Download e-book for iPad: Algebraic Topology: A First Course (Graduate Texts in by William Fulton

By William Fulton

ISBN-10: 0387943277

ISBN-13: 9780387943275

This booklet introduces the $64000 rules of algebraic topology by way of emphasizing the relation of those principles with different components of arithmetic. instead of deciding upon one standpoint of contemporary topology (homotropy thought, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a couple of dimensions, introducing in simple terms as a lot algebraic equipment as worthwhile for the issues encountered. This makes it attainable to determine a greater variety of significant gains within the topic than is usual in introductory texts; it's also in concord with the historic improvement of the topic. The e-book is aimed toward scholars who don't unavoidably intend on focusing on algebraic topology.

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Additional info for Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153)

Example text

The vectors x1 , x2 , . , given by xk = (0, . . , 0, 1, 0, . ) with 1 in the k-th position are easily seen to be linearly independent, but they do not form a spanning set. Indeed, since the span consists only of the finite linear combinations of vectors from the set, the span in this case is the set of all vectors of the form (a1 , . . , ak , 0, . . , 0, 0, . ), namely those infinite sequences of complex numbers that are eventually 0. In other words, the span in C∞ of the vectors x 1 , x2 , .

Scalars will typically be denoted by lower-case Greek letters from the beginning of the alphabet, namely α, β, γ , and so on, while vectors will be denoted by x, y, z, etc. In either case, subscripts or superscripts may be used to enhance readability. 1 Elementary Properties of Linear Spaces We now turn to establish several properties of linear spaces that immediately follow from the axioms. 1 In any linear space V the following statements hold. 1. The neutral element 0 ∈ V is unique. 2. For all x ∈ V , the additive inverse x is unique.

It is completely straightforward to demonstrate that the set {[x] | x ∈ X } of all equivalence classes is a partition of X , and we thus obtain a function P : Equ(X ) → Par(X ). It is quite easy to verify that in fact P is the inverse function of E and thus we have established a bijective correspondence between equivalence classes on X and partitions of X . Given an equivalence relation ∼ on a set X , the set {[x] | x ∈ X } is denoted by X/∼ and is called the quotient set of X modulo ∼. There is also the corresponding function π : X → X/∼, given by π(x) = [x], called the canonical projection.

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Algebraic Topology: A First Course (Graduate Texts in Mathematics, Volume 153) by William Fulton


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