Edited by Nigel Ray
Edited by Grant Walker
Publisher: Cambridge University Press
Print Publication Year: 1992
Online Publication Date:January 2010
Online ISBN:9780511526305
Paperback ISBN:9780521420747
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511526305.014
Subjects: Geometry and Topology
Image View Extract Fullview: Text View | Enlarge Image ‹ Previous Chapter ›Next Chapter
The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some cases completely computable, in a way not possible with the traditional Postnikov systems, or with other models, such as simplicial groups.
These algebraic structures take into account the action of the fundamental group. It follows that the algebra has to be at least as complicated as that of groups, and the basic facts on the use of the fundamental group in 1-dimensional homotopy theory are recalled in Section 1. It is reasonable to suppose that it is these facts that a higher dimensional theory should imitate.
However, modern methods in homotopy theory have generally concentrated on methods as far away from those for the fundamental group as possible. Such a concentration has its limitations, since many problems in the applications of homotopy theory require a non-trivial fundamental group (low dimensional topology, homology of groups, algebraic K-theory, group actions, …). We believe that the methods outlined here continue a classical tradition of algebraic topology. Certainly, in this theory non- Abelian groups have a clear role, and the structures which appear arise directly from the geometry, as algebraic structures on sets of homotopy classes.
It is interesting that this higher dimensional theory emerges not directly from groups, but from groupoids.
pp. i-iv
pp. v-vi
pp. vii-viii
Contents of Volume 2 : Read PDF
pp. ix-xxiii
1 - The work of J. F. Adams : Read PDF
pp. 1-28
2 - Twisted tensor products of DGA's and the Adams-Hilton model for the total space of a fibration : Read PDF
pp. 29-52
3 - Hochschild homology, cyclic homology and the cobar construction : Read PDF
pp. 53-66
4 - Hermitian A∞ rings and their K-theory : Read PDF
pp. 67-82
5 - A splitting result for the second homology group of the general linear group : Read PDF
pp. 83-88
6 - Low dimensional spinor representations, Adams maps and geometric dimension : Read PDF
pp. 89-102
7 - The characteristic classes for the exceptional Lie groups : Read PDF
pp. 103-130
8 - How can you tell two spaces apart when they have the same n-type for all n? : Read PDF
pp. 131-144
9 - A generalized Grothendieck spectral sequence : Read PDF
pp. 145-162
10 - Localization of the homotopy set of the axes of pairings : Read PDF
pp. 163-178
11 - Fibrewise reduced product spaces : Read PDF
pp. 179-186
12 - Computing homotopy types using crossed n-cubes of groups : Read PDF
pp. 187-210
13 - On orthogonal pairs in categories and localization : Read PDF
pp. 211-224
14 - A note on extensions of nilpotent groups : Read PDF
pp. 225-234
15 - On the Swan subgroup of metacyclic groups : Read PDF
pp. 235-240
16 - Fields of spaces : Read PDF
pp. 241-254
17 - Maps between p-completed classifying spaces, III : Read PDF
pp. 255-270
18 - Retracts of classifying spaces : Read PDF
pp. 271-280
19 - On the dimension theory of dominant summands : Read PDF
pp. 281-292