Mod-2 Equivalence of the K-theoretic Euler and Signature Classes

TL;DR

This paper demonstrates that the symbol classes of the de Rham and signature operators are equivalent mod 2 in K-theory on closed even-dimensional manifolds, with an extension to equivariant cases involving Euler characteristic and multi-signature.

Contribution

It establishes a mod 2 equivalence of symbol classes for de Rham and signature operators in K-theory, including an equivariant generalization.

Findings

Symbol classes of de Rham and signature operators are congruent mod 2.

Equivariant generalization relates to Euler characteristic and multi-signature.

Provides new insights into K-theoretic properties of classical differential operators.

Abstract

This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.