Simplicial Homology
- Introduction
- Simplicial Complexes
SimplicialComplex(f): SeqEnum[SetEnum] → SmpCpx- Example: Constructing a Simplicial Complex
SimplicialComplex(G): Grph → SmpCpxFlagComplex(G): Grph → SmpCpxCliqueComplex(G): Grph → SmpCpxDimension(X): SmpCpx → RngIntElt- Example: Dimension of a Simplicial Complex
Faces(X, d): SmpCpx, RngIntElt → SeqEnum[SetEnum]- Example: Faces of a Simplicial Complex
Facets(X): SmpCpx → SeqEnum[SetEnum]- Example: Facets of a Simplicial Complex
Normalization(X): SmpCpx → SmpCpxNormalization(~X): SmpCpxShift(X, n): SmpCpx, RngIntElt → SmpCpxShift(~X, n): SmpCpx, RngIntElt- Example: Normalizing and Shifting Labels
Boundary(X): SmpCpx → SmpCpx- Example: Boundary of a Simplicial Complex
S + T: SmpCpx, SmpCpx → SmpCpx- Example: Disjoint Union of Two Circles
S eq T: SmpCpx, SmpCpx → BoolElt- Example: Equality of Simplicial Complexes
Product(S, T): SmpCpx, SmpCpx → SmpCpx- Example: Torus as a Product of Circles
Join(S, T): SmpCpx, SmpCpx → SmpCpxS * T: SmpCpx, SmpCpx → SmpCpx- Example: Join of Two Edges
AddSimplex(X, s): SmpCpx, SetEnum → SmpCpxAddSimplex(~X, s): SmpCpx, SetEnumAddSimplex(X, s): SmpCpx, SeqEnum → SmpCpxAddSimplex(~X, s): SmpCpx, SeqEnumPrune(X, f): SmpCpx, SetEnum → SmpCpxPrune(~X, f): SmpCpx, SetEnumGlue(X, e): SmpCpx, SeqEnum → SmpCpxGlue(~X, e): SmpCpx, SeqEnum- Example: Connected Sum of Two Tori
BarycentricSubdivision(X): SmpCpx → SmpCpxSkeleton(X, q): SmpCpx, RngIntElt → SmpCpxUnderlyingGraph(X): SmpCpx → GrphUnd, GrphVertSet, GrphEdgeSetCone(X): SmpCpx → SmpCpxSuspension(X): SmpCpx → SmpCpx- Example: Cone and Suspension
- Standard Topological Objects
- Homology Computation
Homology(X): SmpCpx → SeqEnum, SeqEnumHomology(~X): SmpCpxHomology(X, A): SmpCpx, Rng → SeqEnum, SeqEnumHomology(~X, A): SmpCpx, Rng- Example: Homology of a Simplicial Complex
HomologyGroup(X, q): SmpCpx, RngIntElt → ModRngHomologyGroup(X, q, A): SmpCpx, RngIntElt, Rng → ModRngBettiNumber(X, q): SmpCpx, RngIntElt → RngIntEltBettiNumber(X, q, A): SmpCpx, RngIntElt, Rng → RngIntEltTorsionCoefficients(X, q): SmpCpx, RngIntElt → SeqEnum[RngElt]TorsionCoefficients(X, q, A): SmpCpx, RngIntElt, Rng → SeqEnum[RngElt]EulerCharacteristic(X): SmpCpx → RngIntEltBoundaryMatrix(X, q, A): SmpCpx, RngIntElt, Rng → MtrxChainComplex(X, A): SmpCpx, Rng → ModCpx- Example: Chain Complex of a Simplicial Complex
HomologyGenerators(X): SmpCpxHomologyGenerators(X, A): SmpCpx, RngHomologyGenerators(H, M, X): SeqEnum, SeqEnum, SmpCpx- Example: Homology Generators
- Bibliography