By Hajime Sato

The only such a lot tricky factor one faces whilst one starts to benefit a brand new department of arithmetic is to get a think for the mathematical feel of the topic. the aim of this e-book is to assist the aspiring reader gather this crucial logic approximately algebraic topology in a brief time period. To this finish, Sato leads the reader via easy yet significant examples in concrete phrases. furthermore, effects usually are not mentioned of their maximum attainable generality, yet when it comes to the best and such a lot crucial circumstances. according to feedback from readers of the unique variation of this e-book, Sato has additional an appendix of beneficial definitions and effects on units, normal topology, teams and such. He has additionally supplied references.Topics coated contain primary notions comparable to homeomorphisms, homotopy equivalence, primary teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. items and examples thought of within the textual content contain the torus, the Mobius strip, the Klein bottle, closed surfaces, cellphone complexes and vector bundles.

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**Extra resources for Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs, Volume 183)**

**Example text**

Consider the following subspace D ⊂ H D= u∈H σk+1 σk+1 : | |u| = 1, ǫj , u = 0, j = 1, . . , k . 12. Prove that D is homeomorphic to the hemisphere of the dimension σk+1 − k − 1. Thus D is a closed cell of dimension σk+1 −k −1. 2. We define the map f : E(σ1 , . . , σk ) × D −→ E(σ1 , . . , σk , σk+1 ) by the formula f ((v1 , . . , vk ), u) = (v1 , . . , vk , T u) where T is given by (13). We notice that vi , T u = T ǫi , T u = ǫi , u = 0, i = 1, . . , k, and T u, T u = u, u = 1 by definition of T and since T ∈ O(n).

We notice that it is enough to define a map (n) F1 : X ∪ ((A ∪ X (n) ∪ en+1 ) × I) −→ Y extending F (n) to a single cell en+1 . Let en+1 be a (n + 1)-cell such that en+1 ⊂ X \ A. 34 BORIS BOTVINNIK By induction, the map F (n) is already given on the cylinder (¯ en+1 \en+1 )×I since the boundary n+1 n+1 n+1 (n) n+1 (n+1) ∂e = e¯ \e ⊂ X . Let g : D −→ X be a characteristic map corresponding (n) n+1 to the cell e . We have to define an extension of F1 from the side g(S n ) × I and the bottom base g(Dn+1 ) × {0} to the cylinder g(Dn+1 ) × I .

11. 12. Let Y be n-connected CW -complex, and X be an n-dimensional CW complex. Then the set [X, Y ] consists of a single element. A pair of spaces (X, A) is n-connected if for any k ≤ n and any map of pairs f : (Dk , S k−1 ) −→ (X, A) homotopic to a map g : (Dk , S k−1 ) −→ (X, A) (as a map of pairs) so that g(Dk ) ⊂ A. 11. What does it mean geometrically that a pair (X, A) is 0-connected? 1connected? Give some alternative description. 12. Let (X, A) be an n-connected pair of CW -complexes. Prove that (X, A) is homotopy equivalent to a CW -pair (Y, B) so that B ⊂ Y (n) .