HOMOLOGY THEORIES and DUALITY* [X; E.] Satisfy All of the Eilenberg-Steenrod Axioms for a (Reduced)Cohomology

554 MATHEMATICS: G. W. WHITEHEAD PROC. N. A. S.

41 "Uber einige spezielle diophantische Gleichungen," Mathematische Zeitschrift, Verlag-Springer, Berlin, 499-504 (1926). 42 "The Diophantine Equation X2 - DY2 = ZM," Trans. Am. Math. Soc., 38, 447-457 (1935). 43 "Mathematische Abhandlung, enthaltend neue Zahlentheoretische Tabellen," Programm zum Schlusse des Schuljahrs 1855-56 am Koniglichen Gymnasium zu Stuttgart (1856). 44"Some Criteria for the Residues of Eighth and Other Powers," Proc. London Math. Soc. (2), 9, 244-272 (1911). 4C "On 2 as a 16-ic Residue," Proc. London Math. Soc. (1), 27, 85-122 (1895). 46 "Kriterion zum 8. and 16. Potenzcharacter der Reste 2 und -2," Deutsche Math., 4, 44-52 (1939). 47 "The Sixteenth Power Residue Character of 2," Can. J. Math., 6, 364-373 (1954). 48 Werke, 2, 67-92 (1876). 49 Albert, A. A., "The Integers Represented by Sets of Ternary Quadratic Forms," Am. J. Math., 55, 274-292 (1933). 50 "Orthogonal Latin Squares," these PROCEEDINGS, 45, 859-862 (1959). S1 Mathematika, 5, 20-29 (1958). 62 Notices of the Am. Math. Soc., 6, 278-9, 513 (1959). 63 There is an extensive bibliography of articles published along this line in A. A. Albert's Fundamental Concepts of Higher Algebra (Chicago: Univ. of Chicago Press, 1956), pp. 152-155. "4 "On .1-Extensions of Algebraic Number Fields," Bull. Am. Math. Soc., 65, 183-226 (1959). Iwasawa's formula for en, given above, yields the result that for n = 1; e,, = 0, provided e is regular. U' Jacob Westlund, "On the Class Number of the Cyclotomic Number Field," Trans. Am. Math. Soc., 201-212 (1903). 66 Vandiver, H. S., "On the Class Number of the field l(e2te/P' ) and the Second Case of Fermat's Last Theorem," these PROCEEDINGS, 6, 416-421 (1920); "On the First Factor of the Clas Number of a Cyclotomic Field," Bull. Am. Math. Soc., 25, 458-461 (1919).

HOMOLOGY THEORIES AND DUALITY* BY GEORGE W. WHrrEHEAD

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated by Saunders Mac Lane, February 8, 1960 1. It has been known for some years that the integral cohomology group H' (X; II) of a (reasonable) space X with coefficients in an abelian group II can be characterized as the group of homotopy classes of maps of X into a certain space K(H,n). The spaces K(Hl,n)(n = 0, 1, 2,...) are related by the fact that each is the loop-space of the next; and it is this property that is crucial in setting up the apparatus of cohomology theory. In fact it is known that, if E = {E.) is a sequence of spaces, each the loop-space of the next, then the groups H"(X; E) = [X; E.] satisfy all of the Eilenberg-Steenrod axioms for a (reduced) cohomology theory except the dimension axiom. It is natural to ask whether there are corresponding homology theories. Of course, if a cohomology theory is given one can define homology groups (at least for a finite complex X) as the cohomology groups, suitably reindexed, of the complement of X in a sphere of sufficiently large dimension. This definition is awkward to work with and has the disadvantage of failing to be intrinsic. The object of this note is to develop an intrinsic theory of generalized homology groups, and to prove Downloaded by guest on October 2, 2021 VOL. 46, 1960 MATHEMATICS: G. W. WHITEHEAD 555

an Alexander duality theorem, as well as a Poincar6 duality theorem for a suitable class of manifolds. A full account will be published elsewhere. 2. A spectrum' E is a sequence {EI n e Z} of spaces with base point, together with a sequence of maps fn: E. - En + . The spectrum E is said to be an Q- spectrum if and only if each map f, is a weak homotopy equivalence. A conver- gent spectrum is one with the property that, for some N, EN+k is k-connected for all k _ 0. If E is a spectrum, X a base with base point, the reduced points X X En naturally form a spectrum X X E. The n-th homotopy group irn(E) of a spectrum E is the direct limit of the sequence 7rn+k(Ek). If X is a space with base point, and E a spectrum, we define reduced cohomology and homology groups of X with respect to E by ftn(X; E) = lim [SkX, En+k]

