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

摘要

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=F2andπ(T−Dn)=Z*Z*⋯*Zn=Fn+1.Thefundamental group of n-fold torus with one boundary is a free group of rank 2n and with k boundary is a free group of rank 2n+k−1. Namely,π(Tn−D)=F2nandπ(Tn−Dk)=F2n+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(S1)=Zandπ(P−Dn)=Z*Z*⋯*Zn=Fn.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 Dn denote a disc and a disjoint union of n discs, respectively T and Tn denote a torus and n-double torus, respectively S1 and S2 denote a circle and a sphere, respectively.