SOME ASPECTS OF STABLE HOMOTOPY THEORY

Transcription

1 SOME ASPECTS OF STABLE HOMOTOPY THEORY By GEORGE W. WHITEHEAD 1. The suspension category Many of the phenomena of homotopy theory become simpler in the "suspension range". This fact led Spanier and J. H. C. Whitehead in 1953 [26] to introduce the suspension category S and to propose the study of its homotopy theory as a first approximation to the homotopy theory of the category of spaces. The objects of S are spaces, but its maps are equivalence classes of maps: S P X->S Q Y, two such maps being identified if their iterated suspensions of some order are homotopie. Thus the set {X, Y} of $-maps of X into Y acquires the structure of a graded abelian group, whereas the set [X, Y] of homotopy classes of maps of X into Y has no particular structure. In subsequent years, Spanier and Whitehead pursued this program in a series of papers [27, 28, 29], culminating in the Spanier-Whitehead duality. This duality asserts the existence of a natural isomorphism {X, Y} ^ {D n Y,D n X}, where X, Y are subcomplexes of S n and D n X,D n Y are essentially their complements (a more general formulation has been given by Spanier in [25]). Their duality has been applied to a variety of problems, e.g. to the problem of imbeddibility of a complex in S n [24]. 2. Spectra and homology theory For many purposes the suspension category has proved to be too "small"; a more suitable category seems to be the category Sp of spectra. These were introduced by Lima in [20] and have since been studied, under a variety of restrictions, in [1, 25, 31]. A spectrum E in the most general sense is simply a sequence of spaces E n, together with a sequence of maps f n :SE n ->E n+1 (or, equivalently, a sequence of maps g n :E n ->QE n+1 ); E is said to be an 2-spectrum if and only if each map g n is a homotopy equivalence. Examples are the sphere spectrum S = {S n } and the Eilenberg-MacLane spectrum K(U) = {K(U,n)}. The category Sp seems to provide the proper setting for cohomology operations of higher kind [23]. Moreover, it is closely related to homology and cohomology theories of a very general sort. The axiomatic study of homology and cohomology theory was inaugurated by Eilenberg and Steenrod in 1945 [11, 12]. Of their axioms, the first six are of a very general character; on the other hand, their seventh, the dimension axiom, plays a normative role, singling out the classical theories among the multitude of possible theories which satisfy the first six. Examples of such theories are the stable homotopy and cohomotopy groups, the bordism and cobordism groups studied by Atiyah [2] and by Conner and Floyd [7], and the groups of stable vector bundles studied by Atiyah and Hirzebruch [3, 4], Bott [5] and others.

2 STABLE HOMOTOPY THEORY 503 Greneral cohomology theories have become reasonably familiar in recent years (cf. the work of Eckmann and Hilton [10]). For any space Y, the functor [, Y] ot homotopy classes of maps into Y is, in some sense, an "unstable" cohomology theory, and Brown in [6] has given an axiomatic characterization of such theories. One obtains a cohomology theory satisfying the first six axioms of Eilenberg and Steenrod by replacing F by a spectrum E and defining, for any compact polyhedron X, H n (X; E) = {X, E} n = lim k [S k X, E n+k ]. This can be given an alternative description as follows: the function spaces F(X, E n ) of all maps of X into E n are the components of a spectrum F(X,E), and ir(x;e)~rc_ n (F(X;E)). In [6], Brown has shown that, under mild restrictions, every theory satisfying the first six axioms can be obtained in this way, from an essentially unique Q-spectrum E. For example, H n (X; II) ^ H n (X; K(II)) for any abelian group II. On the other hand, homology theories in a general sense have until recently seemed somewhat mysterious. In [31] I proposed the following definition. If X is a compact polyhedron and E a spectrum, then the reduced joins X A E n are the components of a spectrum X A E; let H n (X;E)=7Zn(XAE). One then obtains a homology theory satisfying the first six axioms; moreover H n (X;U)^H n (X;K(U)), while H n (X; S) is the nth stable homotopy group H n (X). 3. Duality theorems Classically, homology and cohomology groups are related by the Alexander and Poincaré duality theorems, and by the universal coefficient theorem. Universal coefficient theorems for general theories have been studied by D. S. Kahn in his thesis (unpublished); the situation is much more complex than in the classical case, a short exact sequence being replaced by a spectral sequence. The isomorphisms of the classical duality theorems are described by certain products, which are induced by a pairing of two coefficient groups to a third. Using an appropriate notion of pairing of spectra, one can define cup, cap, etc. products which have properties completely analogous to the classical ones. Using these, I have shown in [31] that the analogue of the Alexander duality theorem holds in full generality: for any spectrum E, we have the isomorphism Ê Q (X;E)*Hn- Q -i(d n X;E). It then follows that the analogue of Brown's theorem holds for homology. The Poincaré duality theorem is much more delicate. Consider a spectrum

