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l K. Kodaira and D. C. Spencer, "On the Variation of Almost-complex Structures," in Algebraic Geometry and Topology (Princeton, N.J., 1957). 2 A. Frblicher and A. Nijenhuis, "Some New Cohomology Invariants for Complex Manifolds. I, II." Indagationes math., 18, No. 5, 1956 (here referred to as Paper C. I.). 3A Frblicher and A. Nijenhuis, "Theory of Vector-valued Differential Forms. I," ibid. No. 3, pp. 338-359, 1956. 4 K. Kodaira, "On a Differential Geometric Method in Theory of Analytic Stacks," these PROCEEDINGS, 39, 1268-1273, 1953.

ON THE SPECTRAL SEQUENCE OF A FIBER SPACE. II BY WITOLD HUREWICZ AND EDWARD FADELL*

MASSACHUSETTS INSTITUTE OF TECHNOLOGY AND UNIVERSITY OF WISCONSIN Communicated by Richard Brauer, December 18, 1956

1. INTRODUCTION The objective of this paper is to outline a proof of the following result, first con- jectured by Hurewicz, which generalizes a well-known theorem of Leray-Serrel and also extends a previous result of the authors.2 The details will appear else- where. THEOREM. If (X, B, p) is a fiber space in the sense of Hurewicz3 with fiber F, and if B is r - 1-connected, r > 1, then, in the associated spectral sequence {Ej, dj based on singular chains with coefficient group G, we have the following: 1. Ei IAH(B,H(F,G)), 2 < homotopy group on the homology of F. The simple description of dr in terms of cap products was suggested by Norman Steen- rod. 2. PRELIMINARIES 2.1. Fiber Spaces.-We employ the concept of fiber space as given by Hure- wicz., 2.2 The Spectral Sequence Based on Singular Prisms.-Let X denote a topologi- cal space and An = vo, ... , vn|I a fixed n-simplex, for n = 0, 1, ... . A (p, q) singular prism a is a map a: AP X A" -- X. The free Abelian group generated by all such singular prisms will be denoted by C(X). If G is an arbitrary coefficient group, we set C(X, G) = C(X) 0 G. Introducing the usual boundary operator a in C(X, G), we obtain a chain complex. Furthermore, we adopt the usual convention that when the symbol G is omitted, integral coefficients are assumed. Downloaded by guest on September 28, 2021 242 MATHEMATICS: HUREWICZ AND FADELL PROc. N. A. S. Now let (X, B, p) denote a fiber space, and let C(X) denote the singular chains of X based on prisms.5 Let A (X) denote the subcomplex of C"(X) generated by (m, n) prisms a such that pa: Al X Al -/ B depends only upon the second co-ordinate, i.e., pa(x, y) = pa(x', y). A(X) is seen to be chain equivalent to C(X). We filter A (X) as follows. AP(X) = AP is generated by (m, n) singular prisms in A (X) such that n < p. The spectral sequence {Ej, do} associated with (X, B, p) can then be defined as follows. Ej, ',q 2 1, is the image of the homomorphism i*: Hp+,(A P. A P-i) --o Hp+,(A P+J-1, A P-%l where i* is induced by the injection AP C AP+3-l. Then E"=P EjPq and E, q = E Ej". The differential operator di: EJP"qao Ej-PJ q+j-l is given by restrict- p ing a: Hp+,(AP+J-1, AP-') -* Hp+q1(A -1, A"P-J-) to Ej, 2. In what follows, we may use the notation Ej(X) for Ej to avoid confusion when more than one fiber space is involved. Also, if we use a coefficient group other than Z (= integers), we may write Ej(X, G). Let F = p-'(b), b e B, denote a fiber of X. Then C(B), C(F) will denote the singular chains of B and F, respectively, based on simplexes. If B is r-connected, r > 0 then we may assume that C(B) is generated by singular simplexes whose r- skeleton lies at b, and A(X) is generated by singular prisms whose r-skeleton lies in F. We will throughout assume that B is 0-connected and hence that singular elements employed have vertices at b or in F, respectively. If s is a singular (m, n) prism in AP, we associate with s two singular simplexes Fs and Bs as follows. Fs: A"m'- F is given by Fs(x) = f(x, vo), x e Am, where vo is the leading vertex of An. Bs: An -- B is given by Bs(y) = ps(x, y), y e An, and x arbitrary in F if n = p; otherwise, Bs = 0. Set 0(s) = Fs 0 Bs. Thus we obtain a chain map 0: AP/AP-1 C(F) 0 Cp(B), where the boundary operator in C(F) 0 Cp(B) is given by b(c 0 c') = bc 0 c'. 0, of course, depends upon p. 2.3. A Generalized Kunneth Theorem. Let (X, B, p) denote a fiber space and Y a topological space. Furthermore, let t: X X Y -- X denote the natural projec- tion. Then (X X Y, B, pt) is a fiber space. THEOREM. The spectral sequences {Ej(X, H(Y, G)), d,} and {Ej(X X Y, G), do} are isomorphic, i.e., there exist isomorphisms a,: Ej(X, H(Y, G)) -> Ej(X X Y, G) such that djaj = ajdj. In the sequel we will identify the above spectral sequences by reason of this theorem. 2.4. Cap Products.-Suppose that Gi, G2 are coefficient groups paired to G. If X is a space and c: Cq(X) -> G, is a q-cochain and u 092 e Cp+q(X, G2) is a singular p + q simplex with 92 as coefficient, we set c n (u 0 g2) = u P 0 c( u)g2 e Cp(X, G), where uP is the p-face of u determined by the last p + 1 vertices vq, ... , vp+, of AP+Q and Mu is the q-face of u determined by the first q vertices vo, . . . vq. This Downloaded by guest on September 28, 2021 VOL. 43, 1957 MA THEMA TICS: HUREWICZ AND FADELL 243 pairing of C0(X, G,) and C0+q(X, G2) to C,(X, G) induces a pairing of HI(X, GI) and Hp+q(X, G2) to H,(X, G) called the cap product. As usual, h" n hp+q denotes the element of H,(X, G) to which h" and hp+q are paired. This pairing differs by a fac- tor of (-1) P+Q from that given by Eilenberg.6 2.5. Pairing 7r,(B) and H(F, G) to H(F, G). Let (X, B, p) denote a fiber space with fiber F = p-'(b). If X is a lifting function4 for (X, B, p), X induces a map A: B X F - X, where B is the space of paths in B emanating from b. For w e B, f e F, set X(W, f) = (f, w)(1). Then, if A is the loop space of B at b, X: A X F -- F. Hence, employing the Ktinneth theorem, X induces X*: Hr-(A) 0 Hn(F) -> Hn+r-T(F). Thus H,-l(A) and H(F) are paired to H(F). Now we identify 7rr(B) with rr-l(A), and thus the pairing of Tr7(B) and Hn(F) to Hn+r-T(F) is obtained by using the Hure- wicz homomorphism 7r-l (A) -- Hr- (A).

