between TQFTs and cut
and paste invariants introduced by Karras, Kreck, Neumann and Ossa. Cut and
paste invariants, or SK invariants, are functions on the set of smooth
manifolds that are invariant under the cutting and pasting operation. Central
to the work in this paper are also SKK invariants, whose values on cut and
paste equivalent manifolds differ by an error term depending only on the
glueing diffeomorphism. Here we investigate a surprisingly natural group
homomorphism between the group of invertible TQFTs and the group of SKK
invariants and describe how these groups fit into an exact sequence. We
conclude in particular that all positive real-valued SKK invariants can be
realized as restrictions of invertible TQFTs. All manifolds are smooth and
oriented throughout unless stated otherwise.