A common challenge in persistent homology is choosing "good" representative cycles for homology classes in a way compatible with persistence. In this talk, we discuss a geometric framework for codimension-1 persistent homology that addresses this issue using Alexander duality.

For an embedded filtered simplicial complex, connected components of the complement induce cycle representatives for a homology basis. The evolution of these cycles along the filtration can be tracked via the merge tree of the complement and the elder rule. This leads to the notion of cycle progression barcodes, associating to each persistence interval a sequence of representative cycles evolving through the filtration.

Applying geometric functionals to these progressions produces generalized persistence landscapes, which extend classical persistence landscapes and allow geometric information about cycle representatives to be captured without fixing a single filtration value. This provides a way to distinguish data sets with similar persistent homology but different geometric structure.