# mHomology -- Compute the homology.

## Synopsis

• Usage:
mHomology(C)
• Inputs:
• C, a list, of lists with the faces of a complex, sorted by dimension, each face represented by a list of vertices.
• Outputs:

## Description

Returns a list of FinitelyGeneratedAbelianGroups with the homology with closed support of C with integer coefficients.

C does not have to be simplicial.

This uses the Convex functions convHull, simplicialsubdiv, pcomplex and homology.

Remark: One should also implement the relative version from Convex.

 i1 : C={{{}}, {{{-1, -1, -1, -1}}, {{1, 0, 0, 0}}, {{0, 1, 0, 0}}, {{0, 0, 1,0}}, {{0, 0, 0, 1}}}, {{{-1, -1, -1, -1}, {0, 1, 0, 0}}, {{-1, -1, -1,-1}, {0, 0, 1, 0}}, {{1, 0, 0, 0}, {0, 0, 1, 0}}, {{1, 0, 0, 0}, {0, 0, 0,1}}, {{0, 1, 0, 0}, {0, 0, 0, 1}}}, {}, {}, {}}; i2 : mHomology(C) o2 = {} o2 : List

RP2:

 i3 : C= {{{}}, {{{-1, -1, -1, -1, -1}}, {{1, 0, 0, 0, 0}}, {{0, 1, 0, 0, 0}}, {{0, 0, 1, 0, 0}}, {{0, 0, 0, 1, 0}}, {{0, 0, 0, 0, 1}}}, {{{-1, -1, -1, -1,-1}, {1, 0, 0, 0, 0}}, {{-1, -1, -1, -1, -1}, {0, 1, 0, 0, 0}}, {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}}, {{-1, -1, -1, -1, -1}, {0, 0, 1, 0, 0}}, {{1, 0, 0, 0, 0}, {0, 0, 1, 0, 0}}, {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}}, {{-1, -1, -1, -1, -1}, {0, 0, 0, 1, 0}}, {{1, 0, 0, 0, 0}, {0, 0, 0, 1, 0}}, {{0, 1, 0, 0, 0}, {0, 0, 0, 1, 0}}, {{0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, {{-1, -1, -1, -1, -1}, {0, 0, 0, 0, 1}}, {{1, 0, 0, 0, 0}, {0, 0, 0, 0, 1}}, {{0, 1, 0, 0, 0}, {0, 0, 0, 0, 1}}, {{0, 0, 1, 0, 0}, {0, 0, 0, 0, 1}}, {{0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}}, {{{-1, -1, -1, -1, -1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}}, {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}}, {{-1, -1, -1, -1, -1}, {1, 0, 0, 0, 0}, {0, 0, 0, 1, 0}}, {{-1, -1, -1, -1, -1}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, {{-1, -1, -1, -1, -1}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 1}}, {{-1, -1, -1, -1, -1}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 1}}, {{1, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 0, 1}}, {{1, 0, 0, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}, {{0, 1, 0, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}}, {}, {}, {}}; i4 : mHomology(C) o4 = {} o4 : List

## Ways to use mHomology :

• "mHomology(List)"

## For the programmer

The object mHomology is .