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A “product” of the loops ϕ, ψ is the loop ω , difined by the formula: ω(t) = ϕ(2t), for 0 ≤ t ≤ 1/2, ψ(2t − 1), for 1/2 ≤ t ≤ 1. This product operation induces a group structure of π1 (X). It is easy to check that a group operation is well-defined. 4. Write an explicit formula givinig a null-homotopy for the composition ϕ¯ · ϕ. 3. Dependence of the fundamental group on the base point. 2. Let X be a path-connected space, then π1 (X, x0 ) ∼ = π1 (X, x1 ) for any two points x0 , x1 ∈ X . Proof. Since X is path-connected, there exist a path α : I −→ X , such that α(0) = x0 , α(1) = x1 .

Obviously Y is contractible. Now note that X/Y ∼ X ∼ X , and the complex X/Y has only one zero cell. Now we use induction. Let us assume that we already have constructed the CW -complex X ′ such that X ′ ∼ X and X ′ has a single zero cell, and it does not have cells of dimensions 1, 2, . . , k − 1, where k ≤ n. Note that a closure of each k -cell of X ′ is a sphere S k by induction. Indeed, an attaching map for every k -cell has to go to X ′(0) . Since X ′ is still k -connected, then the embedding S k −→ X ′ (corresponding to a cell eki ) may be extended to a map Dk+1 −→ X ′ .

Clearly there is an extension Φ : D2 −→ X of the map ϕ. By the Cellular Approximation Theorem we can assume that Φ(D2 ) ⊂ X (2) . We triangulate D2 in such way that if ∆ is a triangle from this (j) triangulation such that Φ(∆) ∩ d4 = ∅, then (a) Φ(∆) ⊂ d(j) and (b) diam(Φ(∆) < r(j) /5. Let K be a union of all triangles ∆ of our triangulation such that Φ(∆) ∩ (j) d4 j∈J = ∅. Now we make the map Φ′ : K −→ X (2) which concides with Φ on the vertexes of each simplex and is linear on each simplex ∆.

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Analytische Geometrie by Pickert G.

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