18 0. Prequel By Example 0.2.4, π1(Y ) has presentation α, β : α5β−3,β3(αβ)−2 . Thus π1(Y ) has two generators α and β and two relations α5 = β3 = (αβ)2. This is the well-known binary icosahedral group, so called because there is an order-two homomorphism of π1(Y ) onto the group G of rigid motions of the icosahedron. In order to complete the proof we need to show that π1(Y ) is nontrivial. We will do that by exhibiting a homomorphism of π1(Y ) onto a nonabelian subgroup of G. To determine such a homomorphism, send α to the coun- terclockwise rotation through 2π/5 radians about the point x indicated in Figure 0.3 and send β to the counterclockwise rotation through 2π/3 radians about the point y. Then αβ is the rotation through π radians about the point z, so all three of α5, β3, and (αβ)2 represent the identity motion. As a result, this assignment extends to a homomorphism of π1(Y ) to G. Since βα is rotation through an angle of π radians about z , we see that αβ = βα. x y z z Figure 0.3. The icosahedron In the preceding argument it was not necessary to compute π1(Y ) explic- itly to determine its non-triviality, but it is known that G has 60 elements, and that the order of π1(Y ) is 120. Example 0.10.3. There exists a compact, contractible n-dimensional ∂- manifold in Sn, n 5, that is not a ball. Proof. Start with an acyclic 2-complex P (such as that in the preceding example) and embed it in Sn, n 5. Name a regular neighborhood N of P and set M = Sn Int N. General position considerations yield that Sn P is 1-connected the same holds for M, which is a (deformation) retract of Sn P , since N P ∂N × [0, 1). Like P , N is acyclic more importantly, so is M, by duality or a simple Mayer-Vietoris argument (it helps to know ∂N = ∂M is orientable, due to §0.3). Hence M is contractible. Note that ∂M need not be a sphere. In particular, ∂M is 1-connected if and only if P is, for general position implies that the arrow in the line below, π1(∂M = ∂N) π1(∂N × [0, 1)) π1(N P ) π1(N) π1(P ),
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