3 504 G. w. WHITEHEAD B which is ring-like in the sense that there is given a self-pairing (R,R)->R and a map S->R with suitable properties. An H-module M is then a spectrum with a suitable pairing (R, M)->M. If M is an R-module, the cap-product is a pairing H n (X; B) H Q (X; W)->H n - Q (X; M). If X is a compact triangulable manifold and z E H n (X; M) a suitably defined fundamental class then zo:h Q (X;M.)->H n _ Q (X;M) is an isomorphism. However, the question of existence of a fundamental class is extremely complex. For example, if R = S, then X has a fundamental class if and only if X is a homotopy U.-manifold; for the differentiable case this means that the stable normal bundle is fibre-homotopically trivial [21]. Thus the homotopy n-manifolds are precisely those manifolds for which Poincaré duality holds for arbitrary spectra. 4. Semi-simplicial spectra Semi-simplicial complexes were introduced by Eilenberg and Zilber in [31]. In recent years, they have been systematically exploited by Kan [15, 16, 17], Moore [22], Dold [8] and others, and this work has been responsible for much of the great advance that has been made by algebraic topology in recent years. One advantage of this approach is that one can apply to a semi-simplicial complex a functor from the category of sets to a category with a richer algebraic structure, and then to use algebraic machinery to study the resulting object. An example is the F-construction of Milnor (unpublished); F(K), the free group complex generated by K, is the semisimplicial analogue of the space of loops on the suspension of K; thus the map K-+FK is a homotopy equivalence in the stable range. Recently Kan [18] has introduced semi-simplicial spectra. For these the functor F is a weak homotopy equivalence; thus, to all intents and purposes and spectrum E can be replaced by JP(E). In recent unpublished work, Kan and I have applied the semi-simplicial techniques to the question of Poincaré duality. On the one hand, there is a classification of spectra by "degrees of abelianness"; an abelian spectrum is essentially a product of Eilenberg-MacLane spectra, and a spectrum is ^-abelian if it is dominated by an n-stage Postnikov extension whose factors are abelian. On the other hand, there is a classification of manifolds by "degrees of orientability"; a manifold is 1-orientable if and only if it is orientable in the usual sense, and w-orientable if and only if all stable homology operations of the (n l)st kind vanish on the fundamental class. It turns out that a manifold is n-orientable if and only if it satisfies Poincaré duality for all w-abelian spectra. Moreover, there exist n-orientable manifolds for some n, 2 < n < oo. These results were obtained by studying the functor which associates to every group its commutator subgroup. This functor is one of a class of "fully invariant" functors, determined by the fully invariant subgroups of the free group on countably many generators [19, 37]. If T is such a functor, then Il(T) = F(fi)ITF(S) is a ring-like spectrum in the sense

