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By M. Sion

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The group G then acts on the second homotopy group π2 (K), via the standard action of the fundamental group of K (see Rotman [Rot88], p. 342). 38 1. Let P = x | xx−1 x be a presentation of the trivial group. Then KP is the so called Dunce hat ; it is a contractible space. 2. Let P = x | x2 be a presentation of the cyclic group of order 2. Then KP is homotopically equivalent to the 2-dimensional real projective plane. Its second homotopy module is the infinite cyclic group Z with the generator of Z2 acting by change of sign: 1 → −1.

48), we have N ∩ γm (G) = 0; hence, for any l ≥ m, there exists the natural isomorphism γl (G) γl (G), which immediately implies the assertion. 80 If G is a residually nilpotent group with M (n) (G) = 0 for all n ≥ 1, then G is an absolutely residually nilpotent. 14) that if, for a given group G, H1 (G) is torsion-free and H2 (G) = 0, then M (n) (G) = 0 for all n ≥ 1. 81 Let G be a group given by the following presentation: G = a, b, c | a = [c−1 , a][c, b] .

342). 38 1. Let P = x | xx−1 x be a presentation of the trivial group. Then KP is the so called Dunce hat ; it is a contractible space. 2. Let P = x | x2 be a presentation of the cyclic group of order 2. Then KP is homotopically equivalent to the 2-dimensional real projective plane. Its second homotopy module is the infinite cyclic group Z with the generator of Z2 acting by change of sign: 1 → −1. 3. For the one-relator presentation P = x1 , . . , x2n | ni=1 [x2i−1 , x2i ] , n ≥ 1, the standard 2-complex KP is homotopically equivalent to the oriented surface of genus n.

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