$L$-theory Characteristic Classes

TL;DR

This paper proves that the local geometric data of $L$-spectra can be identified with classical characteristic classes, resolving a long-standing open problem from the 1960s and 1970s.

Contribution

It establishes an equivalence between Levitt-Ranicki's connective $L$-orientations and classical $2$-local characteristic classes, connecting geometric and algebraic $L$-theory.

Findings

Equivalence between connective $L$-orientations and classical characteristic classes.

Construction of geometric homotopy equivalences linking $L$-spectra to Eilenberg-MacLane spectra.

Reproof of the local structure of $L$-spectra.

Abstract

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.