$L$-theory Characteristic Classes

Although the local information of the $L$-spectra is well understood, the problem of whether this local information can be identified with the geometric data for bundles remains open for decades, which was originally raised in the 1960s and 1970s by Sullivan, Brumfiel, Taylor-Williams and others independently. In this paper, we provide an affirmative answer by proving that Levitt-Ranicki's theory of connective $L$-orientations for $TOP$ bundles and spherical fibrations is equivalent to the $2$-local characteristic classes constructed by Brumfiel-Morgan's, Madsen-Milgram's and Morgan-Sullivan's, as well as Sullivan's odd-prime-local real $K$-theory orientation. A key step in our proof involves constructing more geometric homotopy equivalences from the $2$-local quadratic, symmetric and normal connective $L$-spectra to products of Eilenberg-Maclane spectra and those from odd-local quadratic and symmetric connective $L$-spectra to the connective real $K$-spectra. This approach reproves the known local structure of $L$-spectra.