17.1

17. Literatur.

ATIYAH-BOTT, On the periodicity theorem for complex vector bundles, Acta mathematica 112 (1964) 229-247.

ATIYAH-HIRZEBRUCH, Vector bundles and homogenuous spaces, Proc. Symp. Diff. Geom. Tuscon 1960.

BARRATT, M.G., Track groups I, II. Proc. London Math. Soc. 5 (1955), 71-106, 285-329.

BROWN, E.H., Cohomology theories, Ann. Math. 75 (1962) 467-484.

CARTAN-EILENBERG, Homological Algebra, Princeton University Press 1956.

CARTAN-SERRE, Espaces fibres et groupes d'homotopie, C.R. Acad. Science Paris 234 (1952) 288-290.

DOLD, A., (a) Partitions of unity in the theory of fibrations, Ann. of Math. 78 (1963) 223-255.

(b) Relations between ordinary and extraordinary ColI. Alg. Topol. Aarhus 1962, 1-9. cohomology,

DOUADY, A., La suite de Adams: structure multiplicative, sem. H. Cartan 11 (1958/59) expo 19.

ECKMANN-HILTON, (a) Group-like structures in general categories I (multiplications and comultiplications),. Math. Ann., 145 (1962), 227-255.

(b) Unions and intersections in homotopy theory, Comment. Math. Helv. 38 (1963/64) 293-307. ElLENBERG-MACLANE, On the groups III, Ann. Math. 60 ( 1954) 513-557.

ElLENBERG-STEENROD, Foundations of algebraic topology, Princeton University Press (1952).

GROTHENDIECK, A., de Chap. 0, § 8 : Vol. III, Publ. Math. 11 (1961).

HILTON, P., An introduction to homotopy theory, Cambridge University Press, 1953.

HU, S.T., Homotopy theory, Academic Press, New York, 1959.

HUREWICZ, W., On the concept of a fibre space, Proc. Nat. Acad. Se. USA 41 (1955), 956-961.

JAMES, 1., (a) Reduced product spaces, Ann. of Math. 62 (1955) 170-197.

(b) On H-spaces and their homotopy groups, Oxford Quart. J. 11 (1960) 161-179.

KAN, D.M., functors, Trans. Amer. Math. Soc. 87, (1958) 294-329.

MASSEY, W. Exact Couples, Ann. of Math. 56 (1952) 363-396.

OLUM, P., Invariants for effective homotopy classification and extension of mappings, Memoirs AMS 37, Providence RI 1961.

PUPPE, D., Homotopimengen und ihre induzierten Abbildungen I, Math. Zeitschr. 69 (1958) 299-344. J.P., Classes de groupes et groupes dthomotopie. Ann. Math. 58 (1953) 258-294.

STASHEFF, J., A classification theorem for fibre spaces, Topology 2 (1963) 239-246.

STEENROD, N., The topology of fibre bundles, Princeton University Press 1951.

WHITEHEAD, G.W., On mappings into group-like spaces, Comment. Math. He1v. 28 (1954) 320-328.

WHITEHEAD, J.H.C., Combinatorial homotopy, Bull. Amer. Math. Soc. 55 (1949) 213-245. A.l

Anhang

Die Theorie ha1bexakter Funktoren t wird erheblich einfacher, wenn die Werte t X und t f Gruppen und -homomorphismen sind oder auch wenn sie in einer abelschen Kategorie liegen. Die Metho- den und Konstruktionen sind jedoch im Prinzip dieselben, sodass dieser einfachere Fall gut als in die allgemeine Theo- rie dienen kann. Aus diesem Grund wird hier im Anhang ein Teil eines Vortrags den einfacheren Fall) reproduziert, den der Autor beim AMS summer institute for topology, Seattle 1963, ge- halten hat. Der Auszug ist der ursprUnglichen Vortrags- fast Zwei Abschnitte fiber Spektral- sequenzen wurden weggelassen, weil sie oben (§§ 14 - 15) formuliert sind. Auf den Haupttext wird mit M verwiesen, also z.B. mit M 1.1 auf "darstellbare Funktoren". -

1. Half exact functors of spaces

Let W be the category whose objects are finite OW-complexes and whose morphisms are homotopy classes of maps. Let A be an abelian category. A (contravariant) functor t:! is called half exact if the sequence

(1.1) t(X/A) tX tA

is exact for every X £ W and subcomplex A(X•

The objects t(Si), where Si denotes the i-sphere i - 0,1,2, ••• A.2

are called the coefficients of t.

