ToricTopology -- homological computations in toric topology

Description

ToricTopology is a package for computing with quasi-toric manifolds and small covers.

A quasi-toric manifold (or small cover) is entirely determined by a pair consisting of a simplicial complex K and a matrix chi 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 chi is characteristic for K if all maximal minors of chi indexed by the facets of K have determinant equal to 1 or -1.

Authors

Alvise Trevisan <[email protected]>
Alexander I. Suciu <[email protected]>

Version

This documentation describes version 1.0 of ToricTopology.

Citation

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

@misc{ToricTopologySource,
  title = {{ToricTopology: A \emph{Macaulay2} package. Version~1.0}},
  author = {Alvise Trevisan and Alexander I. Suciu},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- QuasiToricManifold -- the class of all quasi-toric manifolds
- SmallCover -- the class of all small covers
Functions and commands
- bettiQTM -- Compute the betti numbers of a quasi-toric manifold
- bettiSmallCover -- Compute the betti numbers of a small cover
- chern -- Compute the Chern classes of a quasi-toric manifold
- cohomologyRing -- Compute the cohomology ring of a small cover or quasi-toric manifold
- complexProjectiveSpace -- Complex projective space of dimension n
- hessenbergVariety -- Hessenberg variety asscoiated to the n-permutahedron
- isValidChar -- Whether a matrix is characteristic for a simplicial complex
- quasiToricManifold -- Create a quasi-toric manifold
- realProjectiveSpace -- Real projective space of dimension n
- smallCover -- Create a small cover
- stiefelWhitney -- Compute the Stiefel-Whitney classes of a small cover
Methods
- bettiQTM(QuasiToricManifold) -- see bettiQTM -- Compute the betti numbers of a quasi-toric manifold
- bettiQTM(ZZ,QuasiToricManifold) -- see bettiQTM -- Compute the betti numbers of a quasi-toric 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 quasi-toric manifold
- cohomologyRing(QuasiToricManifold) -- see cohomologyRing -- Compute the cohomology ring of a small cover or quasi-toric manifold
- cohomologyRing(SmallCover) -- see cohomologyRing -- Compute the cohomology ring of a small cover or quasi-toric manifold
- complexProjectiveSpace(ZZ) -- see complexProjectiveSpace -- Complex projective space of dimension n
- 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.m2:378:0.