Transfer and the spectrum–level Siegel—Sullivan KO–orientation for singular spaces

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel–Sullivan orientations, defined when the singular spaces are compact pseudomanifolds satisfying the Witt condition, for example pure-dimensional compact complex algebraic varieties. This holds for bundle transfers associated to block bundles with manifold fibers as well as for Gysin restrictions associated to normally nonsingular inclusions. Our method is based on constructing a lift of the Siegel–Sullivan orientation to a morphism of highly structured ring spectra which factors through L-theory.