L -Theory Characteristic Classes

The p-local homotopy types of the L-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 L-theory orientations for spherical fibrations is equivalent to the 2-local characteristic classes constructed by Brumfiel-Morgan. In [24] 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 ℤ/n manifolds to chain level and to prove that the bordism groups of ℤ/n Poincar’e chain complexes are isomorphic to the homotopy groups of L-spectra with coefficient ℤ/n.