By Botvinnik B.
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Translated from the preferred French variation, the target of the booklet is to supply a self-contained advent to intend topological size, an invariant of dynamical platforms brought in 1999 via Misha Gromov. The publication examines how this invariant was once effectively utilized by Elon Lindenstrauss and Benjamin Weiss to respond to a long-standing open query approximately embeddings of minimum dynamical structures into shifts.
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Let us choose a cell e0 and for each zero cell e0i choose a path si connecting e0i and e0 (these paths may have nonempty intersections). By Cellular Approximation Theorem we can choose these paths inside 1-skeleton. Now for each path si we glue a 2-disk, identifying a half-circle with si , see the picture: NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 41 1 Y e2 111 000 000000 111111 000 111 000000 111111 0000000000000000000000 1111111111111111111111 X 000 111 000000 0000000000000000000000 1111111111111111111111 000111111 111 000000 111111 0000000000000000000 1111111111111111111 0000000000000000000000 1111111111111111111111 1 e1111111111111111111 0000000000000000000 0000000000000000000000 1111111111111111111111 1 0000000000000000000 1111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000000 0 e1111111111111111111111 0000000000000000000 1111111111111111111 0000000000000000000000 1111111111111111111111 0 0000000000000000000 1111111111111111111 s2 0000000000000000000000 1111111111111111111111 s1 0000000000000000000 1111111111111111111 e02 0000000000000000000 1111111111111111111 0 e 1 X Figure 19 We denote the resulting CW -complex by X .
Proof. For each point x ∈ V¯ there exists a simplex ∆n (x) with a center at x and ∆n (x) ⊂ U . By compactness of V¯ there exist a finite number of simplices ∆n (xi ) covering V¯ . 6 to conclude that a union of finite number of ∆n (xi ) has a finite triangulation. 5. 6. We consider carefully our map ϕ : U −→ D . First we construct the disks d1 , d2 , d3 , d4 inside the disk d with the same center and of radii r/5, 2r/5, 3r/5, 4r/5 respectively, where r is a radius of d. Then we cover V = ϕ−1 (d) by finite number of p-simplexes ∆p (j), such that ∆n (j) ⊂ U .
7 that πn (T 2 ) = 0 for n ≥ 2. 17. Let Kl2 be the Klein bottle. Construct two-folded covering space Kl2 −→ T 2 . Compute πn (Kl2 ) for all n. 11. Let M 2 be a two-dimensional manifold without boundary, M 2 = S 2 , RP2 . Then πn (M 2 ) = 0 for n ≥ 2. 18. 11. Hint: One way is to construct a universal covering space over M 2 ; this universal covering space turns our to be R2 . 8 shows that πn (X) = πn (M 2 ). ). Now it remains to make an argument in a general case. 8. Lens spaces. We conclude with important examples.