Homology Groups

Recall the homology version of Cauchy’s theorem in complex analysis of 1-variable ([Shastri, 2009] p.214):

Theorem 4.0.1 Let Ω be an open subset in C and γ be a cycle in Ω. Then the following conditions on γ are equivalent: (a)

∫ γ fdz = 0 for all holomorphic functions f on Ω.