Betti numbers of a simplicial complex

The ranks of the homology groups with rational coefficients: \(\beta_0\) counts connected components, \(\beta_1\) independent loops, \(\beta_2\) voids, and so on.

Usage

hg_betti(sc)

Arguments

sc: A simplicial_complex from simplicial().

Value

A named integer vector c(b0 = , b1 = , ...).

Details

The coefficients are rational, not Z/2. The two agree on most complexes but not on one with torsion: on the six-vertex real projective plane (RP2) hg_betti() gives (1, 0, 0), while hg_homology(), which reduces over Z/2 as the persistence literature does, ends its Betti curve at (1, 1, 1). Use hg_homology() when Z/2 Betti numbers are meant.

References

Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.

Examples

adj <- matrix(c(0, 1, 1, 0,
                1, 0, 1, 1,
                1, 1, 0, 1,
                0, 1, 1, 0), 4, 4, dimnames = list(letters[1:4], letters[1:4]))
hg_betti(simplicial(adj))
#> b0 b1 b2 
#>  1  0  0