  Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the
purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic
curves that cannot be regularized by geometric methods. Their core idea was to
build such a cycle by patching local finite dimensional reductions, given by
smooth sections that are equivariant under a finite isotropy group.
  Building on our notions of topological Kuranishi atlases and perturbation
constructions in the case of trivial isotropy, we develop a theory of Kuranishi
atlases and cobordisms that transparently resolves the challenges posed by
nontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlases
both a virtual moduli cycle (VMC - a cobordism class of weighted branched
manifolds) and a virtual fundamental class (VFC - a Cech homology class).
