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loc. cit., pp. 71-72. Strictly speaking this criterion holds only for an algebraic surface in the usual sense, i.., embedded in a projective space. However, it can be easily extended to an algebraic in the extended sense by observing that if V is mapped by a bi-regular transformation 4b onto an algebraic surface V' in a projective space, then any curve in V is mapped by 4) bi-regularly onto a compact analytic curve S' in the projective space, which must then necessarily be an algebraic curve. See Chow, . ., "On Compact Complex Analytic Varieties," Am. . Math., 71, 893-914 (1949), Theorem V. It is clear that if'S is a rational curve without singularities in V such that I(S, S) = -1, then S' is also a rational curve without singularities in V' such that I(S', S') = -1. It follows then that S' is an exceptional curve of the first kind in V', i.e., there exists a bi-rational transformation I of V' which has S' as a fundamental curve and is bi-regular everywhere else. Then the "product" transformation '4) is a bi-rational transformation of V which has S as a fundamental curve and is bi-regular everywhere else. 6 See van der Waerden, loc. cit., § 3. 7Cf. Zariski, loc. cit., pp. 66-71. 8 Kodaira, ., "The Theorem of Riemann-Roch on Compact Analytic Surfaces," Am. J. Math., 73, 813-875 (1951), Theorem 5.3. 9 See Zariski, loc. cit., pp. 36-41.

COHOMOLOGY GROUPS OF ABELIAN GROUPS AND HOMOTOPY THEORY IV By SAMUEL EILENBERG* AND SAUNDERS MACLANE DEPARTMENTS OF MATHEMATICS, COLUMBIA UNIVERSITY AND THE UNIVERSITY OF CHICAGO Communicated February 27, 1952 The present note is concerned with extension and classification theorems for continuous mappings. 1. Products.-Let K be a simplicial complex with ordered vertices, L,, .. ., L, subcomplexes of K and L = L1 u ... u Lr. Let zn e Zn(K; Hi) ( _ 1) be an n-cocycle with values in an abelian group . If cq is a - simplex of K then zn determines a cocycle in Zn(A, H[) which may be re- garded as a q-simplex of the complex K(II, n).2 There results a simplicial map (zn): K K(ll, n). If Zn is zero on Li then T(zn) maps LI into the subcomplex K(, n) of K(ll, n). Cohomologous cocycles yield chain- homotopic maps. Let xi Hn(K, L,; fl),i = 1, .. .,randlety e H#(Hl,n; ). Wedefine the product [Xi, . sXr; ] e- Hq(K, L; G) as follows: Let zi e xi be cocycles. Consider the chain transformation s: (K, L) -- (K(H, n), K(O, n)) defined by Downloaded by guest on September 26, 2021 326 AIA THEMA TICS: EILENBERG A ND MA C LA NE PROC. N. A. S.

s = (1) kT(zi, + .+ zi), 1 ii < . < ik = r; then

[Xl, ..*, Xr; y] = Sty- This product is additive in y independent of the order of xi, ..., xr, and is natural with respect to simplicial maps of K, homomorphisms of 11 and homomorphisms of G. Further properties are:

