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Constructing the virtual fundamental cycle

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SYGW05 - Symplectic geometry - celebrating the work of Simon Donaldson

Consider a  space $X$, such as a compact space of $J$-holomorphic stable maps with closed domain, that is the zero set of a Fredholm operator. This note explains how to define the  virtual fundamental class of $X$ starting from a finite dimensional reduction in the form of a Kuranishi atlas, by  representing $X$ as the zero set of a section of a (topological) orbibundle that is constructed from the atlas.     Throughout we assume that the   atlas satisfies Pardon's topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem.

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