Simplicial Betti numbers over F_2 — morie_betti_numbers

\(b_k = \dim\ker\partial_k - \mathrm{rank}\,\partial_{k+1}\), every rank exact by mod-2 elimination. Before any rank is trusted, \(\partial_k \partial_{k+1} = 0\) is verified entry by entry, and the Euler-Poincare identity is computed from both sides independently and compared.

Usage

morie_betti_numbers_gf2(maximal_simplices, check_boundary_squared = TRUE)

Arguments

maximal_simplices: A list of integer vectors.
check_boundary_squared: Verify d^2 = 0 before reporting.

Value

A list with betti, f_vector, boundary_ranks, chi_from_faces, chi_from_betti, euler_poincare_holds, boundary_squared_zero, warnings.

References

Munkres JR (1984) Elements of Algebraic Topology.

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
morie_betti_numbers_gf2(V)
#> $betti
#> [1] 8
#> 
#> $f_vector
#> [1] 8
#> 
#> $boundary_ranks
#> [1] 0 0
#> 
#> $chi_from_faces
#> [1] 8
#> 
#> $chi_from_betti
#> [1] 8
#> 
#> $euler_poincare_holds
#> [1] TRUE
#> 
#> $boundary_squared_zero
#> [1] TRUE
#> 
#> $estimate
#> [1] 8
#> 
#> $n
#> [1] 8
#> 
#> $warnings
#> character(0)
#> 
#> $method
#> [1] "Simplicial homology over F_2 (Munkres 1984)"
#>