[Xl, . *, Xr-1 -+ Xr': y] = [xi, ** *, Xr-.I xr; y] +

[xlp ..* Xr-1) Xr ; Y] + [Xl, *** Xr-bs Xr, Xr'; Y], (1) [xi, .. ., xr; y] = O, if q < rn, (2) [xI, . ., xr; y] = xl U ... u xr,if q = rn. (3) In (3) the cup product is to be taken relative to the pairing 1 0 ... 01 -_ G which to each system al, . . ., ar e H assigns the value of y on the cocycle arl* ... ar* where * denotes the product of chains in the complex K(H, n).3 For = 1 we have the following additional properties. Let bn e Hn (II, n; II) be the basic. cohomology class defined by bn(a) = a, ac e H. Then y = [bn; y]. (4) Let x e Hn(L; H), ex e Hn+'(K, L; H), y e H"+'(H, n + 1; G). Then [ax; y] = 6[x; Sy], (5) where S: Hl+1(H, n + 1; G) -IJHQ(11, n; G) is the suspension homomor- phism. 2. Depressed Products.-Let K X I be the cartesian product of K with the closed unit interval, subdivided simplicially in the usual manner. For a subcomplex L of K we define L =LXI u KXO u KX 1, L1=LXI u KX 1. We define maps p: (K X I, L X I) ---(K, L), 1: (K, L) (L,L) by p(u, t) = u, (u) = (u, 0). Both p* and 1* are isomorphisms for coho- mology groups. Combining 1*1 witlh 5:Hg(L, LI) H"+(K X I,L) we obtain an isomorphism ,p: H"(K, L) HIH+1(K X I, L) for any coefficient group. Now, let xi E Hn(K, L; H) i = 1, ..., r - 1, xr E Hn-I (K, Lr; H1),Ye Downloaded by guest on September 26, 2021 VOL. 38, 1952 MA THEMA TICS: EILENBERG A ND MA C LA NE 327

HI(, n; G). We define the depressed product (xi, . . Xr-1, Xr; Y)) = l[P1*X1, Pr-.Xr. pXr; Y] which is an element of H6-1 (K, L; G). As in (2) and (3) we have (xi, ..., X7; Y) = ° if q < rn (6) (xI, * X*,x7; y) = xi u ... u xr if q = rn.5 (7) Further (x; y) = [x; Sy]. (8) The products above may be applied to the case when K = S(X) is the singular complex of a topological space, thus carrying over the products to topological spaces. The invariance of the products in a simplicial complex may be established similarly. 3. Obstruction Theorems.-Let Y be a pathwise connected topological space with 7ri(Y) = Ofor 1 < i < n and n < i < q where 1 < n < q. We consider the natual homomorphisms6 K: HI(7r,,, n; G) -*1 HI(Y; G), 7r6 = 7r (Y), and introduce the basic cohomology class

In = K*b n Hn(Y; 7r-). The space Y has an obstruction kg+' = kg+1(Y) e Hg+l(rw, n; ir).

Let : Kn u L -- Y be a map extendable to Kn+' u L Y. Then f admits an extension f': K6 u L -- Y. The obstruction cg+1 (f1) f H1+1 (K, L; 7rq) is then independent of f' and is denoted by zg+1(f). This is the secondary obstruction7 of f. The homomorphisms fi*: Hn(Y; G) -G) Hn(K; G) are also independent off' are denoted byf*. If g: Kn u L -0 G is another map with extension g': Kg u L - Y such thatf| L = gI L, then the difference homomorphism (f' - g')*: Hn(Y; G) -. Hn(K, L; G) is independent off' and g' and will be written as (f -g)*. THEOREM I. Letf, g: Kn u L -- Y be maps extendable to Kn+1 u L Y and such that f L = g L. Then Zq+'(f)-I z+I(g) = [(f - g)*Lr; k6+1] + [g*Ln, (f - g)*tfn; k6+'] (9) THEOREM II. Let fo,fi: Kq u L- YbetwomapswithfoIKe-l u L = Downloaded by guest on September 26, 2021 328 MATHEMATICS: EILENBERG AND MACLANE PROC. N. A. S.

