A categorified excision principle for elliptic symbol families

Markus Upmeier* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle that allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications will appear in companion papers Joyce–Tanaka–Upmeier [16], Joyce–Upmeier [17] and Cao–Gross–Joyce [8]
Original languageEnglish
Pages (from-to)1099–1132
Number of pages34
JournalQuarterly Journal of Mathematics
Issue number3
Early online date13 Jan 2021
Publication statusPublished - 1 Sept 2021

Bibliographical note


The author was funded by DFG grant UP 85/3-1, by grant UP 85/2-1 of the DFG priority program SPP 2026 ‘Geometry at Infinity’, and by the ‘Centre for Quantum Geometry of Moduli Spaces’ of the DNRF. The author would like to thank Dominic Joyce for numerous discussions. The author would also like to thank Arkadij Bojko, Simon Donaldson, Sebastian Goette, Jacob Gross, Yuuji Tanaka and Thomas Walpuski for helpful conversations.


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