Simplicial Structures in Topology

38 I Fundamental Concepts
satisfies the equality
(∀x ∈ G) θ( f (x)g(x) −1 )=1H ;
by Lemma (I.2.5), there is a unique homomorphism
that extends the function θ and such that
θ : G → H
θ f = h2 , θg = h1. �
The group G, also denoted with G1 ∗ f ,g G2, istheamalgamated product of the
groups G1 and G2 with respect to the homomorphisms f and g.
Exercises
1. Prove that the relation of based homotopy in Top ∗(X,Y ) is an equivalence
relation.
2. Let A be a subspace of X and i: A −→ X be the inclusion map; then A is a
deformation retract of X if there exists a continuous function r : X → A such that
ri = 1A : A → A and ir ∼ 1X. In particular, if A ⊂ X is a deformation retract and
A = {x0}, we say that X is contractible to x0. In this case, the identity 1X : X →
X is homotopic to the constant map c: X →{x0}. The space A is called strong
deformation retract of X if ri = 1A and ir ∼A 1X. Intuitively, a subspace A of X
is a strong deformation retract of X if X can be deformed over A with continuity,
keeping A fixed during the deformation. Clearly, a strong deformation retract is a
deformation retract. It follows from the definitions that, if A is a (strong or not)
deformation retract of X,thenAand X are of the same homotopy type.
(i) Prove that the circle
S 1 = {(x,y) ∈ R 2 | x 2 + y 2 = 1}
is a strong deformation retract of the cylinder
(ii) Prove that the disk
C = {(x,y,z) ∈ R 3 | x 2 + y 2 = 1 , 0 ≤ z ≤ 1}.
D 2 = {(x,y) ∈ R 2 | x 2 + y 2 ≤ 1}
is contractible to (0,0).
3. Prove that for every X,Y ∈ Top ∗, the function
is a natural bijection.
[Φ]: [ΣX,Y ]∗ ∼ = [X,ΩY ]∗ , [ f ] ↦→ [Φ( f )]