Luciano Melodia

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math.AT2026

Magnitude Homology Is the Associated Graded of the Length Filtration

Luciano Melodia

Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one const…

math.AT2026

Universal Coefficients and Mayer-Vietoris Sequence for Groupoid Homology

Luciano Melodia

We study homology of ample groupoids via the compactly supported Moore complex of the nerve. Let $A$ be a topological abelian group. For $n \geq 0$ set $C_n(\mathcal G;A) := C_c(\mat…

math.AT2026

Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids

Luciano Melodia

The homology of an ample groupoid is computed from the complex of compactly supported continuous functions on the nerve. Two hypotheses routinely imposed on this complex behave in…

math.AT2026

Groupoid Homology and Classifying-Space Homology Are Not Isomorphic: The Cantor Unit Groupoid

Luciano Melodia

For an ample groupoid $G$ , Matui-type groupoid homology $H_{∙} (G; Z)$ is built from the nerve $G_{∙}$ via the Moore complex of compact…

math.AT2024

Algebraic and Topological Persistence

Luciano Melodia

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as…