fi K'-I u L and with the difference cohomology class7 "(fo, fi) e H(K, L; 7rq). Then fo and fi are homotopic rel. L if and only if dg(f,ofi) + [el-'; SkQ+i] + (fo*Ln, en-l; ke+1) = 0 (10) for some en-1 e Hn-1 (K, L; 7rs). The assumption that 7ri( Y) = 0 for i < n may be dropped provided one assumes that the various maps carry K"'- into the base point yo of Y. The case n = 1, lri( Y) abelian, may be included provided the cohomology groups are taken with local coefficients. In the nonabelian case further changes are necessary and only partial results are obtained. These include earlier known results.8 4. The Case q = n + 1, n > 1.-A better understanding of this case de- pends upon a closer examination of the products [XI; Y], (1 [xI, x2; y], (12) [X3; Sy], (13) (x, x2; Y), (14) where xi e Hn (K, L2; II), x2 e Hn (K, L2; H), X3 e H -1 (K, L3; H), y e Hn+2 (H, n; G). This is possible because of our good knowledge of the group Hn+2(II, n; G). Indeed the elements of this group are in a 1-1-correspondence with func- tions t: H G (called traces) satisfying t(-a) - t(a) = 0, (15)

t(a + ,B + y) - t( + y) - t(a + y) -t(a + A) + t(a) + t() + t(y) = 0, if n = 2, (16) t(a + B)- t(a) - t() = 0, if'n > 2. (17) The products (12) and (14) are easiest to interpret. They are zero for n > 2 and are, respectively, the cup products xi U x2 and xi u X3 if n = 2. The cup product is taken relative to the pairing t(a + ) -t(a) - t(f,) where t is the trace of y. If n > 2, then there exist a bilinear function d: II ® II G such that d(a, a) = t(a). Relative to such a bilinear extension d of t one may con- sider the Steenrod products.7 It turns out that (11) and (13) reduce to Sq2xi and Sq2x3, respectively. For n = 2, we consider an exact sequence Downloaded by guest on September 26, 2021 VOL. 38, 1952 MA THEMA TICS: EILENBERG AND MAC LANE 329

0 II' -'H-4. H- 0 and define a generalized bilinear extension of t to be a bilinear function d: IH' IH' - G such that d(a, a) = tQ(pa) for a e II d(#(13), At(zy)) = 0 for (3, y e IIf Such a d always exists if HI' is free and H is finitely generated. Let U E C2 (K, LI, '), v E C' (K, L3, H') be cochains such that .pu and (pv are cocycles in the cohomology classes xi and X3, respectively. Then the expressions u u u-Uuu1 au, v u 5v

where u and u i are taken relative to d, are cocycles in the class (11) and (13), respectively. In the particular case y = kn+1(Y), the trace t of y coincides with J. H. C. Whitehead's function i: 7rn -O 7n+l obtained by combining each map Sn -o Y with the map Sn+" __ Sn which is the Hopf map for n = 2 and its (n - 2)-fold suspension for n > 2.9 The above results include those of Steenrod,7 Whitney,'0 Postnikov" and J. H. C. Whitehead."2 * Essential portions of the study here summarized were done during the tenure of a John Simon Guggenheim Fellowship by one of the authors. 1 The appropriate class of complexes to work with in §§1, 2 and 4 are the "semi- simplicial" complexes of Eilenberg and Zilber, Ann. Math., 51, 499-513 (1950). 2 Eilenberg, S., and MacLane, S., PROC NATL. ACAD. Sci., 36, 443-447 and 657-663 (1950); 37, 307-310 (1951); subsequently referred to as Notes I, II and III. I Note III, p. 308. 4 Note III, p. 307, or Note I, pp. 444-445. 5 The process of depressing a product may be iterated and may be applied to other products besides those of §1. In particular, the cup product depressed is again the cup product. 6 Eilenberg, S., and MacLane, S., Ann. Math., 51, 514-533 (1950). 7 Steenrod, N. E., Ibid., 48 290-320 (1947). 8 Robbins, H., Trans. Am. Math. Soc., 49, 308-324 (1941). Olum, P. (to appear). 9 Cf. G. W. Whitehead, PRoC. NATL. ACAD. Sci., 34, 207-211 (1948), Theorem 5. 10 Ann. Math., 50, 270-284 (1949). 11 C. R. (Doklady) Acad. Sci. URSS N. S., 67, 427-430 (1949). 12 Ann. Math., 54, 68-84 (1951). Downloaded by guest on September 26, 2021