By Hajime Sato

ISBN-10: 0821810464

ISBN-13: 9780821810460

The only so much tough factor one faces while one starts to profit a brand new department of arithmetic is to get a suppose for the mathematical experience of the topic. the aim of this booklet is to assist the aspiring reader collect this crucial good judgment approximately algebraic topology in a quick time period. To this finish, Sato leads the reader via uncomplicated yet significant examples in concrete phrases. in addition, effects will not be mentioned of their maximum attainable generality, yet by way of the easiest and so much crucial situations.

In reaction to feedback from readers of the unique variation of this publication, Sato has additional an appendix of precious definitions and effects on units, basic topology, teams and such. He has additionally supplied references.

Topics coated comprise primary notions akin to homeomorphisms, homotopy equivalence, basic teams and better homotopy teams, homology and cohomology, fiber bundles, spectral sequences and attribute periods. items and examples thought of within the textual content comprise the torus, the Möbius strip, the Klein bottle, closed surfaces, mobile complexes and vector bundles.

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**Additional info for Algebraic Topology: An Intuitive Approach**

**Example text**

Am 1 + b m 1 am 2 + b m 2 . . define their sum a1 n + b1 n a2 n + b2 n .. am n + bm n . For any matrix A = (ai j ), we let −A be the matrix (−ai j ). Given a scalar λ ∈ K, we define the matrix λA as the matrix C = (ci j ) such that ci j = λai j ; that is a1 1 a1 2 . . a1 n λa1 1 λa1 2 . . λa1 n a2 1 a2 2 . . a2 n λa2 1 λa2 2 . . λa2 n λ .. .. = .. .. . . . . . . . . am 1 am 2 . . am n λam 1 λam 2 . . 1. MATRICES Given an m × n matrices A = (ai k ) and an n × p matrices B = (bk j ), we define their product AB as the m × p matrix C = (ci j ) such that n ci j = ai k b k j , k=1 for 1 ≤ i ≤ m, and 1 ≤ j ≤ p.

A1n a21 a22 . . a2n .. .. . . . am1 am2 . . amn and v1 v2 .. vn g(u1 ) g(u2 ) b11 b12 b21 b22 .. . bn1 bn2 . . g(up ) . . b1p . . b2p .. .. . . . 1. MATRICES for all k, 1 ≤ k ≤ n, and for every y = y1 v1 + · · · + yn vn , letting f (y) = z = z1 w1 + · · · + zm wm , we have n zi = ai k y k (3) k=1 for all i, 1 ≤ i ≤ m. Then, if y = g(x) and z = f (y), we have z = f (g(x)), and in view of (2) and (3), we have p n zi = bk j x j ) ai k ( j=1 k=1 n p = ai k b k j x j k=1 j=1 p n = ai k b k j x j j=1 k=1 p n = ai k bk j )xj .

8 implies that |I| ≤ |J|. A symmetric argument yields |J| ≤ |I|. 9 also holds for vector spaces that are not finitely generated. When E is not finitely generated we say that E is of infinite dimension. The dimension of a finitely generated vector space E is the common dimension n of all of its bases and is denoted by dim(E). Clearly, if the field R itself is viewed as a vector space, then every family (a) where a ∈ R and a = 0 is a basis. Thus dim(R) = 1. Note that dim({0}) = 0. If E is a vector space of dimension n ≥ 1, for any subspace U of E, if dim(U ) = 1, then U is called a line; if dim(U ) = 2, then U is called a plane; if dim(U ) = n − 1, then U is called a hyperplane.

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