In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" Levitt mentions that if one restricts to compact PL $4k$-manifolds then the signature is "locally defined" - i.e. that there is a real-valued function $d$ on triangulated $(4k-1)$-spheres with the property that $$ \sigma(M) = \sum_{v \in M^0} d( \text{link}(v)). $$

What is this function $d$? I am particularly interested in the case where $k=1$.