4 STABLE HOMOTOPY THEORY 505 explained above, and the R(T)-mdoules are those spectra which are dominated by F(E)ITF(E) for some spectrum E. One then has the classes of R(T) orientable manifolds; the relationships among these classes remain an object for future study. 5. Conclusion This lecture has not aimed at completeness. Some of the most interesting developments have been reported on by others at this Congress. It may be remarked that most of this work has examined stable homotopy theory for its own sake. On the other hand, the viewpoint of Spanier and Whitehead toward stable homotopy as a first approximation to homotopy theory suggests the possibility of higher-order approximations. There are a few examples of "semi-stable" phenomena. Among them are the exact sequence involving the homology suspension which I found in [30]; the exact sequence of James [14] involving the homotopy suspension; and the analogous exact sequences studied by Dold and Puppe in [9]. In this connection, the special role played by quadratic functors in [9] is suggestive. REFERENCES [1]. ADAMS, J. F., Theorie de l'homotopie stable. Bull. Soc. Math. France, 87 (1959), [2]. ATIYAH, M. F., Bordism and cobordism. Proc. Cambridge Philos Soc, 57 (1961), [3]. ATIYAH, M. F. & HIRZEBBUCH, F., Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc, 65 (1959), ] t Vector bundles and homogeneous spaces. Proc Symp. Pure Math., Vol. Ill, Providence, [5]. BOTT, R., Quelques remarques sur les théorèmes de périodicité. Bidl. Soc Math. France, 87 (1959), [6]. BROWN, E. H., Cohomology theories. Ann. Math., 75 (1962), [7]. CONNER, P. E. & FLOYD, E. E., Differentiable periodic maps, Bull. Amer. Math. Soc, 68 (1962), [8]. DOLD, A., Homology of symmetric products and other functors of complexes. Ann. Math., 68 (1958), [9]. DOLD, A. & PUPPE, D., Homologie nicht-addiviter Funktoren. Ann. Inst. Fourier, 11 (1961), [10]. ECKMANN, B. & HILTON, P. J., Groupes d'homotopie et dualité, C. R. Acad. Sci. Paris, 246 (1958), , , [11]. EILENBERG, S. & STEENROD, N. E., Axiomatic approach to homology theory. Proc Nat. Acad. Sci. U.S.A., 31 (1945), [12]. Foundations of Algebraic Topology. Princeton Univ. Press, [13]. EILENBERG, S. & ZILBER, J. A., Semi-simplieial complexes and singular homology. Ann. Math., 51 (1950), [14]. JAMES, I. M., On the suspension triad. Ann. Math., 63 (1956), [15]. KAN, D. M., On c.s.s. complexes. Amer. J. Math., 79 (1957), [16]. A combinatorial definition of homotopy groups. Ann. Math., 67 (1958), [17]. On homotopy theory and C.S.S. groups. Ann. Math. 68 (1958), [18]. Semi-simplicial spectra. Illinois J. Math. (To appear.)

5 506 G. w. WHITEHEAD [19]. KUROSH, A. G., The Theory of Groups. New York, [20]. LIMA, E. L., The Spanier-Whitehead duality in new homotopy categories- Summa Bras. Math., 4 (1958-9), [21]. MILNOR, J. & SPANIER, E. H., Two remarks on fibre homotopy type- Pacific J. Math., 10 (1960), [22]. MOORE, J. C, Semi-simplicial complexes and Postnikov systems. Symp.. International Topologia Algebraica. Mexico City, 1958, [23]. PETERSON, F. P., Functional cohomology operations. Trans. Amer.,. Math. Soc, 86 (1957), [24]. Some non-embedding problems. Bol. Soc. Mat. Mexicana, 1957, [25]. SPANIER, E. H., Function spaces and duality. Ann. Math., 70 (1959),, [26]. SPANIER, E. H. & WHITEHEAD, J. H. C, A first approximation to homotopy theory. Proc. Nat. Acad. Sci. U.S.A., 39 (1953), [27]. Duality in homotopy theory. Mathematika, 2 (1955), [28]. The theory of carriers and is-theory. Algebraic Geometry and Topology, Princeton, [29]. Duality in relative homotopy theory. Ann. Math., 67 (1958), [30]. WHITEHEAD, G. W., On the homology suspension. Ann. Math., 62 (1955), [31]. Generalized homology theories. Trans. Amer. Math. Soc, 102 (1962)*