I will discuss joint work with McLean and Smith, lifting the results of Seidel, Lalonde, and McDuff concerning the topology of Hamiltonian fibrations over the 2-sphere from rational cohomology to complex cobordism. In addition to the use of Morava K-theory (as in the earlier work with Blumberg on the Arnold Conjecture), the essential new ingredient is the construction of global Kuranishi charts for genus 0 pseudo-holomorphic curves; i.e. their realisation as quotients of zero loci of sections of equivariant vector bundles on manifolds.

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