Filtrationsoberfläche

Filtrationsoberfläche is a term used in German mathematical literature to denote a geometric realization or visualization of a filtration on a space, group, or algebraic object. A filtration is a nested sequence of subobjects F0 ⊆ F1 ⊆ … ⊆ Fn (or an infinite sequence) whose final member is the object being studied. The Filtrationsoberfläche is not a single standard object, but a conceptual construction: a stratified surface that encodes the successive layers of the filtration by associating to each level a sheet of the surface and gluing the sheets along the inclusions Fi ⊆ Fi+1.

Construction: Given a finite filtration, one may realize the Filtrationsoberfläche by taking for each i a sheet representing the quotient Fi/Fi−1 and placing it at height i in a product with an interval or line. The interfaces between consecutive heights model the inclusions, producing a connected surface that projects to the height axis. In practice the construction is chosen to emphasize the graded pieces Ai = Fi/Fi−1 while preserving information about how the filtration is built up. The surface is typically not canonical: different embeddings or gluings yield different visualizations, but all reflect the same filtration data.

Interpretation and use: The Filtrationsoberfläche serves as a visual aid in topology, algebraic geometry, and data analysis contexts such as persistent homology, where filtrations organize building blocks of the object. It helps communicate how complex structures arise from simpler layers, and how invariants relate to successive quotients.

See also: Filtration, Associated graded, Spectral sequence, Stratified space, Persistent homology.