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INIDEPENDENT UNIVERSITY OF MOSCOW

7. FUNDAMENTAL GROUP.

TOPOLOGY-1, SPRING 2012 Г.

is associative. (b) Let the topological space X be a topological group (i.e. there is a multiplication with the usual group properties; and multiplication and taking an inverse are continuous maps). Prove that 1 (X ) is commutative. Problem 2. For the following spaces construct their cell decompositions and compute (write down presentations of ) their fundamental groups: (a) a wedge product of n circles; (b) a connected sum of g tori (of dimension 2); (c) a connected sum of g copies of RP 2 ; (d) a genus g surface with n holes; (e) a connected sum of g copies of RP 2 with n holes; (f ) R3 \ {(x; y; 0) | x2 + y2 = 1}. Problem 3. (a) Prove that if G = 1 (gT2 ), then G=G = Z2g . (Here G is a commutant, i.e. G is a subgroup generated by all elements of the type aba-1 b-1 for a, b G.) (b) Prove that if G = 1 (gRP 2 ), then G=G = Zg-1 Z2 . Problem 4. Prove that the boundaries of the following surfaces are not their retracts: (a) a Moebius band; (b) a handle (a torus with one hole); (c) a Klein bottle with one hole.

Problem 1. (a) Write out a formula for a homotopy which proves that multiplication in the fundamental group

Problem 5. Let Xn be a set of all cardinality n subsets U

strands. (a) Prove that X2 is homotopy equivalent to a circle, and the group Bn is in nite. (c) Prove that for all n 3 the group Bn Problem 6. (a) Prove that X3 is homotopy equivalent to the set is homotopy equivalent to the set {(p; q) C2 | p2 + q3 = 0; |p|2 + fundamental group of R3 \ K where K is the trefoil knot. Hint. For 6(a): the space Xn is homeomorphic to the set of degree n complex polynomials without multiple roots. Now consider cubic polynomials t3 + pt + q. For 6(c): a trefoil knot is de ned as a curve on a standard torus T2 R3 making 2 turns along the meridian and 3 turns along the parallel of the torus.

R2

. 1 (Xn ) def Bn is called a braid group on n = therefore B2 = Z. (b) Prove that for all n 2 is not commutative. {(p; q ) C2 | p2 + q 3 = 0}. (b) Prove that X3 2 |q | = 1}. (c) Prove that B3 is isomorphic to a

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