PH-DECK Reference Library

This page exists because the deck cards are for spaced practice, not first exposure. Use this as the lookup layer for terms and assumptions that otherwise make the cards feel like magic.

Core Objects

FiltrationA nested sequence of spaces or complexes, usually written K_0 ⊆ K_1 ⊆ .... Time means inclusion order: once a simplex appears, it stays.

Simplicial ComplexA combinatorial space built from vertices, edges, triangles, tetrahedra, and higher simplices, closed under taking faces.

ChainA formal sum of simplices. Over F₂, coefficients are 0 or 1, so adding the same simplex twice cancels.

BoundaryThe operation taking a simplex to its faces. The boundary of an edge is its two endpoints; the boundary of a triangle is its three edges.

CycleA chain with zero boundary. A loop of edges is a 1-cycle when every vertex appears an even number of times.

Boundary ChainA cycle that is itself the boundary of a higher-dimensional chain. Boundaries are considered topologically trivial holes.

Homology ClassA cycle modulo boundaries. Two cycles represent the same class if their difference is a boundary.

Persistence Terms

BirthThe first filtration value where a homology class appears.

DeathThe filtration value where a class becomes a boundary or merges into an older class. Intervals are usually half-open: [birth, death).

Persistence ModuleA sequence of vector spaces plus linear maps between them, one vector space per filtration time.

Interval ModuleA persistence module that is one-dimensional during one interval and zero outside it.

BarcodeThe multiset of intervals in the interval decomposition of a one-parameter persistence module over a field.

Persistence DiagramThe same barcode drawn as points (birth, death) above the diagonal.

Essential BarA class with no death in ordinary persistence, written with death ∞. It records homology of the final space, not a transient feature.

Distances and Stability

Diagonal CostSending a point (b,d) to the diagonal costs (d-b)/2 in bottleneck distance.

Bottleneck DistanceThe best possible worst-case matching cost between two persistence diagrams, allowing unmatched points to go to the diagonal.

InterleavingAn approximate isomorphism of persistence modules with delay ε. It is the algebraic form of “the same up to time error.”

Isometry TheoremFor q-tame modules, interleaving distance equals bottleneck distance.

StabilityIf functions differ by at most ε in sup norm, their diagrams differ by at most ε in bottleneck distance.

Complexes

Cech ComplexA simplex appears when the corresponding radius-r balls have a common intersection point. Meaningful via the nerve theorem.

Vietoris-Rips ComplexA simplex appears when all its edges are present. It is a flag complex, hence computationally convenient.

Good CoverA cover whose every nonempty finite intersection is contractible. The nerve theorem needs this hypothesis.

Jung's TheoremA diameter-bounded point set in Euclidean space fits inside a ball of controlled radius. It gives the Cech-Rips comparison constant.

Recommended References

Edelsbrunner and Harer, Computational Topology.
Ghrist, Barcodes: The Persistent Topology of Data.
Oudot, Persistence Theory.
Chazal, de Silva, Glisse, Oudot, The Structure and Stability of Persistence Modules.

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