The Mayer-Vietoris theorem decomposes the homology of a space into contributions from overlapping pieces — if X = A ∪ B, the homology of X is determined by the homologies of A, B, and their intersection A ∩ B, assembled via a long exact sequence. In practice, you need to compute homology for three spaces instead of one, which helps only if the pieces are simpler.
Gonzalez-Lorenzo et al. (arXiv:2603.13143) make Mayer-Vietoris effective for discrete Morse theory by eliminating the intermediate homology computations entirely. Given gradient vector fields on subcomplexes A and B — the discrete Morse data that encodes which cells cancel and which are critical — their algorithm computes the homology of X directly from the trajectory data of these gradient flows. No homology groups of A, B, or A ∩ B are ever computed.
The key structural insight: the gradient vector fields on A and B need not be coherent on their intersection. Where A and B overlap, the two gradient flows may disagree — pairing different cells, pointing in different directions. Classical Mayer-Vietoris requires compatible algebraic data; the discrete Morse version requires only that each piece separately admits a valid gradient field. The assembly happens at the level of trajectories, not homology classes.
This incoherence tolerance is what makes the method practical. In applications where the decomposition arises from data partitioning — parallel computation, sensor coverage, overlapping image patches — requiring coherent Morse data on overlaps would demand global coordination that defeats the purpose of decomposition. The algorithm lets each piece be processed independently, then assembles the global topology from local flow data.