Algebraic Topology: An Intuitive Approach (Translations of by Hajime Sato

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.

Show description

Read Online or Download Algebraic Topology: An Intuitive Approach (Translations of Mathematical Monographs, Volume 183) PDF

Similar topology books

Introduction to Topological Manifolds (Graduate Texts in Mathematics)

This ebook is an creation to manifolds at first graduate point, and available to any pupil who has accomplished an effective undergraduate measure in arithmetic. It includes the fundamental topological principles which are wanted for the extra learn of manifolds, quite within the context of differential geometry, algebraic topology, and comparable fields.

Braids and Coverings: Selected Topics (London Mathematical Society Student Texts)

This publication is predicated on a graduate direction taught through the writer on the collage of Maryland. The lecture notes were revised and augmented by means of examples. the 1st chapters increase the simple idea of Artin Braid teams, either geometrically and through homotopy conception, and talk about the hyperlink among knot conception and the combinatorics of braid teams via Markou's Theorem.

Convergence and Uniformity in Topology. (AM-2) (Annals of Mathematics Studies)

The outline for this e-book, Convergence and Uniformity in Topology. (AM-2), can be drawing close.

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) .

Download PDF sample

Rated 4.73 of 5 – based on 43 votes

About the Author