Algebraic Topology: A First Course (Graduate Texts in - download pdf or read online

By William Fulton

ISBN-10: 0387943277

ISBN-13: 9780387943275

This publication introduces the $64000 principles of algebraic topology through emphasizing the relation of those principles with different components of arithmetic. instead of settling on one perspective of recent topology (homotropy conception, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a number of dimensions, introducing basically as a lot algebraic equipment as priceless for the issues encountered. This makes it attainable to work out a greater variety of significant gains within the topic than is usual in introductory texts; it's also in concord with the ancient improvement of the topic. The booklet is geared toward scholars who don't inevitably intend on focusing on algebraic topology.

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

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A basic property of convex geometries is that the convex hull of the union of two convex sets C,D is the union of the convex hulls of all sets {c,d}, where c e C and d e D. Already in this general setting it is possible to establish several properties which were first observed for convex sets in a real vector space. In a convex geometry also, extreme points and faces may be characterized more simply than in an arbitrary alignment Four more axioms are then introduced, each of which on its own ensures some useful additional property of a convex geometry.

Thus [T] r. S s T, which is all that requires proof. Conversely, a set Sis independent if for each Ts S there exists a convex set C such that T = C r. S . For suppose some x e S is not an extreme point of S. Thenx e [S\x]. By hypothesis there exists a convex set C such thatS\x =Cr. S. Then [S\x] s C. Hencex e Cr. S, which is a contradiction. Again, if S and T are sets such that [SJ = [T] and if S is finite, then [S] =[Sr. T]. For, by Proposition 1 l(ii), Sand T have the same set E of extreme points.

3 ADDITIONAL AXIOMS We now examine the consequences of imposing various additional axioms on a convex geometry. These axioms may seem arbitrary at first sight, but their natural role will become apparent in the next chapter. We first consider the axiom (L2) if be [a,c], c e [b,d] and b "* c, then be [a,d]. If the convex geometry X satisfies (L2}, then the alignment of convex sets is an anti-exchange alignment. PROPOSITION 10 Proof We will show that if e is an extreme point of S, then it is also an extreme point of [S].

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

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