The signature operator at 2 Jonathan

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: what is this class localized at 2, and what special properties does it have? Our answers include the following: • the K-homology class ∆M of the signature operator is a bordism invariant; • the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant; • the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.