If we apply 1.1 to X ""' A = a point we find tea point) = 0 If we apply it to X ""' A v B (wedge) we find teA v B) ""' tA 9 tB more generally

(1.2)

I.e., t takes sums into products. There is, of course, no difference betwean (finite) sums and products in !; we chose to write the product sign in 1.2 because this is the adequate

form if one wants to deal with infinite complexes. Indeed, every•

thing that follows holds for half exact functors on infinite CW• complexes (of finite dimension.) provided the "infinite analogon"

of 1.2 is satisfied.

If X £ W is arbitrary and A (X contains exactly one point in every component of X then X/A is connected and 1.1, 1.2 imply

(1.3) tX ""' t(X/A) $ tA ,

a natural splitting. Moreover, A ""' V.So , tA = • We shall therefore look at J J connected CW­complexes only, having a single O­cell, and shall con­

sider it as obvious from 1.3 how to deal with the general case.

Another simplifying remark concerns base points. One could consider half exact which are defined on homotopy classes with base points. However, it turns out that these are the same

as those defined on free homotopy classes, the reason being that

we assume an additive domaine category. (cf. M, 5.27) For set valued functors the situation is quite different. (cf. M, § 4). (1.4) Examples: Let H be a homotopy-associative, homotopy-commutative H-space (e.g. H =1t2y) . Then tX noH (homotopy classes modulo those of constant maps) is half exact. If A is the category of abelian groups then every half exact t with countable coefficients is of this form [2].

(1.5) Let L be a fixed space (or a spectrum) and put tX = {X,L) = = group of stable homotopy classes.

(1.6) Put tX = group of stable vector bundles (real or complex) over X i.e., t = resp.

If h is a cohomology theory on pairs (X,A) with values in A (satisfying the Eilenberg-Steenrod axioms except possibly the dimension axiom) then tX = Coker[hq(point) hqX] = 1qx = reduced q-th cohomology of X is half exact. Examples can be found in [1], [4] et ale

(1.8) If r:! is an exact functor or cr: a functor which takes cofibrations into cofibrations then composi- tion rt resp. tcr with every half exact t is again half exact. Examples for cr are: the suspension functor the '-product with a fixed space, the passage to the n-dual in the sense of Spanier-Whitehead (the last works in the stable category only).

2. Comparing half exact functors

2.1 Proposition. Let -r: t t' be a natural transformation of half exact functors. 1! f(Si) : t(Si) t'(Si) is an equivalence for all i n then tX t'X for all X of dimension n • A.4

Proof: The Puppe sequence [M, 5.6]

A B Cf lJl LB •••

is defined for every continous map f, and every term is (up to homotopy equivalence) the cofibre of the preceding map. Applying any half exact t therefore gives an exact sequence

(2.2) tA tB tCf tlJl tLB

Every CW-complex is obtained by successively attaching wedges of cells; let v(X) denote the number of such operations which are required to construct X. For instance, v(X) if and

only if X is a wedge of spheres; further V (Lx) < V (X) • The proposition is proved by induction on v. We can then assume X - Cf where f: AB and Y(A) = 1 = V(lJl) , VCB) < v(X) hence Y(2::B) < V(X) . We apply 'to the exact sequence 2.2 and get tX = tX' by the five lemma, qed.

