Algebraic Topology – A Student's Guide
A Student's Guide
By J. F. Adams
General editor G. C. Shepherd
Publisher: Cambridge University Press
Print Publication Year: 1972
Online Publication Date:May 2010
Online ISBN:9780511662584
Paperback ISBN:9780521080767
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662584.014
Subjects: Geometry and topology
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The next piece is a summary on generalised homology and cohomology written by me especially for the present work.
It is generally accepted that a functor can be called an ordinary homology or cohomology theory if it satisfies the seven axioms of Eilenberg and Steenrod (see Paper no. 2). These axioms serve to describe the behaviour of the functor on finite complexes. If we want to describe the behaviour of the functor on infinite complexes, we add suitable axioms about limits (see Paper no. 15).
A generalised homology or cohomology theory is a functor which satisfies all of the axioms of Eilenberg and Steenrod except for the dimension axiom. In addition, we may impose suitable axioms about limits. In recent years several such functors have been found useful in algebraic topology. The most important are κ-theory and various sorts of bordism and cobordism theory (see Papers no. 19, 20, 21, 23, 24). Various forms of stable homotopy and cohomotopy also satisfy the axioms, but are hard to calculate.
Obviously the beginning of the subject involves one in setting up various such examples and proving that they satisfy the axioms. This overlaps with topics to be considered below.
The next part of the subject, which is rather formal, consists in exploiting the consequences of the six axioms of Eilenberg and Steenrod which one does assume. For example, one has the Mayer- Vietoris sequence, and one has the spectral sequence of Atiyah and Hirzebruch (see § 12 and Papers no. 14, 19). If one wishes to study also infinite complexes, one exploits the consequences of whatever axiom one has on limits (see §12 and Papers no. 15, 16, 17).
pp. i-ii
pp. iii-vi
pp. 1-31
PAPERS ON ALGEBRAIC TOPOLOGY
1 - Combinatorial homotopy : Read PDF
pp. 32-45
2 - An axiomatic approach to homology theory : Read PDF
pp. 46-50
3 - La suite spectrale. 1: Construction générale : Read PDF
pp. 51-65
4 - Exact couples in algebraic topology : Read PDF
pp. 66-73
5 - The cohomology of classifying spaces of H-spaces : Read PDF
pp. 74-78
6 - Cohomologie modulo 2 des complexes d'Eilenberg-MacLane : Read PDF
pp. 79-100
7 - On the triad connectivity theorem : Read PDF
pp. 101-112
8 - On the Freudenthal theorems : Read PDF
pp. 113-115
9 - The suspension triad of a sphere : Read PDF
pp. 116-117
10 - On the construction FK : Read PDF
pp. 118-136
11 - On Chern characters and the structure of the unitary group : Read PDF
pp. 137-139
12 - Espaces fibrés et groupes d'homotopie. I, II : Read PDF
pp. 140-145
13 - Generalised homology and cohomology theories : Read PDF
pp. 146-165
14 - Relations between ordinary and extraordinary homology : Read PDF
pp. 166-177
15 - On axiomatic homology theory : Read PDF
pp. 178-183
16 - Characters and cohomology of finite groups : Read PDF
pp. 184-185
17 - Extract from thesis : Read PDF
pp. 186-187
18 - Relations between cohomology theories : Read PDF
pp. 188-195
19 - Vector bundles and homogeneous spaces : Read PDF
pp. 196-222
20 - Lectures on K-theory : Read PDF
pp. 223-238
21 - Vector fields on spheres : Read PDF
pp. 239-241
22 - On the groups J(X). IV : Read PDF
pp. 242-259
23 - Summary on complex cobordism : Read PDF
pp. 260-273
24 - New ideas in algebraic topology : Read PDF
pp. 274-300