By James W. Vick

ISBN-10: 0127212507

ISBN-13: 9780127212500

This booklet is designed to be an creation to a few of the elemental rules within the box of algebraic topology. specifically, it really is dedicated to the rules and purposes of homology thought. the single prerequisite for the coed is a easy wisdom of abelian teams and aspect set topology. The necessities of singular homology are given within the first bankruptcy, in addition to probably the most very important functions. during this manner the scholar can speedy see the significance of the fabric. The successive subject matters comprise attaching areas, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology items, manifolds, Poincare duality, and glued element thought. during the booklet the process is as illustrative as attainable, with various examples and diagrams. Extremes of generality are sacrificed once they are inclined to vague the basic recommendations concerned. The ebook is meant to be simply learn via scholars as a textbook for a direction or as a resource for person research. the second one variation has been considerably revised. It encompasses a new bankruptcy on protecting areas as well as illuminating new routines.

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I 2 1 is a map, then d ( C f ) d ( f ) . 1 0 Note that if f ( x l ,. . ,. Y " + ~ ) -- (-s1 ,. . ul,.. , x,,+~) (--sl,. . 20. 21 COROLLARY I f f : S" - S" is given by then d( f ) = -1. Let h : S" + S" be the map that exchanges the first coordinate and the ith coordinate. Then h is a homeomorphism (h-l = h), so d(h) = * l . Let g ( x l , . . Y,+,) = (-&,. . , x,+~) so that d ( g ) = -1. Then PROOF d ( f ) = d ( h o g 0 h ) = d(h)"(g) = (*l)*(-1) = 0 -1. 22 COROLLARY The antipodal map A : Sn + S" A ( x l , ..

Since f is one to one, it follows that c - c’ = db and c and c’ are homologous cycles. Therefore, the correspondence induced on the homology groups is well defined and obviously must be a homomorphism. This homomorphism is denoted by d : H,(E) + H,-,(C) and called the connecting homomorphism for the short exact sequence 0 + C t D+ E + 0. 13 THEOREM If 0 -+ C f a g D+ E + 0 is a short exact sequence of chain complexes and degree zero chain maps, then the long exact sequence - - - J* * * H,(D) Q* A H,(E) H,-l(C) - -f.

Denote by fi: Dn+l + RP(n 1) the composition of the maps - + - + 5 E T + ~E sn+l-. ~ n + l h RP(n + 11, where h is the quotient map on S"+'. Thus, we have a mapping of the union Dn+l u RP(n) 3RP(n + 1). It is not difficult to check thatf, u i is onto; in fact,fl is onto. Note that l), f,-'(z) is either a single point of Dn+l - Sn or a pair for z E RP(n {x, -x} in Sn,the latter being true if and only if z lies in the subspace RP(n). 5 are satisfied and we conclude that RP(n 1) is homeomorphic to Dn+l u nRP(n), the space given by attaching an (n 1)-cell to RP(n) via z.

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