3. THE BASIC MAPS 3.1. The Map Up. Let B-denote a 0-connected space, and fix b e B. Further- more, let B denote the space of paths in B emanating from b. Then, if t: B -> B is given by t(w) = w(l), (B, B, t) is a fiber space, with fiber A = p-'(b) the space of loops. Let C(B) denote the singular chains of B based on prisms and 0(B) the singular chains of B based on simplexes. Then, using the notation of section 2.2, set 4 = A (B), 4P = A P(B) . 4 is then filtered, and we have an associated spectral sequence { Ej(B), dj}. Our objective in this section is a dimension-preserving homomorphism (not, in general, a chain map) so: 0(B) -i 4. Therefore, let s: An- B denote a singular n-simplex. Then define s: Al X Al B by s(x, y) = (OX)s(x) (x y) E An X An, where vo is the leading vertex of An, and tox, Xy are line segments with indicated end points. For 0 < q < n, let ,An and A"n, n- denote q and n - q faces of An determined, respectively, by the first q vertices of An and the last n - q vertices of An. Let f3qn: Al X An-q - an X Ann'- denote the natural barycentric map. Then set (p"(s) = s'f3,n, and note that spg(s) E .4n". Finally, set (p(s) = E (p,(s). We note here that sp depends on n, but this fact will not be dis- played in the notation. Thus p: 0(B) -X 44 is a dimension-preserving homo- morphism. so can also be interpreted as arising from a chain approximation to the diagonal map An An X An as follows: Interpreting An as a finite complex with ordered vertices in the usual manner, let aP denote a p-face of 'An, and %Jp, COP P-q faces of aP spanned by the first q + 1 and last p - q + 1 vertices, respectively. Then the correspondence p ap E ap X (aptV 2 q=o gives rise to a chain mapping D: C(An) C(0An X A"), which is a chain approxi- mation to the diagonal map in the sense of Lefshetz. Then op(s) can be given by restricting s to the simplicial chain D(An) in the product complex A" X>A". Downloaded by guest on September 28, 2021 244 MATHEMATICS: HUREWICZ AND FADELL PROC. N. A. S.