Application: A contravariant functor t: W is called stable if there is a natural factorization

t

where s is the passage to stable homotopy classes (1.5); the examples 1.5, 1.6, 1.7 are stable. If t is half exact then T is a homomorphism of abelian groups (see 3.3), and is therefore adjoint to a morphism

e(X,y,t) tx, e tY tX • A.5

In particular/we have a natural transformation (2.4)

the "Hurewicz-map" •

2.5 Proposition: If etSi

(a) is an exact functor (Le. tSi is flat), (b) kills finite groups

then the morphism 2.4 is an isomorphism.

i Proof: (a) implies by 1.8 that u(X) = E&i X,Si3 @ tS is half exact. It therefore suffices to show that E(Sj,t) is isomorphic for all j • If j i then is finite, hence i j j e tS = 0 by (b), hence u(sj) = e tS tS , qed •

More generally we have

Proposition: Assume t is as in 2.5 and t l is an arbitrary half exact (additive suffices) stable functor. If ,i : tSi tlSi is a sequence of then there is a unique natural transformation f: t t' such that f(Si) = fi • Proof: f is the unique jfiller of the following diagram E9 {X,sj t Isj . tS j

tX - t !'t'X , qed.

(*)A more elaborate argument shows that 2.6 holds also for non- stable t,tl; cf' , M, §IO) A.6

.., j (2.7) Example: If' tX = Q then tS = f'or even j > 0 , and is zero otherwise = lieven(sj,Q) where Hev en = @H4< There is a r: unique f : t fIeven(_,Q) , the Chern-character, which extends the identity on spheres, and it is an equivalence by 2.1 •

3. Lemmas on homotopy groups.

3.1 Lemma. Assume a diagram

["'n o Cf = Cf/B = Sn+l =

(3.2) gJ 'I' glB 19 n L B' eft ) Cf'/B' = V.sn+l = , n > 0 V·SJ J J

is given where g: Cf Cf' is such g(B) ( B' , and g is the induced map on quotientsi the 2n d and 3r d square are then commutative.

g' has been so chosen that g •

£ differ only Conclusion: The two elements [fIg'], [(g\B)f] nnB' by a sum of elements of the form x - xy where x e nnB' , y e nl B'

i.e., the first square in 3.2 may not be commutative (it isn't I in general) but the defect is not too bad. The proof relies on the fact that the kernel of the Hurewicz map nn+l(Cf',B') Hn+l(Cf',Bf) is generated by elements of the form

x - xy with x E nn+l(Cff,Bf) and y E nIB' (compare M,12.3).

3.3 Lemma: If t w A is half exact then the map

is a homomorphism of' abelian groups. More generally, if' Y is an H'-space then Hom(tX,ty) , t¥

is a homomorphism.

This holds because addition on both sides of 3.4 t is based on the diagram

3.5 Corollary; If X :: Sn ,and has degree r then n. tf .0 tSn --, tS lS muIt"lPI"lcat"lon byr.

3.6 Corollary: Under the assumptions of lemma 3.1 the diagram

is commutative, i.e. t(gt)t(ft) = t(f)t(glB) •

This holds because x and xy are freely homotopic, hence t(x - xy) = 0 •

3.7 Corollary: Let X be a CW-complex, Xn its n-skeleton, Vjsn Xn an attaching map for the (n+l)-cells (hence = xn+l ). c(X). t(VjSn) is a functor on cellular maps, and tf: t(Xn) t(VjSn) is a natural transformation. I.e., although the attaching map is not unigue, is.

This follows from lemma 3.1 and Corollary 3.6 •

4. Cochains C(X,tSn), cohomology H*(X,tSn)

A slight generalization of a result of Eilenberg and Watts shows:

If K E A then there is a unique contravariant functor from A.a

finitely generated abelian groups to A which (a) is left exact,

(b) takes Z into K. It is denoted by hom(-,K) (P. Freyd calls it the"symbolic hom-functor".) This extends to arbitrary abelian

groups provided one requires =irAhom(G ,K) whenever all G Z• It is in fact obvious how to construct hom(L,K) A from a free resolution of L.

