# Algebraic Topology: Homology and Cohomology by Andrew H. Wallace

By Andrew H. Wallace

This self-contained textual content is acceptable for complex undergraduate and graduate scholars and should be used both after or at the same time with classes mostly topology and algebra. It surveys numerous algebraic invariants: the basic crew, singular and Cech homology teams, and various 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, despite the fact that, the writer advances an figuring out of the subject's algebraic styles, leaving geometry apart to be able to research those styles as natural algebra. various routines seem through the textual content. as well as constructing scholars' pondering by way of algebraic topology, the routines additionally unify the textual content, given that lots of them characteristic effects that seem in later expositions. vast appendixes provide necessary experiences of heritage material.

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

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Theorem 1-12. For any coefficient group the following diagram is commu- tative : Cp(E) -) Cp+I (E x I) i Cp(E') fi P Cp+ (E' x I) 26 Singular Homology Theory Proof. This expresses exactly what would be expected from geometric intuition : given a simplex in E = E x {0} then the result of constructing a prism over it and then mapping into E' x I by f' is the same as that of first carrying the simplex over to E', by means of f, and then constructing the prism over the result. The proof is a straightforward verification.

Thus, the kernel of i* is zero and the proof is complete. I 2 Singular and Simplicial Homology The main theorems of the last chapter are the basic properties of the singular homology groups. With these properties as a foundation, this chapter develops a systematic method for calculating the homology groups of certain types of space. At the same time, this procedure indicates the possibility of an axiomatic approach to the subject. 2-1. : (E, F) -* (E', F') a homomorphism f* : HH(E, F) -+ HH(E', F') for each p.

Let Gi G, + 1 > be a sequence of groups and homomorphisms with the property that the kernel of Oi is equal to the image of O i _ 1 for all i. This is called an exact sequence. The sequence can be assumed to extend infinitely in either direction, or it may terminate. ) ascend, as here, or descend is clearly immaterial. Now let is j: Y) be inclusion maps, that is, maps defined by i(x) = x and j(x) = x. These maps are continuous; hence they induce homomorphisms on the chain groups and in turn induce homomorphisms al Cp(Y) Cp(X) CC(X) Cp(X) CC(Z) CC(Y) Cp( Z) C p(Z ) For convenience write CC(X, Y) for CC(X)/CC(Y), and use similar notations for the other pairs.