(4) In this problem, you will show that the fundamental groups of most compact, orientable surfaces are non-Abelian. Wri

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4 In This Problem You Will Show That The Fundamental Groups Of Most Compact Orientable Surfaces Are Non Abelian Wri 1 (23.3 KiB) Viewed 36 times

(4) In this problem, you will show that the fundamental groups of most compact, orientable surfaces are non-Abelian. Write E,= #,72 for the g-fold torus, also known as a surface of genus g. (a) Calculate the fundamental group of S¹ v S¹ (a bouquet of two circles, also known as the figure-eight space), and show that it is non-Abelian. (b) Let 72 V 7² denote the quotient space obtained from a union of two disjoint tori by identifying a point in each. Construct a map T2 VT² S¹v S¹ inducing a surjection on fundamental groups. (c) Construct a map E₂ →TVT2 inducing a surjection on fundamental groups (hint: collapse an appropriate diameter of the octagon used to build ₂). (d) Construct a map , E inducing a surjection on fundamental groups. (e) Use the preceding results to show that (E,) is non-Abelian for n > 1.