If now

is a chain complex of abelian groups then

is a cochain complex in A whose homology is denoted by H*(C;K) Applying this to cellular chains C of a CW-pair (X,A) defines Cq(X,A;K) and Hq(X,A;K) •

4.1 Proposition: Let (X,A) be a CW-pair, Xn the n-skeleton of X,

= VjSn Xnv A an attaching map for the (n+l)-cells of X-A. There are natural (with respect to cellular maps) isomorphisms

(4.2) Cn(X,A;tSn) t(Xnv A/Xn-lv A)

(4.3) Cn+l(X,A; tSn) t( VjSn)

which take the coboundary b: Cn Cn+ l into the composite t(Xnv A/Xn-lv A) t(Xnv A) t(lf) ) t( VjSn) •

n Proof: 4.2 is clear because X v A/Xn-lv A = VkSn with one ) sphere for each n-cell in X-A. Further, 4.3 follows from Corollary 3.7. The last statement is true because the incidence coefficients of C are the various degrees of the composite A.9

and because of 3.5.

5. The obstruction sequence

5.1 Proposition: There exists a natural (with respect to (X,A) and t ) exact sequence

(5.2) t(Xn+lvA)

In words: the map 0" associates with every "eLemen't" x of t(Xn-lv A) which admits an extension to Xnv A an obstruction , and • 0 if and only if x extends to Xn+lv A•

Proof. If If: V.sn Xnv A is an attaching map for the J (n+l)-cells of X-A then we have an exact sequence

Dividing by the image of K = t(XnvA/Xn-lvA) we get an exact sequence

and by proposition 4.1, the exact sequence

where B denotes coboundaries. It remains to show that maps into cocycles, i.e., that 0 • This amounts to showing that the composite

t(XnvA) t{If» Cn+l(X,A;tSn) -f...... ::, Cn+2

is zero. A.IO

n 2 Let Y \/, e + X be the characteristic map for the = k n+2 (n+2)-cells of X-A, where e is the standard (n+2)-cell; note that the n-skeleton yn consists of a single point. Naturali-

ty gives a commutative diagram

t(Xnv A) Cn+l(X,A) -L Cn+ 2(X,A) It '1 n+2(y) o = t(yn) ) cn+l(y) c

which proves the assertion.

n) 5.' Corollary: If Hn+l(X,A;tS = 0 for all n then tX tA 0 is exact

- because all obstructions vanish.

n) 5.4 Corollary: If Hn(y,tS = 0 for all n then ty = 0 .

Proof: Take A = y, X = CA = cone over A, remark that = 0 apply 5.3 , and use tX = tCA = 0 •

5.5 Corollary: If f: AX is a continuous map such that for all n f*: Hi(X,tSn) Hi(A,tSn) is epimorphic for i = n-l , ) isomorphic for i = n , and monomorphic for i = n+l then tf : tx tA. In particular, this applies if ordinary integral homology is mapped isomorphically because the universal coefficient theorem then implies the assumption.

Proof: We can assume f is an inclusion. The assumptions then mean Hn(X,A;tSn) • 0, Hn+l(X,A;tSn) = 0 for all n. Therefore tX tA is monomorphic by 5.4 and epimorphic by 5.3 • A.ll

5.6 Proposition: If :tSi• 0 for i < n then there -exists a natural transformation

Proof: For every X Corollaries 5.5 and 5.3 give

Im[ t{Xnv A) t{Xn-lv A)] • t{Xn-lv A) a: tA , so the obstruction map becomes tA • Putting X = CA this is tA Hn+l (CA,A;tSn) Hn{A,tSn) , as required.

5.7 Corollary: (Hopf-Whitney). If tSi• 0 for i < nand dim{A) i n then

5.8 Corollary: (Uniqueness of half exact functors satisfying a dimension aXiom) If tSi• 0 for i. n then tA. Hn{A;tSn) for all A (and becomes the coboundary homomorphism $*). Both Corollaries follow from 5.6 and 2.1 •

Another typical application of 5.1 gives a result of F. Peterson, as follows.

