ToricTopology -- homological computations in toric topology

Description

ToricTopology is a package for computing with quasitoric manifolds, small covers and moment-angle complexes.

A quasitoric manifold (or small cover) is entirely determined by a pair consisting of a simplicial complex K and a matrix lambda which is characteristic for K.

If K has n vertices, we can think of its k-faces as sets of integers between 1 and n. A matrix lambda is characteristic for K if all maximal minors of lambda indexed by the facets of K have determinant equal to 1 or -1.

Authors

Alvise Trevisan <a.trevisan@enpicom.com>
Alexander I. Suciu <a.suciu@neu.edu>
Kumar Sannidhya Shukla <kshukla5@uwo.ca>

Version

This documentation describes version 1.1 of ToricTopology, released November 7, 2025.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ToricTopologySource,
  title = {{ToricTopology: A \emph{Macaulay2} package. Version~1.1}},
  author = {Alvise Trevisan and Alexander I. Suciu and Kumar Sannidhya Shukla},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

Types
- MomentAngleComplex -- the class of all moment-angle complexes
- QuasiToricManifold -- the class of all quasitoric manifolds
- SmallCover -- the class of all small covers
Functions and commands
- bettiMAC -- compute the Betti numbers of a moment-angle complex
- bettiQTM -- compute the Betti numbers of a quasitoric manifold
- bettiSmallCover -- compute the Betti numbers of a small cover
- chern -- compute the Chern classes of a quasitoric manifold
- cohomologyRing -- compute the cohomology ring of a small cover or quasitoric manifold
- complexProjectiveSpace -- complex projective space of dimension n
- equivariantCohomology -- compute the equivariant cohomology of a moment-angle complex
- eulerMAC -- compute the Euler characteristic of a moment-angle complex
- hessenbergVariety -- Hessenberg variety asscoiated to the n-permutahedron
- isValidChar -- whether a matrix is characteristic for a simplicial complex
- momentAngleComplex -- create a moment-angle complex
- quasiToricManifold -- create a quasitoric manifold
- realProjectiveSpace -- real projective space of dimension n
- smallCover -- create a small cover
- stiefelWhitney -- compute the Stiefel-Whitney classes of a small cover
Methods
- bettiMAC(MomentAngleComplex) -- see bettiMAC -- compute the Betti numbers of a moment-angle complex
- bettiMAC(ZZ,MomentAngleComplex) -- see bettiMAC -- compute the Betti numbers of a moment-angle complex
- bettiQTM(QuasiToricManifold) -- see bettiQTM -- compute the Betti numbers of a quasitoric manifold
- bettiQTM(ZZ,QuasiToricManifold) -- see bettiQTM -- compute the Betti numbers of a quasitoric manifold
- bettiSmallCover(SmallCover) -- see bettiSmallCover -- compute the Betti numbers of a small cover
- bettiSmallCover(ZZ,SmallCover) -- see bettiSmallCover -- compute the Betti numbers of a small cover
- chern(QuasiToricManifold) -- see chern -- compute the Chern classes of a quasitoric manifold
- cohomologyRing(QuasiToricManifold) -- see cohomologyRing -- compute the cohomology ring of a small cover or quasitoric manifold
- cohomologyRing(SmallCover) -- see cohomologyRing -- compute the cohomology ring of a small cover or quasitoric manifold
- complexProjectiveSpace(ZZ) -- see complexProjectiveSpace -- complex projective space of dimension n
- equivariantCohomology(MomentAngleComplex) -- see equivariantCohomology -- compute the equivariant cohomology of a moment-angle complex
- eulerMAC(MomentAngleComplex) -- see eulerMAC -- compute the Euler characteristic of a moment-angle complex
- hessenbergVariety(ZZ) -- see hessenbergVariety -- Hessenberg variety asscoiated to the n-permutahedron
- realProjectiveSpace(ZZ) -- see realProjectiveSpace -- real projective space of dimension n
- stiefelWhitney(SmallCover) -- see stiefelWhitney -- compute the Stiefel-Whitney classes of a small cover
Symbols
- QTMCharacteristicMatrix (missing documentation)
- QTMDimension (missing documentation)
- QTMSimplicialComplex (missing documentation)

For the programmer

The object ToricTopology is a package, defined in ToricTopology.m2.

The source of this document is in ToricTopology.m2:408:0.