Consider I have boundary operators $\partial_1$, $\partial_2$: $\partial_1 \circ \partial_2 = 0$. Then if interested in $\text{ker}\,\partial_1 / \text{im}\,\partial_2$ one can study $\text{ker}\,(\partial_1^T\partial_1 + \partial_2\partial_2^T)$. However this is only true if I can define non-degenerate inner product in order to relate $(\text{im})^{\perp}$ and $\text{ker}$. But can I some how generalise this if I have $Z_2$ coefficients?
2026-03-27 18:14:54.1774635294
Combinatorial Laplacian for homology with $Z_2$ coefficients
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