By Kenji Ueno, Koji Shiga, Shigeyuki Morita
This booklet will convey the sweetness and enjoyable of arithmetic to the study room. It deals critical arithmetic in a full of life, reader-friendly type. integrated are routines and lots of figures illustrating the most techniques.
The first bankruptcy provides the geometry and topology of surfaces. between different themes, the authors talk about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of elements of the concept that of measurement, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it really is proved that the three-d sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a chain of lectures given via the authors at Kyoto collage (Japan).
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This article suits any path with the be aware "Manifold" within the name. it's a graduate point e-book.
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The group π1 (Mg2 (1)) is isomorphic to a group on generators c1 , . . , c2g+1 wit a single relation c21 · · · c22g+1 = 1. 2. The group π1 (Mg2 (2)) is isomorphic to a group on generators c1 , . . , c2g+2 wit a single relation c21 · · · c22g+1 c22g+2 = 1. 13. 7. 14. Compute π1 (RPn ), π1 (Kl2 ). 15. Compute the group π/[π, π] for the groups π = π1 (Mg2 (1)), π1 (Mg2 (2)). 16. 7 are pairwise nonisomorphic. Prove that any two manifolds above are not homeomorphic and even are not homotopy equivalent to each other.
There is a natural question: Question: Does there exist a map f : Z −→ T covering the map f : Z −→ X , such that f (z0 ) = x0 ? In other words, the lifting map f should make the following diagram commutative: T f (20) Z f ✒ p ❄ ✲ X NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 55 where f (z0 ) = x0 , f (z0 ) = x0 . Clearly the diagram (20) gives the following commutative diagram of groups: π1 (T, x0 ) f∗ (21) ✒ p∗ ❄ f∗ ✲ π1 (Z, z0 ) π1 (X, x0 ) It is clear that commutativity of the diagram (21) implies that f∗ (π1 (Z, z0 )) ⊂ p∗ (π1 (T, x0 )).
Let a base point x0 ∈ Y1 ∩ Y2 ⊂ X , and ρ1 : π1 (Y1 ) −→ π1 (X), ρ2 : π1 (Y2 ) −→ π1 (X). Then π1 (X) ∼ = π1 (Y1 ) ∗π (Z) π1 (Y2 ). 20. 8 in the case of finite CW -complexes using induction on the number of cells of Y1 ∩ Y2 . Remark. There is more general version of Van Kampen Theorem, see [Massey, Ch. 2]. 52 BORIS BOTVINNIK 7. 1. Definition and examples. A path-connected space T is a covering space over a pathconnected space X , if there is a map p : T −→ X such that for any point x ∈ X there exists a path-connected neighbourhood U ⊂ X , such that p−1 (U ) is homeomorphic to U × Γ (where Γ is a discrete set), futhermore the following diagram commutes ∼ = p−1 (U ) ❅ (18) ❅ ✲ U ×Γ p ❅ ❅ ❘ ❅ ✠ pr U The neighbourhood U from the above definition is called elementary neighborhood.