Catalog of simplicial setsΒΆ

This provides pre-built simplicial sets:

  • the n-sphere and n-dimensional real projective space, both (in theory) for any positive integer n. In practice, as n increases, it takes longer to construct these simplicial sets.
  • the n-simplex and the horns obtained from it. As n increases, it takes much longer to construct these simplicial sets, because the number of nondegenerate simplices increases exponentially in n. For example, it is feasible to do simplicial_sets.RealProjectiveSpace(100) since it only has 101 nondegenerate simplices, but simplicial_sets.Simplex(20) is probably a bad idea.
  • n-dimensional complex projective space for n \leq 4
  • the classifying space of a finite multiplicative group or monoid
  • the torus and the Klein bottle
  • the point
  • the Hopf map: this is a pre-built morphism, from which one can extract its domain, codomain, mapping cone, etc.

All of these examples are accessible by typing simplicial_sets.NAME, where NAME is the name of the example. Type simplicial_sets.[TAB] for a complete list.

EXAMPLES:

sage: RP10 = simplicial_sets.RealProjectiveSpace(8)
sage: RP10.homology()
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: C2, 6: 0, 7: C2, 8: 0}

sage: eta = simplicial_sets.HopfMap()
sage: S3 = eta.domain()
sage: S2 = eta.codomain()
sage: S3.wedge(S2).homology()
{0: 0, 1: 0, 2: Z, 3: Z}