By Andrew H. Wallace

This self-contained textual content is appropriate for complex undergraduate and graduate scholars and should be used both after or at the same time with classes usually topology and algebra. It surveys numerous algebraic invariants: the elemental staff, singular and Cech homology teams, and quite a few cohomology groups.

Proceeding from the view of topology as a sort of geometry, Wallace emphasizes geometrical motivations and interpretations. as soon as past the singular homology teams, although, the writer advances an figuring out of the subject's algebraic styles, leaving geometry apart in an effort to learn those styles as natural algebra. a variety of routines seem in the course of the textual content. as well as constructing scholars' considering when it comes to algebraic topology, the routines additionally unify the textual content, considering lots of them characteristic effects that seem in later expositions. vast appendixes provide worthy experiences of heritage material.

Reprint of the W. A. Benjamin, Inc., ny, 1970 variation.

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**Example text**

Referring back to the construction of a this means that if a is represented by a E Cp(X, Y) then a = j1p(p) and dp /3 = i i -1(y) and here y is homologous to 0. Thus y = d p e with e E C,(Y, Z ). Hence dp j3= i i -1 d p e =dpi1 pe (by commutativity). Hence dp(,8 - i1pO) = 0. Thus j3 - i1p9 represents an element fi of HH(Y. Z) and j1 p(P - i1 PO) = j1 P$ = a, so that a = j *p. Hence the kernel of a is contained in the image of j,k. j3 E CC(X, Z). Parts (3) and (4) imply that the image of j* is equal to the kernel of a.

It follows that f is a continuous function on Kr' which is equal to 1 at points of Kr'- 1. It is clear that the points of Kr'- 1 are the only points at which f takes the value 1. Let V be the set of points of Kr for which f (x) > I and let W be the set for which f (x) > J. Since f is continuous on Kr, the sets V and W are open in Kr. It is also easy to see that W is the set of points for which f (x) >_ J (the closure being taken in Kr' ). Since f (x) = 1 only on K'r_ 1, it follows that K,_1cWc We V=V The essential properties of V and W are that there is a deformation of Kr which collapses V into Kr'- 1, and that the removal of W leaves a disjoint union of r simplexes.

Relation The homomorphisms d, P, f1, g1 are connected by the g1 - f1 = dP + Pd It is now possible to prove the homotopy theorem in its second formulation, namely, Theorem 1-11. Proof of Theorem 1-11. Let a E HH(E, F) and let a be a relative cycle of E modulo F representing a. Then, by Theorem 1-16, g1(a) - f1(a) = dPa + Pda Since doe c- Cp _ 1(F), Pda E Cp(F x I). Hence this equation means that g1(a) and f1(a) are homologous in E x I modulo F x I. Since these are relative cycles representing g*(a) and f*(a), respectively, it follows that g*(a) = f*(a).