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In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" on page 61 in the penultimate paragraph, 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 (up to simplicial-isomorphism) 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 $k=1$.

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  • $\begingroup$ Are you certain the claim about signature is made in that paper? I'm scanning it over and I have not found the claim. Could you give a page and line number? $\endgroup$
    – Ryan Budney
    Commented Jan 7, 2018 at 4:09
  • $\begingroup$ @RyanBudney perhaps the OP was thinking of what's written on page 61 after Theorem B? gdz.sub.uni-goettingen.de/dms/load/img/… . I'm not sure this is what is written in the question though. $\endgroup$
    – j.c.
    Commented Jan 7, 2018 at 4:41
  • $\begingroup$ As far as I know, such a formula has not been worked out. It's one of those things that "in principle" exists but expressing $d$ explicitly as a function of triangulated spheres is likely a fair bit of work. Perhaps someone working on invariants of triangulated manifolds might have some additional insights. $\endgroup$
    – Ryan Budney
    Commented Jan 7, 2018 at 16:20

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