Paths and cylinders
For , the -Betti number of a chain complex (of modules over a ring ) is the rank
b_n(V) := rk_R H_n(V)
of its th homology group, regarded as an -module.
For a topological space, its -Betti number is that of its singular homology-complex
b_n(V) = rk_R H_n(X, R)
The alternating sum of all the Betti numbers is – if it exists – the Euler characteristic.