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Sheaves on Graphs and Their Homological Invariants
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Abstract:
We introduce a notion of a sheaf of vector spaces on a graph,
and develop the foundations of homology theories for such sheaves.
One sheaf invariant,
its ``maximum excess,'' has a number of remarkable
properties.
It has a simple definition, with no reference to homology theory,
that resembles graph expansion.
Yet it is a ``limit'' of Betti numbers, and hence has a
short/long exact sequence theory and resembles the $L^2$ Betti numbers
of Atiyah.
Also, the maximum excess is defined via a
supermodular function, which gives the maximum excess
much stronger properties than
one has of a typical Betti number.
The
maximum excess
gives a simple interpretation of an important graph invariant, which
will be used
to study the Hanna Neumann Conjecture
in a future paper.
Our sheaf theory can be viewed as a vast generalization of algebraic graph
theory: each sheaf has invariants associated
to it---such as Betti numbers and Laplacian matrices---that generalize
those in classical graph theory.
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