Cohomology

A cochain complex is a sequence of linear maps with . Elements of are -cochains and the coboundary; the cocycles are , the coboundaries , and the cohomology is the quotient

A cochain is exact if it is a coboundary, so says every cocycle is exact. On the nerve of a graph or category, are vertex functions and edge functions, with . A 1-cochain is exact, meaning , iff its signed sum around every cycle vanishes, hence iff (Kirchhoff’s voltage law).

Theorem. Combinatorial Hodge decomposition

With inner products on the cochain spaces, splits orthogonally into exact ⊕ curl ⊕ harmonic,

where the harmonic part satisfies . Projecting a measured 1-cochain onto gives the least-squares exact fit, whereas a non-zero harmonic component is a true obstruction.

Definition. The sheaf Laplacian

Degree-zero cohomology of a cellular-sheaf uses and the sheaf Laplacian . A state-dependent metric weights it, , whose kernel is the weighted global sections. See laplacian for the Hodge Laplacian that this specialises.

References

• Xiaoye Jiang, Lek-Heng Lim, Yuan Yao & Yinyu Ye, “Statistical ranking and combinatorial Hodge theory”, Math. Programming (2011), combinatorial Hodge decomposition
• Jakob Hansen & Robert Ghrist, “Toward a spectral theory of cellular sheaves”, J. Applied and Computational Topology (2019), sheaf Laplacian