Bundle transfer of L–homology orientation classes for singular spaces

We consider transfer maps on ordinary homology, bordism of singular spaces and homology with coefficients in Ranicki's symmetric L-spectrum, associated to block bundles with closed oriented PL manifold fiber and compact polyhedral base.We prove that if the base polyhedron is a Witt space, for example a pure-dimensional compact complex algebraic variety, then the symmetric L-homology orientation of the base, constructed by Laures, Mc-Clure and the author, transfers to the L-homology orientation of the total space.We deduce from this that the Cheeger-Goresky-MacPherson L-class of the base transfers to the product of the L-class of the total space with the cohomological L-class of the stable vertical normal microbundle. CONTENTS1. Introduction 1 2. Stable Vertical Normal Block Bundles 4 3.The PL Umkehr Map 8 4. The Thom Homomorphism, Mock Bundles and Witt Spaces 9 5. Block Bundle Transfer 12 5.1.Block Transfer on Ordinary Homology 13 5.2.Block Transfer on Witt Bordism 14 5.3.Block Transfer on L • -Homology 15 6.Geometric Pullback Transfer on Bordism 18 7. Transfer of the L • -Homology Fundamental Class 32 8. Behavior of the Cheeger-Goresky-MacPherson L-class Under Transfer 33 9. Normally Nonsingular Maps 34