3.2. The Basic Lemma.-Suppose that s is a singular n-simplex in B whose r - 1 skeleton lies at b E B 1 K r < n. Let ts denote the r-face of s given by the composi- tion A7 -A- B, where the first map is the injection AT C An. Furthermore, we denote by rS the r - 1 singular simplex in A given by the composition Ar-l ArT r-l - A, where the first map is the natural barycentric map and the second map assigns to each x e ATr t-1 the arc rs(vox) from the leading vertex v0 to x. 1s deter- mines an element of 7r.il(A) which we denote by [rT]. Then the following lemma is valid. LEMMA. If s is a singular n-simplex in B whose r - 1 skeleton (1 < r < n) lies at b e B, then a<(s) - (pa(s) = a- mod 4n-r-l where a and 3 are singular (r - 1, n - r) prisms in An-r having the following proper- ties: (i) Ba = BO = sf-r, the n - r face of s determined by the last n - r + 1 vertices of An. (ii) Aa e [rs ]; Ad is degenerate, having the constant loop at b as constant value. Of course, in the above lemma, if r > n, ba(s) = fp(s) trivially. On the basis of the above lemma, the following theorem is valid. THEOREM. If B is r - 1-connected, r > 1, the map ap: C(B) X induces p1: C(B, G) ElE1(B, G), (ps: H(B, G) Ei(B, G), i = 2, ... such that di(pi = (plb and dispi = 0, i = 2, . . ., r -1. 3.3 The Map #/.-Let (X, B, p) denote a fiber space with B r - 1-connected and F = p-l(b) a fixed fiber. If X is a lifting function for (X, B, p), X induces a map X: B X F -. X (see sec. 2.5). If 71: B X F -- B is the natural projection, then (B X F, BY h) is a fiber space, and pX> = . Therefore, X induces homo- morphisms Xi: Ei(B X F, G) -- Ei(X, G) such that di~i = Xidi, i > 1. Now, applying the theorem of section 3.2 with G = H(F, G)' and identifying Ei(B, H(F, G)) with Ei(B X F, G), we set VOf = kipi, 1 < i < r. Therefore, A1: C(B, H(F, G)) -- E1(X, G) and 4ti: H(B, H(F, G)) -> Ei(X, G) for 2 < i < r. It is known that C(B, H(F, G)) is isomorphic to E1(X, G), and it is not difficult to show that ,&1 is such an isomorphism, where the inverse of 4/1 is induced by the homomorphism 0 of section 2.2. Therefore, this fact, together with the theorem of section 3.2, implies that the 41i are isomorphisms for 1 < i < r and di = 0 for 2 < i < r - 1. 3.4. LEMMA. If (X, B, p) and (X', B, p') are fiber spaces over B, which is r - 1- connected, and if f: X -* X' is a fiber-preserving map, then the following diagram is commutative: H(B, H(F, G)) H(B, H(', G)) Er(XI G) r Er(X" G) Downloaded by guest on September 28, 2021 VOL. 43, 1957 MATHEMATICS: HUREWICZ AND FADELL 245

where F, F' are fibers over b e B, the f's are naturally induced by -f, and Vr/, 4i'/ are the maps defined in section 3.3.

4. THE STRUCTURE OF dr Consider again the fiber space (B, B, t), B r - 1-connected. Applying section 3.3, we have isomorphisms 1: C(B, H(A)) E1(B) and is: H(B, H(A)) -Es(B ) 2 < i < r. Let ap: C(B) A denote the map of section 3.1. Then, if s is a singu- lar n-simplex in B, cbp(s) -8p(s) is a cycle mod An-r-1 and hence determines an element [-)p(s) -sp(s) ] e EEl-r' r-l(B). Then, employing the lemma of section 3.2, it is easy to see that ',(,(rT)sT-r) = [b(s) - pb(s) ]1, where y e Cr(B, 7r,(B)) is the characteristic cocycle of B, 's and s8'- are as in section 3.2, and irT(B) is identified with Hri-(A). It follows, then, that if z is an n-cycle in C(B, G), then so(z) determines an element [bo(z) ] e E1n-r, T-1(B, G) and {&(y n z) = [6o(z) ]h, where the cap product is derived from the pairing of 7r,(B) and G to ir,(B) 0 G, and hence 'y n z e C(B, 7rr(B)) ®) G) - C(B, Hr-l(A, G). Hence we have 'k(i& n h) = dr H(B H(F)) is induced by X: A X F -- F. But it is clear that X*(I7 n h) is then just the cap product y n h, where 7rr(B) ( =H,-1(A)) and H(F) are paired to H(F) according to section 2.5. Therefore, #rT(Y n h) = drTr(h) h e H(B, H(F)). Since VI, is an isomorphism, we may identify Er(X) with H(B, H(F)), and hence dr: E,(X) -- Er(X) is given by dr(h) = nA h. Thus our outline is complete for the proof of the main result with integral coefficients. The use of an arbitrary coefficient group offers no additional difficulties. * This paper was written by the junior author following the untimely death of Professor Hure- wicz. It was supported, in part, by the Wisconsin Alumni Research Foundation. 1 J. P. Serre, Ann. Math., 54, 425-505, 1951. 2 W. Hurewicz and E. Fadell, these PROCEEDINGS, 41, 961-964, 1955. 3 The junior author has also proved this theorem for fiber spaces in the sense of Serre. 4 W. Hurewicz, these PROCEEDINGS, 41, 956-961, 1955. 5 An alternate proof of the theorem based on cubical singular chains will appear in the forth- coming "Symposium in Algebraic Topology" publication (Mexico City, 1956). 6 S. Eilenberg, Ann. Math., 45, 407-447, 1944. Downloaded by guest on September 28, 2021