The fundamental group of an oriented surface of genus n with k boundary surfaces

Abstract

The fundamental group of a surface with boundary is always a free group. The fundamental group of torus with one boundary is a free group of rank two and with n boundary is a free group of rank n+1. Namely, π(T−D)=Z* Z=F 2 and π (T−D n)= Z* Z* ⋯ * Z n=F n+1. Thefundamental group of n-fold torus with one boundary is a free group of rank 2 n and with k boundary is a free group of rank 2 n+ k−1. Namely, π(T n−D)=F 2n and π (T n−D k)=F 2n+k−1. Thefundamental group of a real projective plane with one boundary (Mobius band) is a free group of rank one and with n boundary is a free group of rank n, Namely, π(P−D)= π(S 1)=Z and π (P−D n)= Z* Z* ⋯ * Z n=F n. Thus, the problem 4–6 on page 134 [Bozhuyuk, Genel Topolojiye Giris. Ataturk Universitesi, Yaymlan No: 610, Erzurum,1984] and exercise 5.1 on page 135 [Crowell, Fox, Introduction to Knot Theory, Blaisdell-Ginn, New York, 1963] is solved complete details. Here, D and D n denote a disc and a disjoint union of n discs, respectively T and T n denote a torus and n-double torus, respectively S 1 and S 2 denote a circle and a sphere, respectively.