5.9 Proposition: Assume H2n+l{X,A;Z) has no torsion dividing en-I)! • Then a stable bundle KCA which extends to v A has an

extension to if and only if where cn = n-th Chern class, and i*. coboundary homomorphism.

5.10 Corollary: If H2i+l(x,A;Z) has no torsion dividing (i-I)! ,

i • 3,4,... , then a stable bundle "l KCA extends to X if and only if I*c(£). 0 , where c = total Chern class.

5 •11 Coro11ary: If H2i (A, Z) has no torsion dividing (i-I)! , i = 3,4, ••• , then 1 KCA is the trivial bundle if and only if c = 1 • A.12

This follows from 5.10 by taking X = CA = cone over A.

Proof of 5.9: It is clear that the condition is necessary: if 2n l extends to X + v A then cn (7) extends to X hence I*cn = 0 • 2n-1 For the converse, put t = "'K t ,and 1et I)'l co t (X v .. an extension of The exact sequence 2n t (X v A) t (X2n-l v A) t ( S2n-l) = 0 shows V.:L 2n "1.' e rm[ t(X v A) t(12n-lv A)] , so we can use the obstruction sequence 5.2 • We map it into the corresponding sequence for 2n(_,Z) t' = H via the Ohern class cn: t t' . As remarked in the proof of 5.6 we have

so we get a commutative diagram

8. Generalizations.

One can generalize the preceding results to functors t : where A is not necessarily abelian. For instance, one can take A = ENS, the category of sets, provided we *eplace half-exactness by the Meyer-Vietoris condition (e) of Brown [2] also for most results we then have either to restrict ourselves

to simply connected OW-complexes X or we have to assume that t A.13 is n-simple for all n; this means: tea) = t(a.y) for all a E nnX, y E nIX , and all X. Generalizing both abelian categories and ENS one can probably take for A any category such that the category of abelian group objects of A is an abelian category.

On the other hand W can be replaced by more general categories. As a gUiding model consider the following: Let u : be half exact, let n : E B be a fibration, and for every a: XB let n E X be the induced a: a fibration. For every a: X Band YCX put t(X,Y,a) = u(Ea/EaI Y). This is a functor on triples (X,Y,a) (i.e. on the category of pairs of spaces over B) to which the preceding results generalize. For instance, one has a spectral sequence such that = F/* x'F)) for p-q So 1 , ,-q = 0 for p-q > 1 ; and -q , for p-q So 0 , is the graded object associated with a filtration of • l(*) Here F = n- is the fibre, * = basepoint and u indicates a local coefficient system (the group nIB operates on u(sq x F/* x' F) = $uSq) •

Other possiblities for Ware the category of spectra, or the category of (free) chain complexes over a fixed ring. Dual to section 1, one could consider half-exact functors t*, i.e., /\ functors which are exact on fibrations (instead of Guided by these examples one is lead to axiomatize too, probably (following a suggestion of S. Eilenberg) by distinguish- ing in W a certain class of morphism X t X X" as "short exact sequences", which the functor would have to carry into exact sequences. A.14

References'

[1] AT IYAH-HIRZEBRUCH, Vector bundles and homogenuous spaces, Proc. Symp. Diff. Geom. Tucson 1960.

AT I YAH , M. Characters and cohomology of finite groups, Inst. Hautes Etud., Publ. Math., Paris 1961.

[2] BROWN, E., Jr. , Cohomology theories, Annals of Math. 75(1962) 467-484.

CARTAN-EILENBERG, Homological Algebra, Princeton Univ. Press 1956.

DOLD, A., Relations between ordinary and extraordinary cohomology, Coll. on Algebr. Topology Aarhus 1962, p. 1-9. I

[5] PUPPE, D. Homotopiemengen und ihre induzierten Abbildungen I, Math. Zeitschr. 69(1958) 299-344. Offsetdruck: Julius Beltz, WeinheimJBergstr. Printed in Germany