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/CBO9780511662584.005
Subjects: Geometry and topology
Image View Extract Fullview: Text View | Enlarge Image ‹ Previous Chapter ›Next Chapter
Introduction
The main purpose of this paper is to introduce a new algebraic object into topology. This new algebraic structure is called an exact couple of groups (or of modules, or of vector spaces, etc.). It apparently has many applications to problems of current interest *in topology. In the present paper it is shown how exact couples apply to the following three problems: (a) To determine relations between the homology groups of a space X, the Hurewicz homotopy groups of X, and certain additional topological invariants of X; (b) To determine relations between the cohomology groups of a space X, the cohomotopy groups of X, and certain additional topological invariants of X; (c) To determine relations between the homology (or cohomology) groups of the base space, the bundle space, and the fibre in a fibre bundle.
In each of these problems, the final result is expressed by means of a Leray- Koszul sequence. The notion of a Leray-Koszul sequence (also called a spectral homology sequence or spectral cohomology sequence) has been introduced and exploited by topologists of the French school. It is already apparent as a result of their work that the solution to many important problems of topology is best expressed by means of such a sequence. With the introduction of exact couples, it seems that the list of problems, for which the final answer is expressed by means of a Leray-Koszul sequence, is extended still further.
pp. i-ii
pp. iii-vi
pp. 1-31
PAPERS ON ALGEBRAIC TOPOLOGY : Read PDF
pp.
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