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

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 11 boundary is a free group of rank n + 1. Namely, pi(T-D) = Z* Z=F-2 and pi(T-D-n) = Z* Z* (...) * Z = Fn+1. The fundamental group of n-fold torus with one boundary is a free group of rank 211 and with k boundary is a free group of rank 2n + k - 1. Namely, pi(T-n - D) = F-2n and pi(T-n - D-k) = F2n+k-1. The fundamental 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, pi(P - D) = pi(S-1) = Z and pi(P - D-n) = Z * Z * (...) * Z = 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 it 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. (C) 2003 Published by Elsevier Inc.