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 normally topology and algebra. It surveys a number of algebraic invariants: the elemental team, singular and Cech homology teams, and a number of cohomology groups.

Proceeding from the view of topology as a kind of geometry, Wallace emphasizes geometrical motivations and interpretations. as soon as past the singular homology teams, even though, the writer advances an realizing of the subject's algebraic styles, leaving geometry apart with the intention to learn those styles as natural algebra. various workouts look 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 a lot of them function effects that seem in later expositions. large appendixes provide invaluable reports of heritage material.

Reprint of the W. A. Benjamin, Inc., manhattan, 1970 version.

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1-10. The Homotopy Theorem 23 Note. Some special attention should be given to the lower end of the homology sequence. The last terms will be - H0(X, Z) H0(X, Y) An element a c- H0(X, Y) is represented by an element a c- C0(X, Since Y). j1 ° is onto, a = j1 °/3 for some j3 E C0(X, Z) but since there are no elements of dimension -1, dp j3 must be 0. Thus j3 represents an element /3 E H0(X, Z) and a = j* p. Thus j* is onto at dimension 0. The homology sequence can thus be completed by adding a 0 at the lower end: - H0(X , Z) '- H0(X, Y) -- 0 and it will be exact up to the last term.

Prove that for any space X, H,(X, X) = 0 for all r. 1-10. THE HOMOTOPY THEOREM Intuitively, if a one cycle on a space is thought of as a closed curve, then a continuous deformation of the curve from one position to another will trace out a surface whose boundary is formed by the initial and final positions of the curve. In this way, homologous cycles, should be represented by the initial and final positions of the curve. , if one can be continuously deformed into the other) then for any cycle a on E the cycles f1(a) and g1 (a) in E' can be thought of as obtained from each other by continuous deformation.

1-12. 1 1) and let P be a point. Let f: ESP map Eon the point P and let g : P --*E map P on, for example, the center of E. Show that fg and gf are both homotopic to the identity. Then show that, for any coefficient group, Hr(E) = 0(r > 0) and that Ho(E) is isomorphic to the coefficient group. 1-13. Exercise 1-12 illustrates, in a simple way, the notion of deformation retrac- Let E be a space and F a subspace. F such that fg and gf are both homotopic to the identity. Then F is called a deformation retract of E.