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.

Show description

Read or Download Algebraic Topology: Homology and Cohomology PDF

Similar topology books

Selectors

Although the hunt for strong selectors dates again to the early 20th century, selectors play an more and more very important position in present examine. This ebook is the 1st to gather the scattered literature right into a coherent and stylish presentation of what's identified and confirmed approximately selectors--and what continues to be discovered.

From Topology to Computation: Proceedings of the Smalefest

A unprecedented mathematical convention used to be held 5-9 August 1990 on the collage of California at Berkeley: From Topology to Computation: solidarity and variety within the Mathematical Sciences a world learn convention in Honor of Stephen Smale's sixtieth Birthday the subjects of the convention have been many of the fields during which Smale has labored: • Differential Topology • Mathematical Economics • Dynamical structures • conception of Computation • Nonlinear practical research • actual and organic functions This publication includes the lawsuits of that convention.

Applications of Contact Geometry and Topology in Physics

Even though touch geometry and topology is in brief mentioned in V I Arnol'd's booklet "Mathematical equipment of Classical Mechanics "(Springer-Verlag, 1989, 2d edition), it nonetheless continues to be a site of study in natural arithmetic, e. g. see the new monograph by way of H Geiges "An creation to touch Topology" (Cambridge U Press, 2008).

Why Prove it Again?: Alternative Proofs in Mathematical Practice

This monograph considers numerous recognized mathematical theorems and asks the query, “Why turn out it back? ” whereas studying replacement proofs. It explores the several rationales mathematicians could have for pursuing and proposing new proofs of formerly confirmed effects, in addition to how they pass judgement on even if proofs of a given outcome are various.

Additional info for Algebraic Topology: Homology and Cohomology

Sample text

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.

Download PDF sample

Rated 4.30 of 5 – based on 5 votes