MATH 74 HOMEWORK 10
Due Wednesday May 6th at 3:10pm.
(1) Prove that homotopy defines an equivalence relation on the set of based loops in a topological space.
(2) Prove that multiplication in the fundamental group is welldefined. That is, given equivalence classes
of loops [
γ
] and [
η
] we choose representatives
γ
and
η
so that [
γ
]
*
[
η
] = [
γ
*
η
]. If we choose different
representatives, ˜
γ
and ˜
η
, show that [
γ
]
*
[
η
] = [˜
γ
*
˜
η
].
HINT: There are homotopies
H
1
and
H
2
between
γ
and ˜
γ
and
η
and ˜
η
respectively. Use these to build a homotopy between
γ
*
η
and ˜
γ
*
˜
η
.
(3) Show that following are equivalent:
(a) Every map
S
1
→
X
is homotopic to the constant map.
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 Spring '09
 DANBERWICK
 Logic, Topology, Division, Multiplication, Homeomorphism, Equivalence relation, Manifold, Topological space, homotopy

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