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SimplicialHomology

compute the simplicial homology of a top simplices list

Syntax
SimplicialHomology(A: LIST): RECORD
SimplicialHomology(A: LIST, B: LIST): RECORD

Description
This function computes the simplicial homology of a simplicial complex described by a list of top faces. With 2nd argument only with the second list of vertices.

Package GeomModelling, by Elisa Palezzato.

Example
/**/ Use QQ[x[1..5]], DegLex;  --> DegLex ?

/**/ L := [x[1]*x[2]*x[3], x[2]*x[3]*x[4], x[3]*x[4]*x[5]]; -- list top faces
/**/ indent(SimplicialHomology(L));
record[
  H_0 := record[betti := 1, lambda := []],
  H_i := [record[betti := 0, lambda := []]],
  H_max := record[betti := 0, lambda := []]
]
-- 1 connected component (betti in H_0)

/**/ L := [x[1]*x[2]*x[3], x[2]*x[3]*x[4]]; -- list top faces
/**/ -- indent(SimplicialHomology(L)); --!!! ERROR !!! as expected: missing x[5]

/**/ L := [x[1]*x[2]*x[3], x[2]*x[3]*x[4], x[5]];
/**/ indent(SimplicialHomology(L));
record[
  H_0 := record[betti := 2, lambda := []],
  H_i := [record[betti := 0, lambda := []]],
  H_max := record[betti := 0, lambda := []]
]
-- 2 connected components

/**/ L := [x[1]*x[2]*x[3], x[2]*x[3]*x[4]];
/**/ indent( SimplicialHomology(L, [x[1], x[2], x[3], x[4]]) );
record[
  H_0 := record[betti := 1, lambda := []],
  H_i := [record[betti := 0, lambda := []]],
  H_max := record[betti := 0, lambda := []]
]