# 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.

## Version

This documentation describes version 1.0 of ToricTopology.

## Source code

The source code from which this documentation is derived is in the file ToricTopology.m2.

## Exports

• Types
• Functions and commands
• 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 .