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Covering Maps and the Monodromy Theorem-The Fundamental Group of the Circle
المؤلف:
David R. Wilkins
المصدر:
Algebraic Topology
الجزء والصفحة:
...
21-6-2017
1717
Theorem 1.6 π1(S1, b) ≅Z for any b ∈ S1.
Proof We regard S1 as the unit circle in R2. Without loss of generality, we can take b = (1, 0). Now the map p: R → S1 which sends t ∈ R to (cos 2πt,sin 2πt) is a covering map, and b = p(0). Moreover p(t1) = p(t2) if and only if t1 − t2 is an integer; in particular p(t) = b if and only if t is an integer.
Let α and β be loops in S1 based at b, and let ˜α and β˜ be paths in R that satisfy p ◦ α˜ = α and p ◦ β˜ = β. Suppose that α and β represent the same element of π1(S1, b). Then there exists a homotopy F: [0, 1] × [0, 1] → S1
such that F(t, 0) = α(t) and F(t, 1) = β(t) for all t ∈ [0, 1], and F(0, τ ) = F(1, τ ) = b for all τ ∈ [0, 1]. It follows from the Monodromy Theorem (Theorem 1.5) that this homotopy lifts to a continuous map G: [0, 1]×[0, 1] → R satisfying p ◦ G = F. Moreover G(0, τ ) and G(1, τ ) are integers for all τ ∈ [0, 1], since p(G(0, τ )) = b = p(G(1, τ )). Also G(t, 0)−α˜(t) and G(t, 1)− β˜(t) are integers for all t ∈ [0, 1], since p(G(t, 0)) = α(t) = p(α˜ (t)) and p(G(t, 1)) = β(t) = p(β˜(t)). Now any continuous integer-valued function on [0, 1] is constant, by the Intermediate Value Theorem. In particular the functions sending τ ∈ [0, 1] to G(0, τ ) and G(1, τ ) are constant, as are the functions sending t ∈ [0, 1] to G(t, 0) − α˜(t) and G(t, 1) − β˜(t). Thus
G(0, 0) = G(0, 1), G(1, 0) = G(1, 1),
G(1, 0) − α˜(1) = G(0, 0) − α˜(0), G(1, 1) − β˜(1) = G(0, 1) − β˜(0).
On combining these results, we see that
α˜(1) − α˜(0) = G(1, 0) − G(0, 0) = G(1, 1) − G(0, 1) = β˜(1) − β˜(0)
We conclude from this that there exists a well-defined function λ: π1(S1, b) → Z characterized by the property that λ([α]) = α˜ (1)−α˜(0) for all loops α based at b, where α˜: [0, 1] → R is any path in R satisfying p ◦ α˜ = α.
Next we show that λ is a homomorphism. Let α and β be any loops based at b, and let α˜ and β˜ be lifts of α and β. The element [α][β] of π1(S1, b) is represented by the product path α.β, where
(Note that σ(t) is well-defined when t =1/2.) Then p ◦ σ = α.β and thus
λ([α][β]) = λ([α.β]) = σ(1) − σ(0) = α˜ (1) − α˜(0) + β˜(1) − β˜(0)
= λ([α]) + λ([β]).
Thus λ: π1(S1, b) → Z is a homomorphism.
Now suppose that λ([α]) = λ([β]). Let F: [0, 1] × [0, 1] → S1 be the homotopy between α and β defined by
F(t, τ ) = p( (1 − τ ) α˜ (t) + τβ˜(t) ),
where α˜ and β˜ are the lifts of α and β respectively starting at 0. Now β˜(1) = λ([β]) = λ([α]) = α˜ (1), and β˜(0) = α˜ (0) = 0. Therefore F(0, τ ) = b = p(α˜ (1)) = F(1, τ ) for all τ ∈ [0, 1]. Thus α ≃β rel {0, 1}, and therefore [α] = [β]. This shows that λ: π1(S1, b) → Z is injective.
The homomorphism λ is surjective, since n = λ([γn]) for all n ∈ Z, where the loop γn: [0, 1] → S1 is given by γn(t) = p(nt) = (cos 2πnt,sin 2πnt) for all t ∈ [0, 1]. We conclude that λ: π1(S1 , b) → Z is an isomorphism.
We now show that every continuous map from the closed disk D to itself has at least one fixed point. This is the two-dimensional version of the Brouwer Fixed Point Theorem.
Theorem 1.7 Let f: D → D be a continuous map which maps the closed disk D into itself. Then f(x0) = x0 for some x0 ∈ D Proof Let ∂D denote the boundary circle of D. The inclusion map i: ∂D → D induces a corresponding homomorphism i#: π1(∂D, b) → π1(D, b)of fun damental groups for any b ∈∂D.
Suppose that it were the case that the map f has no fixed point in D.
Then one could define a continuous map r: D → ∂D as follows: for each x ∈ D, let r(x) be the point on the boundary ∂D of D obtained by continuing the line segment joining f(x) to x beyond x until it intersects ∂D at the point r(x). Note that r|∂D is the identity map of ∂D.
Let r#: π1(D, b) → π1(∂D, b) be the homomorphism of fundamental groups induced by r: D → ∂D. Now (r ◦ i)#: π1(∂D, b) → π1(∂D, b) is the identity isomorphism of π1(∂D, b), since r ◦ i: ∂D → ∂D is the identity map. But it follows directly from the definition of induced homomorphisms that (r ◦ i)# = r# ◦ i#. Therefore i#: π1(∂D, b) → π1(D, b) is injective, and r#: π1(D, b) → π1(∂D, b) is surjective. But this is impossible, since π1(∂D, b) ≅ Z (Theorem 1.6) and π1(D, b) is the trivial group. This contradiction shows that the continuous map f: D → D must have at least one fixed point.
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