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 ),
