Implicit atlases and virtual fundamental cycles

Implicit atlases and virtual fundamental cycles

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John Pardon, Stanford University
Fine Hall 322

An implicit atlas on a (moduli) space consists of certain auxiliary (moduli) spaces satisfying a precise set of axioms. We will summarize the construction of implicit atlases on moduli spaces of J-holomorphic curves, under the assumption of a precise "strong gluing" theorem. We will also describe an algebraic "theory of virtual fundamental cycles" (which does not use perturbation) in the abstract setting of spaces equipped with implicit atlases. This "VFC package" is sufficient to define Floer-type homology theories from a collection of (moduli) spaces equipped with a compatible system of implicit atlases.