HOMEWORK 1
Example 1. Show that SSk
= Sk+1
. For that consider the map f : Sk
× I → Sk+1
:
f(x, t) = ( 1 − (2t − 1)2x, 2t − 1)
Example 2. Let the pair (X, A) satisfy HEP and let g : A → Y . Show that (X∪gY, Y )
satisfies HEP.
Example 3. Let f : X → Y .
(1) Prove that Y is a deformation retract of Mf . In particular describe the retraction
r : Mf → Y
(2) Given the standard inclusion iY : Y → Mf , prove that iY ◦ f ∼ iX.
Example 4. Show that the cone CX of a space X is homotopy equivalent to a point.
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