L-theory characteristic classes

The [Formula: see text]-local homotopy types of the [Formula: see text]-spectra are well understood. However, the problem of identifying this local information with geometric data for bundles remained open for decades. Specifically, it has been an open question whether Levitt–Ranicki’s theory of connective [Formula: see text]-theory orientations for spherical fibrations is equivalent to the [Formula: see text]-local characteristic classes constructed by Brumfiel–Morgan. In [L. Taylor and B. Williams, Surgery spaces: Formulae and structure, in Algebraic Topology Waterloo 1978 (Springer, 1979), pp. 170–195], Taylor and Williams resolved part of this problem. In this paper, we provide a complete affirmative answer for the remaining part. Our approach is to generalize the [Formula: see text] manifolds to the chain level and to prove that the bordism groups of [Formula: see text] Poincaré chain complexes are isomorphic to the homotopy groups of [Formula: see text]-spectra with coefficient [Formula: see text].