Journal of Symplectic Geometry

Volume 18 (2020)

Number 1

$\mathrm{K}$-theoretic invariants of Hamiltonian fibrations

Pages: 251 – 289



Yasha Savelyev (CUICBAS, University of Colima, Mexico)

Egor Shelukhin (Department of Mathematics and Statistics, University of Montreal, Québec, Canada)


We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted $\mathrm{K}$-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $\mathit{Spin}^c$‑Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.

Received 19 March 2018

Accepted 7 January 2019

Published 25 March 2020