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Geometry And Topology

Algebraic Topology Notes(2010 version,complete,175 pages) by Boris Botvinnik

By Boris Botvinnik

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P : S n −→ RPn , where p maps a point x ∈ S n to the line in Rn+1 going through the origin and x. 2. Theorem on covering homotopy. The following result is a key fact allowing to classify coverings. 1. Let p : T → X be a covering space and Z be a CW -complex, and f : Z → X , f : Z → T such that the diagram T f (19) p ❄ ✲ X f Z ✒ commutes; futhermore it is given a homotopy F : Z × I −→ X such that F |Z×{0} = f . 2. For any path s : I −→ X and any point x0 ∈ T , such that p(x0 ) = x0 = s(0) there exists a unique path s : I −→ T , such that s(0) = x0 and p ◦ s = s.

Vq+1 ) | tj = 0 } . 7. A finite triangulation of a subset X ⊂ Rn is a finite covering of X by simplices {∆n (i)} such that each intersection ∆n (i) ∩ ∆n (j) either empty, or ∆n (i) ∩ ∆n (j) = ∆n−1 (i)k for some k = 0, . . , n. 5. Let ∆n1 , . . , ∆ns be a finite set of n-dimensional simplexes in Rn . Prove that the union K = ∆n1 ∪ ∆n2 ∪ · · · ∪ ∆ns is a finite simplicial complex. 6. Let ∆p1 , ∆q2 be two simplices. Prove that K = ∆p1 × ∆q2 is a finite simplicial complex. A barycentric subdivision of a q -simplex ∆q is a subdivision of this simplex on (q +1)!

So we can say that a map S 1 −→ X to a path-connected space X determines an element of π1 (X, x0 ) up to conjugation. Let X be a CW -complex with a single zero-cell e0 = x0 , one-cells e1i , i ∈ I , and twocells e2j , j ∈ J . Then we identify the first skeleton X (1) with i∈I Si1 . The inclusion map Si1 → i∈I Si1 determines an element αi ∈ π1 (X (1) , x0 ). 4 π1 (X (1) , x0 ) is a free group on generators αi , i ∈ I . The characteristic map gj : D2 −→ X of the cell e2j determines attaching map fj : S 1 −→ X (1) which determines an element βj ∈ π1 (X (1) , x0 ) up to conjugation.

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