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

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.

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Received 19 March 2018

Accepted 7 January 2019

Published 25 March 2020