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user101010
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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 $k=1$.

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 $k=1$.

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 $k=1$.

add link to freely-available and legal version of paper and add link to claim; deleted 120 characters in body
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j.c.
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In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes""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 $$ \sigma(M) = \sum_{v \in M^0} d( \text{link}(v)) $$$$ \sigma(M) = \sum_{v \in M^0} d( \text{link}(v)). $$

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

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 $k=1$.

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 $k=1$.

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user101010
user101010
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Local formula for the signature of $4k$-manifold

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 $k=1$.