Here is our boundary matrix representing map (Subscript[δ, 1]):

bdry1=(({{1, 1, 1, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 1, 0, 0, 0, 0}, {0, 1, 0, 1, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 1, 1, 1}}))

Compute rank((Subscript[δ, 1])) with arithmetic mod 2:

MatrixRank[bdry1,Modulus->2]