Non-Additive Functors, Their Derived Functors, and the Suspension HomomorphismAuthor(s): Albrecht Dold and Dieter PuppeSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 44, No. 10 (Oct. 15, 1958), pp. 1065-1068Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/89954 .

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3 E. L. Post, J. Symb. Logic, 12, 1-11, 1947. 4 In the sense of Post, the word problem for Z is reducible to that for (p (Z). 5 W. Magnus, Math. Annalen, 106, 295-307, 1932.

Dana Scott, abstract, J. Symb. Logic, 21, 111-112, 1956. 7 Marshall Hall, J. Symb. Logic, 14, 115-118, 1949. s G. Higman, B. H. Neumann, and H. Neumann, J. London Math. Soc., 24, 247-254, 1949. 9 As defined, UG, Part I, p. 234. 0 Also provable by Theorem I of Higman, Neumann, and Neumann, op. cii., as pointed out to

by Higman. 1 The set of words Al, A2, .. ., AK on 3 is not independent if there is a product of the A's, with

adjacent A's inverses of each other, which equals 1 in the free group on 3. 2 The 32 defining relation group mentioned above has one relation which is astronomical in

gth. 3 Included in a report filed with the National Science Foundation on contract G-1974, May 28, i6-but in a form more akin to UG, Parts V-VI, than Result a as presently shown (cf. W. Craig, Symb. Logic, 18, 30-32, 1953, and B. H. Neumann, J. London Math. Soc., 12, 125, Theorem (13), :7). 4 For related material see W. W. Boone, abstract, Bull. A.M.S., 62, 148, 1956.

NON-ADDITIVE FUNCTORS, THEIR DERIVED FUNCTORS, AND THE SUSPENSION HOMOMORPHISM

BY ALBRECHT DOLD AND DIETER PUPPE

MATHEMATISCHES INSTITUT DER UNIVERSITAT HEIDELBERG

Communicated by Saunders Mac Lane, July 3, 1958

1. Derived Functors of Non-additive Functors.-Let T be a (covariant) functor of )dules over a ring A to modules over a ring A'. If T is additive i';s derived func- rs have been defined by Cartan-Eilenberg.2 Additivity is used to show that T plied to a chain homotopy again gives a chain homotopy (cf.2 IV, 5, and V, 3). ing FD-complexes4 instead of chain complexes, we define left derived functors for )itrary functors T. 1.1. Definition.-A projective FD-resolution of type n of the module M is an )-module P such that (i) Pj = 0 forj < n, (ii) Pj is projective for allj, (iii) Hn(P) M, Hj(P) = O forj s n.

Passing from an FD-module to its normal(ized) chain module establishes a 1 to 1 rrespondence (up to natural equivalences) between FD-modules, FD-maps, FD-

motopies and chain modules, chain maps, chain homotopies.3 In particular, it

;ablishes such a correspondence between projective FD-resolutions P of type n of and chain modules C for which Cj = 0 for j < n and

0 - M = Hn(C) -Cn C +- Cn+1 <- C+2 ... (1.2)

an ordinary projective resolution2 of M. From the corresponding properties of

linary resolutions it follows: Every module has a projective FD-resolution of any -en type n. If f: M ->- M' is a homomorphism and P,P' are projective FD-

solutions of type n of M,M' resp., then there exists an FD-map F: P ->- P' such at F*: Hn(P) -> Hn(P') equals f. Moreover, if F' is another such map, then

F' are FD-homotopic (cf. Cartan-Eilenberg,2 V, 1).

3 E. L. Post, J. Symb. Logic, 12, 1-11, 1947. 4 In the sense of Post, the word problem for Z is reducible to that for (p (Z). 5 W. Magnus, Math. Annalen, 106, 295-307, 1932.

Dana Scott, abstract, J. Symb. Logic, 21, 111-112, 1956. 7 Marshall Hall, J. Symb. Logic, 14, 115-118, 1949. s G. Higman, B. H. Neumann, and H. Neumann, J. London Math. Soc., 24, 247-254, 1949. 9 As defined, UG, Part I, p. 234. 0 Also provable by Theorem I of Higman, Neumann, and Neumann, op. cii., as pointed out to

by Higman. 1 The set of words Al, A2, .. ., AK on 3 is not independent if there is a product of the A's, with

adjacent A's inverses of each other, which equals 1 in the free group on 3. 2 The 32 defining relation group mentioned above has one relation which is astronomical in

gth. 3 Included in a report filed with the National Science Foundation on contract G-1974, May 28, i6-but in a form more akin to UG, Parts V-VI, than Result a as presently shown (cf. W. Craig, Symb. Logic, 18, 30-32, 1953, and B. H. Neumann, J. London Math. Soc., 12, 125, Theorem (13), :7). 4 For related material see W. W. Boone, abstract, Bull. A.M.S., 62, 148, 1956.

