George W. Whitehead

On Mappings into Group-like Spaces

This Chapter continues the discussion of H-spaces which was begun in Chapter III. If G is a group-like space, then [X, G] is a group for any space X. This group need not be abelian (although it is if X is the suspension of a space Y). It is natural to ask whether it may be nilpotent. It turns out that the degree of nilpotence is closely related to the Lusternik-Schnirelmann category of the space X. Indeed, a mild change in the definition of the latter in order to relate it more closely to homotopy notions, results in the theorem that, if cat X < c and G is a 0-connected group-like space, then [X, G] is nilpotent of class < c.

Homotopy Theory of CW-complexes

In Chapter II we proved that if X is an n-cellular extension of the pathwise connected space A, then the pair (X, A) is (n — l)-connected, so that π i (X, A) = 0 for all i < n. The next step is the determination of π n (X, A). By the results of §2 of Chapter II, H n (X, A) is a free abelian group with one basis element for each n-cell of (X, A). We have seen that the Hurewicz map ρ: π n (X, A) → H n (X, A) is an epimorphism whose kernel is generated by all elements of the form α — τ’ ξ (α) with α ∈ π n (X, A), ξ ∈ π 1 (A). If II = π 1 (A) operates trivially on π n (X, A), then ρ is an isomorphism. But this condition is not easy to verify a priori.

Fifty years of homotopy theory

The subject of homotopy theory may be said to have begun in 1930 with the discovery of the Hopf map. Since I began to work under Norman Steenrod as a graduate student at Chicago in 1939 and received my Ph.D. in 1941, I have been active in the field for all but the first ten years of its existence. Thus the present account of the development of the subject is based, to a large extent, on my own recollections. I have divided my discussion into two parts, the first covering the period from 1930 to about 1960 and the second from 1960 to the present. Each part is accompanied by a diagram showing the connections among the results discussed, and one reason for the twofold division is the complication of the diagram that would result were we to attempt to merge the two eras into one. The dating given in this paper reflects, not the publication dates of the papers involved, but, as nearly as I can determine them, the actual dates of discovery. In many cases, this is based on my own memory; this failing, I have used the date of the earliest announcement in print of the result (for example, as the abstract of a paper presented to the American Mathematical Society or as a note in the Comptes Rendus or the Proceedings of the National Academy). Failing these, I have used the date of submission of the paper, whenever available. Only in the last resort have I used the actual publication date. I wish to thank my many friends who have made pertinent comments, and helped refresh my memory on a number of points. Particular thanks are due to Saunders Mac Lane, William S. Massey, and Franklin P. Peterson. I also wish to acknowledge that my exposition of the solution of the immersion conjecture was based on a seminar talk by Professor Peterson on the same subject.

Homology of Fibre Spaces

We conclude this volume with an introduction to the method of spectral sequences for studying the homology of a fibration. In §1 we consider a filtered pair (X, A). The homology groups of the triples (X p , Xp-1, A) are linked together in an intricate way. They can, however, be assembled into two graded groups which are connected by an exact triangle Open image in new window Such a diagram is called an exact couple; the notion is due to Massey [1; 2]. The basic operation on exact couples, that of derivation, gives rise to a new exact couple Open image in new window Hence the process can be iterated to obtain an infinite sequence of diagrams.

Generalities on Homotopy Classes of Mappings

The set [X, Y] of homotopy classes of maps between two compactly generated spacesX, Y has no particular algebraic structure. This Chapter is devoted to the study of conditions on one or both spaces in order that [X, Y] support additional structure of interest. Guided by the fact that π1 (X, 0 ) = [S 1 , y 0 ; X, x 0 ] is a group, while[S 1 , X] is in one-to-one correspondence with the set of all conjugacy classes inπ 1 (X, x0), and the latter set has no algebraic structure of interest, we discuss in §1 the way in which [X, x 0 ; Y,y 0 ] depends on the base points. It turns out that under reasonable conditions the sets [X, x 0 ; Y 0 , y 0 ] andX, x 0 ; Y, y 1 ] are isomorphic. However, there is an isomorphism between them for every homotopy class of paths inY fromy 1 toy 0 . In particular, the group π1(Y, y 0 ) operates on[X, x 0 ;Y,y 0 ], and [X, Y] can be identified with the quotient of the latter set under the action of the group. This action for the caseX = Sn was first studied by Eilenberg [1] in 1939; it, and an analogous action ofπ 1 (B, y 0 ) on the set [X, A, x0;B, y 0 ], are discussed in §1.

Homology of Fibre Spaces: Elementary Theory

The relations among the homotopy groups of the fibre F, total space X and base space B of a fibration are rather simple, as we saw in §8 of Chapter IV. The behavior of the homology groups is much more complicated. In the simplest case, that of a trivial fibration, the relationship is given by the Kimneth Theorem. The general case will be treated in Chapter XIII with the aid of the complicated machinery of spectral sequences. In this Chapter we shall treat by more elementary methods certain important special cases. There are two reasons for this. The first is the hope that the geometrical considerations of this Chapter will help motivate the spectral sequence. The second is that the present route leads quickly to certain applications we have in mind, e.g., the homology of the classical groups.

Stable Homotopy and Homology

The adjoint of the identity map of SW is a map λ 0 : W → ΩSW. For any CW-complex K, [K, ΩSW] ≈ [SK, SW]; moreover, the injection [K, W] → [K, ΩSW] corresponds under this isomorphism to the suspension operator

Elements of Homotopy Theory

S_* :[K,W] \to [SK,SW]