LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...

LECTURES ON ZARISKI VAN-KAMPEN THEOREM 3
0 q
s s s s
✻
u ′ v ′ ✻
u
✻ ✻
u v w
❆
❆
❈
❈
F G
❆
❆
❈ ❈
❆
❆
❆
❆
✲ t ❆ ✲ t ❆✁ ❈ ❈
❈ ❈
✲ t ❈ ❈ ✲
u v
t
0 p u ū u u v w
(1)
(2)
(3)
(4)
Figure 2.2. Proof of Lemma 2.2.1.
a homotopy H : I × I → X stationary on ∂I from uv to u ′ v ′ by
{
F (2t, s) if 0 ≤ t ≤ 1/2
H(t, s) :=
G(2t − 1,s) if 1/2 ≤ t ≤ 1.
(2) We can construct a homotopy F : I × I → X stationary on ∂I from 0 p u to
u by
{
p
if 0 ≤ t ≤ (1 − s)/2
F (t, s) :=
u(1 − 2(1 − t)/(s + 1)) if (1 − s)/2 ≤ t ≤ 1.
A homotopy stationary on ∂I from u0 p to u can be constructed in a similar way.
(3) We can construct a homotopy F : I × I → X stationary on ∂I from ūu to
0 q by
⎧
⎪⎨ ū(2t)
if 0 ≤ t ≤ (1 − s)/2
F (t, s) := ū(1 − s) =u(s) if (1− s)/2 ≤ t ≤ (1 + s)/2
⎪⎩
u(2t − 1) if (1 + s)/2 ≤ t ≤ 1.
A homotopy stationary on ∂I from uū to 0 p can be constructed in a similar way.
(4) We can construct a homotopy F : I × I → X stationary on ∂I from u(vw)
to u(vw) by
⎧
⎪⎨ u(4t/(2 − s))
if 0 ≤ t ≤ (2 − s)/4
F (t, s) := v(4t + s − 2)
if (2 − s)/4 ≤ t ≤ (3 − s)/4
⎪⎩
w((4t + s − 3)/(s + 1)) if (3 − s)/4 ≤ t ≤ 1.
The following is obvious from the definition:
Lemma 2.2.2. Let u and v be paths on X with u(1) = v(0), and φ : X → Y a
continuous map. Then φ ◦ u and φ ◦ v are paths on Y with (φ ◦ u)(1) = (φ ◦ v)(0)
and they satisfy φ ◦ (uv) =(φ ◦ u)(φ ◦ v).
We fix a point b of X, and call it a base point of X. A path from b to b is
called a loop with the base point b. Let π 1 (X, b) denote the set of homotopy classes
(relative to ∂I) of loops with the base point b. We define a structure of the group
on π 1 (X, b) by
[u] · [v] :=[uv].
From Lemma 2.2.1 (1), this product is well-defined; that is, [uv] does not depend
on the choice of the representatives u of [u] and v of [v]. By Lemma 2.2.1 (4), this