Note

An Integral Comparison of Signature Classes in K- and L-Theory

Author: Wei Yang · Added: September 26, 2026

Abstract

Let $M$ be a closed oriented smooth manifold of dimension $n$ . We prove that the comparison map from connective complex $K$ -theory to the $L$ -theory of $\mathbb{C}$ sends the signature class of $M$ to $2^{\lfloor n/2\rfloor}$ times its Sullivan-Ranicki fundamental class, integrally. This answers Problem 8.4 of Land, Nikolaus, and Schlichting. The proof lifts both classes to the Thom spectrum of the virtual bundle $\varepsilon^n-\gamma_n$ over $BSO(n)$ . Ebert’s universal signature symbol determines their rational comparison. At the prime $2$ , the remaining difference is detected by rationalization and reduction modulo $2$ , using the integral cohomology of $BSO(n)$ and the Eilenberg-MacLane decomposition of connective $L$ -theory. The result gives an integral comparison for classes mapped to an arbitrary space, a specified divisibility statement, and a corresponding equality of assembled signature classes.