# Download e-book for kindle: Algebraic Topology Notes(2010 version,complete,175 pages) by Boris Botvinnik By Boris Botvinnik

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Discrete differential geometry is an energetic mathematical terrain the place differential geometry and discrete geometry meet and have interaction. It presents discrete equivalents of the geometric notions and techniques of differential geometry, equivalent to notions of curvature and integrability for polyhedral surfaces.

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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 ∆.