The Multiparameter Dual

In one-parameter persistent homology, a classical duality relates the barcode of absolute homology to the barcode of relative cohomology. The birth of a homology class corresponds to the death of a cohomology class, and vice versa. This duality is computationally useful — it lets you extract information about one from the other — and conceptually clean.

Multiparameter persistence, where the filtration varies along N independent axes simultaneously, has resisted such clean statements. There is no barcode (the algebraic structure is too complex for a complete discrete invariant), and the relationship between homology and cohomology is tangled by the multigraded algebra.

This paper establishes a multiparameter duality anyway. For chain complexes of free N-parameter persistence modules with acyclic colimit, the pointwise dual of the q-th homology is isomorphic to the (N+q)-th cohomology of the global dual complex. The degree shift by N is the price of multiple parameters — each additional filtration axis adds one to the cohomological degree.

Two duality functors are in play: the pointwise dual (which dualizes at each point of the parameter space separately) and the global dual (which dualizes the entire chain complex as a multigraded module). The theorem relates them through the homology-cohomology bridge.

A corollary recovers the correspondence between minimal free resolutions of a module and its pointwise dual — resolutions being the algebraic objects that encode the full structure of multiparameter persistence. This feeds directly into algorithms for computing Vietoris-Rips bifiltrations, where the duality converts a hard computation into an equivalent one that may be easier to perform.