NON-ADDITIVE FUNCTORS, THEIR DERIVED FUNCTORS, AND THE SUSPENSION HOMOMORPHISM

BY ALBRECHT DOLD AND DIETER PUPPE

MATHEMATISCHES INSTITUT DER UNIVERSITAT HEIDELBERG

Communicated by Saunders Mac Lane, July 3, 1958

1. Derived Functors of Non-additive Functors.-Let T be a (covariant) functor of )dules over a ring A to modules over a ring A'. If T is additive i';s derived func- rs have been defined by Cartan-Eilenberg.2 Additivity is used to show that T plied to a chain homotopy again gives a chain homotopy (cf.2 IV, 5, and V, 3). ing FD-complexes4 instead of chain complexes, we define left derived functors for )itrary functors T. 1.1. Definition.-A projective FD-resolution of type n of the module M is an )-module P such that (i) Pj = 0 forj < n, (ii) Pj is projective for allj, (iii) Hn(P) M, Hj(P) = O forj s n.

Passing from an FD-module to its normal(ized) chain module establishes a 1 to 1 rrespondence (up to natural equivalences) between FD-modules, FD-maps, FD-

motopies and chain modules, chain maps, chain homotopies.3 In particular, it

;ablishes such a correspondence between projective FD-resolutions P of type n of and chain modules C for which Cj = 0 for j < n and

0 - M = Hn(C) -Cn C +- Cn+1 <- C+2 ... (1.2)

an ordinary projective resolution2 of M. From the corresponding properties of

linary resolutions it follows: Every module has a projective FD-resolution of any -en type n. If f: M ->- M' is a homomorphism and P,P' are projective FD-

solutions of type n of M,M' resp., then there exists an FD-map F: P ->- P' such at F*: Hn(P) -> Hn(P') equals f. Moreover, if F' is another such map, then

F' are FD-homotopic (cf. Cartan-Eilenberg,2 V, 1).

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Now let T be an arbitrary fulnctor from modules to modules. Its prolongation3 FD-modules (defined by applying T is every dimension and also to the face- and generacy operators) preserves homotopy, i.e., F _ F' implies TF _ TF' for all )-maps F,F', even if T is not additive.3 Hence it follows, as in Cartan-Eilen- -g2 (V, 3), that Hq(TP) depends only on M and n (not on P), and (TF): Hq( TP) Hq(TP') depends only on f and n. We denote H,(TP), (TF)*q by L,(n)TM, r)Tf resp., and call L,q()T the qth left derived functor of T of type n.

[f T is additive, L.( )T coincides with the left derived functor Lq_nT as defined in .rtan-Eilenberg.2 For arbitrary T we shall exhibit a natural transformation -: n) T - L(n+l)T, the suspension (cf. (2.4)). a is an isomorphism for additive but not in general. 1.3. Examples.-Let T,T' be the functors from abelian groups to A-modules ined by TM = N 0 Z(M), T'M = N 0 SA(M), where N is a fixed &-module, MI) is the group ring, and SA (M) is the symmetric algebra of the abelian group M. .en L,(')TM for all n, and Lq(n)T'M for n > 0 are the Eilenberg-Mac Lane mod- Is HQ(M, n; N).4 We do not know how to answer the interesting question as the nature of the derived functors of the r-functor of Eilenberg-Mac Lane.5 2. The Suspension.-Define the cone over an FD module K to be the FD-module L given by

(CK)q = Kq +Kq_l K+Kq-2 + . . .+ Ko, (a,, aq_, . . ., ao) = (6iaq, 6b-laq-, ..., la_q-+l, 6oaa-i +

aq_-i1, a__i-2, . . ., ao), i < q, (2.1) ),(a,, a-1, . . ., ao) = (6,aq, 6,-la,-, . . ., blal), si(aq, a_l-, . . ., ao) = (siaq, si-laq-1, .. . ., soaq-i O, a-i-, . . ., ao),

.ere 56 and si are the face- and degeneracy operators. There is a natural injection K -* CK, and the FD-module SK = CK/iK is called the suspension of K. The ict sequence

i p

0 -- K CK --SK 0 (2.2)

its if we disregard the FD-structures, i.e., (CK), - K, + (SK)q for all q. CK is the same homotopy type as the trivial complex 0 (notation: CK = 0), hence (K) - Hq+i(SK). If P is a projective resolution of type n, then SP is a projec- e resolution of type n + 1 of the same module. Let T be a functor from modules to modules prolonged to FD-modules as above, d assume T(0) = 0. Then (Tp)(Ti) = T(pi) = 0, i.e., im(Ti) c ker(Tp), and have (Ti)*: Hq(TK) -* Hq(ker(Tp)). Since (2.2) splits, Tp is an epimorphism, ice TSK - TCK/ker(Tp). The boundary homomorphism A: H+,(TSK) -

