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Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to\mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi}\to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphicsections of $\pi$, Lalonde, McDuff and Polterovich proved that if $\phi$ liftsto the Hamiltonian group $\mathrm{Ham}(X,\omega)$, then the rational cohomologyof $P_{\phi}$ splits additively. We prove, with the same assumptions, that the$\mathbb{E}$-generalised cohomology of $P_{\phi}$ splits additively for anycomplex-oriented cohomology theory $\mathbb{E}$, in particular the integralcohomology splits. This class of examples includes all complex projectivevarieties equipped with a smooth morphism to $\mathbb{CP}^1$, in which case theanalogous rational result was proved by Deligne using Hodge theory. Theargument employs virtual fundamental cycles of moduli spaces of sections of$\pi$ in Morava $K$-theory and results from chromatic homotopy theory. Ourproof involves a construction of independent interest: we build globalKuranishi charts for moduli spaces of pseudo-holomorphic spheres in $X$ in aclass $\beta \in H_2(X;\mathbb{Z})$, depending on a choice of integralsymplectic form $\Omega$ on $X$ and ample Hermitian line bundle over the modulispace of one-pointed degree $d = \langle \Omega,\beta\rangle$ stable genus zerocurves in $\mathbb{CP}^d$.

Complex cobordism, Hamiltonian loops and global Kuranishi charts