fln(X; E) = 7rn(X X E) The corresponding non-reduced groups of a space X (without base-point) are defined to be the reduced groups of the space X+ obtained from X by adjoining a point P, the base point of X+. With suitable definitions of relative groups and connecting homomorphisms 5*, a*, we have THEOREM 1. The homology and cohomology theories with respect to a spectrum E satisfy the Eilenberg-Steenrod axioms, except for the dimension axiom, on the category of compact polyhedra. Clearly ftn(SO; E) ; 7r-n(E), Rn(SO; E) T(E).7r Hence, if E is an abelian group, and K(H) is the spectrum of the spaces K(ll, n) we have COROLLARY. ftn(X; K(fl)) f:S(X; II) 3. In order to prove duality theorems, we shall need to introduce cup- and cap- products, and therefore a notion of pairing of two spectra to a third. A bispectrum e is a double sequence { Emn4, together with maps fmn :Emxn - QEm+1yn and gmin - Em,n Q12Em,n+1 such that the diagrams fm)n Emyn )UlEm+lyn gmen Ir I Qgm+1yn QEmmn+i )U2Em+1yn+1 Qfm n+i are anticommutative. A map of a bispectrum e into a spectrum C is a double sequence of maps 6mmn :Emn Cm+n such that the top triangle in the diagram fmn/12Em+1n \Qf12qm+4yn Emyn'- > Cm+n 12gCm+n+l 4+m n 72nhm+n/

is commutative, while the bottom is commutative up to a suitable sign. Downloaded by guest on October 2, 2021 556 MATHEMATICS: G. W. WHITEHEAD PROC. N. A. S.

If A and B are spectra, then the spaces Am X B,, form in a natural way a bispectrum A X B; a pairing of A with B to C is a map of the bispectrum A X B into the spectrum C. Let S be the spectrum of spheres; then there is a natural pairing of A with S to A for any spectrum A. Let X be a simplicial complex, and let A be a subcomplex of X. Let N(A) be the regular neighborhood of A in X, and let X . A be the closure of the complement of N(A); thus A is a deformation retract of N(A) and X .- A of X - A. If further B is a subcomplex of A, let rl:N(A) -o N(A)/N(B), 7r2:X .B --X + B/X + A be the natural maps. Then the reduced diagonal map A: X -N(A) N(B) X X - B/X . A is defined by ) A(X) = ri(X) X 7r2(X) (x N(A)nX . B) base point otherwise Given a pairing of A with B to C. we can then define cup- and cap-product pairings HP(A, B; A) X Hq(X + B, X .A; B) Hp+q(X; C) -- HP(A, B; A) 0 Hn(X; B) Hn-,(X *B, X . A; C) and these have many of the formal properties of the usual cup- and cap-products. 4. An orientable n-manifold X is called a H-manifold if and only if its fundamental class is stably spherical; i.e., the natural homomorphism Hn(X; S) -+ H,,- (X; K(Z)) is onto. Let X be a triangulated II-manifold and let z E H,(X; S) be an element mapping onto a fundamental class. The cap product with z induces homomorphismsqop: HP(A, B; A) - HZ-(X, . B, X . A; A) for any subcomplexes A n B of X. THEOREM 2. The homomorphisms p are isomorphism. COROLLARY 1. (Alexander duality theorem). If A is a subcomplex of Sn, then, for any spectrum A, 17P(A; A) ftfnp_(Sn - A; A) COROLLARY 2. (Poincar6 duality theorem for HI-manifolds). If X is a tri- angulable H-manifold of dimension n, then HP(X; A) ; Hn-,(X; A) for any spectrum A. Remark: The isomorphism of Corollary 2 is natural for maps of spectra (once the class z has been chosen). Conversely it is easy to see that, if the orientable n- manifold satisfies Poincare duality functorially, then X is a II-manifold. * This research was sponsored by Office of Ordnance Research, U. S. Army. 1 The notion of spectrum was introduced by Lima (University of Chicago dissertation, 1958); cf. Spanier, E. H., Ann. Math., 70, 338-378 (1959). Downloaded by guest on October 2, 2021