(ker(Tp)) is an isomorphism because CK - 0 implies TCK = T(O) = 0 (T pre- ves homotopy). So we can form the suspension homomorrhism,

a = A-1(Ti)*: H,(TK) - H+,(TSK). (2.3)

particular, if K = P is a projective FD-resolution of type n of M, this gives a tural transformation,

aq () (n): L() n)TM r L(n + )TM. (2.4)

If T does not satisfy T(O) = O, then TM = T'M + Mo, where T'(O) = 0, and

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is a fixed module. In order to apply our constructions, replace T by the functor

.Properties of the Suspension.-Eilenberg and Mac Lane5 defined the kth ss effect functor of a functor T with T(0) = 0. These functors, also denoted by satisfy

T(M1 + M2 + . . . + Mr) = El<i<i2<... <ik<rT(Mi,Mtui . .. [Mik). (3.1)

;ting all variables in the kth cross-effect be equal gives a functor in one variable,

T[rM = T(MIMl. ..M) (k variables). (3.2)

Let CkM denote the direct sum of k copies of M. Define ai': ZkM - C-k-lM I /i': Ek-IM -- EkM for 0 < i < k by

al'(ml, m2, , k)= (ml, . . ., m) = ( mi + mi+l, mi+2, . . ., mk) (33) i'(ml, m2, . ., mk-1) = mi, *. .M i-, mil m, , mi+l, . . ., mk-l).

restriction and projection (cf. 3.1) we obtain from Tai' and T3i' the natural nsformations

a,: Takl M - Trk-.M afi T7l]M T[kl]M 0 < i < k. (3.4) 0': T[rc-l]3M-- T kT]M

and 3i are prolonged to FD-modules by applying them in every dimension. Let K be an FD-module. The elements in the image of a,*: H(T(KIK)) -- TK) resp. in the kernel of 1,*: H(TK) -> H(T(KIK)) are called decomposable p. primitive. 3.5 PROPOSITION. The decomposable elements of H(TK) lie in the kernel of the ,pension a: H(TK) -- H(TSK). The image of a consists of primitive elements of TSK). ro obtain further results on the suspension (in particular a partial inverse of 3.5: 3.9) we consider the sequence

di d2 dk-i dk

TK = TK *- TT[2K - .. <- T[k1K <- . ., (3.6)

ered = dk = Z='(--l)jaj. Then dd = 0, and :?d, together with the "internal" ferential b = (- 1) jl of T[klK, turns SK = E= 1T[k]K into a double complex. T stands for "group ring" or "symmetric algebra" (cf. 1.3) ZK is very much like bar construction.4

3.7. THEOREM. There is a natural isomorphism HQ+1(TSK) H Hq(ZK) which nsforms the suspension ar: Hq(TK) - Hqg+(TSK) into the map t*: Hq(TK) -

(SK) induced by the inclusion L: TK -> ZK. (H(TK) is the homology of the gle complex associated with the double complex ZK; cf. Cartan-Eilenberg,2 IV,

There are two spectral sequences associated with the double complex SK (cf. rtan-Eilenberg,2 XV, 6). One of them can be used to prove 3.7, the other gives 3.8. COROLLARY. There is a spectral sequence Er such that E1 together with its ferential d1 is the (single) complex

d*T d d* d, d

H(TK) - H(T21K) -.. .. <--H(T[k,K) --...,

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d E' is the graded module associated with a certain filtration of H(TSK). It follows from a generalization of the Eilenberg-Zilber theorem5 by P. Cartier it H,(T[kIK) = 0 for q < kn if (An): K, = 0 for q < n. Using Dold,3 Section 3, e can replace the hypothesis (An) by (Bn): K is a projective FD-module over a reditary ring and H,(K) = 0 for q < n. By standard arguments on spectral se- ences one then obtains from 3.8 3.9. COROLLARY. If K satisfies (An) or (Bn) then there is an exact sequence

Cns a 13 a1*

H,(T[21K)-- H(TK) - Hq+i(TSK) - H,q-(T[21K) - Hq(TK) (3.10)

' q _ 3n; in particular, a: Hq(TK) - Hq+,(TSK) is an isomorphism for q < 2n d an epimorphism for q = 2n. It can be shown that the homomorphism f in (3.10) is the composition of 3l*: +1(TSK) -> Hq+(T[lSK) and an isomorphism H,q+(T21SK) -

Hq(T2K). T is a quadratic functor, i.e., T(MI M21M3) = 0 for all M1, M2, M3, there is an ict sequence (3.10) without any assumption on K and withoat restriction on q. All results on the suspension have obvious applications to the left derived func- Is of T. Corollary 3.9 gives, in particular,

): L()TM _ Lj+l TM for q < 2n. (3.11)

There is a "dual" to the construction (3.6) which resembles the cobar construc- n.' Together with the details about the preceding results, it will appear else- Lere.

J. F. Adams, "On the Cobar Construction," these PROCEEDINGS, 42, 409-412, 1956. H. Cartan and S. Eilenberg, Homological Algebra (Princeton, N. J., Princeton University

ess, 1956). A. Dold, "Homology of Symmetric Products and Other Functors of Complexes," to appear in

n. Math., 68, 1958. S. Eilenberg and S. Mac Lane, "On the Groups H(7, n). I," Ann. Math., 58, 55-106, 1953. S. Eilenberg and S. Mac Lane, "On the Groups H(zr, n). II," ibid., 60, 49-139, 1954.

DIFFERENTIABLE MAPPINGS IN THE SCHOENFLIES PROBLEM

BY MARSTON MORSE

INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY

Communicated July 14, 1958

1. Introduction.-The recent remarkable contributions of Mazur, of Princeton liversity, to the topological theory of the Schoenflies problem lead naturally to estions of fundamental importance in the theory of differentiable mappings. particular, if the hypothesis of the Schoenflies Theorem is stated in terms of

gular differentiable mappings of class Cm, m > 0, can the conclusion be stated in -ms of regular differentiable mappings of the same class? This paper presents an tline of an answer to this question. It shows how to develop a differential theory rresponding to Mazur's process.' An abstract r-manifold ;, r > 1 of class Cm is understood in the usual sense.2"3

d E' is the graded module associated with a certain filtration of H(TSK). It follows from a generalization of the Eilenberg-Zilber theorem5 by P. Cartier it H,(T[kIK) = 0 for q < kn if (An): K, = 0 for q < n. Using Dold,3 Section 3, e can replace the hypothesis (An) by (Bn): K is a projective FD-module over a reditary ring and H,(K) = 0 for q < n. By standard arguments on spectral se- ences one then obtains from 3.8 3.9. COROLLARY. If K satisfies (An) or (Bn) then there is an exact sequence

Cns a 13 a1*

H,(T[21K)-- H(TK) - Hq+i(TSK) - H,q-(T[21K) - Hq(TK) (3.10)

' q _ 3n; in particular, a: Hq(TK) - Hq+,(TSK) is an isomorphism for q < 2n d an epimorphism for q = 2n. It can be shown that the homomorphism f in (3.10) is the composition of 3l*: +1(TSK) -> Hq+(T[lSK) and an isomorphism H,q+(T21SK) -

Hq(T2K). T is a quadratic functor, i.e., T(MI M21M3) = 0 for all M1, M2, M3, there is an ict sequence (3.10) without any assumption on K and withoat restriction on q. All results on the suspension have obvious applications to the left derived func- Is of T. Corollary 3.9 gives, in particular,

): L()TM _ Lj+l TM for q < 2n. (3.11)

There is a "dual" to the construction (3.6) which resembles the cobar construc- n.' Together with the details about the preceding results, it will appear else- Lere.

J. F. Adams, "On the Cobar Construction," these PROCEEDINGS, 42, 409-412, 1956. H. Cartan and S. Eilenberg, Homological Algebra (Princeton, N. J., Princeton University

ess, 1956). A. Dold, "Homology of Symmetric Products and Other Functors of Complexes," to appear in

n. Math., 68, 1958. S. Eilenberg and S. Mac Lane, "On the Groups H(7, n). I," Ann. Math., 58, 55-106, 1953. S. Eilenberg and S. Mac Lane, "On the Groups H(zr, n). II," ibid., 60, 49-139, 1954.

DIFFERENTIABLE MAPPINGS IN THE SCHOENFLIES PROBLEM

BY MARSTON MORSE

INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY

Communicated July 14, 1958

1. Introduction.-The recent remarkable contributions of Mazur, of Princeton liversity, to the topological theory of the Schoenflies problem lead naturally to estions of fundamental importance in the theory of differentiable mappings. particular, if the hypothesis of the Schoenflies Theorem is stated in terms of

gular differentiable mappings of class Cm, m > 0, can the conclusion be stated in -ms of regular differentiable mappings of the same class? This paper presents an tline of an answer to this question. It shows how to develop a differential theory rresponding to Mazur's process.' An abstract r-manifold ;, r > 1 of class Cm is understood in the usual sense.2"3

This content downloaded from 62.122.77.88 on Mon, 5 May 2014 07:06:39 AMAll use subject to JSTOR Terms and Conditions