VOL. 44, 1958 ERRATA: M. MORSE 1285 ERRATA: DIFFERENTIABLE MAPPINGS IN THE SCHOENFLIES PROBLEM BY MARSTON MORSE In the article of the above title appearing in these PROCEEDINGS, 44, pp. 1068- 1072, 1958, the following corrections and additions should be made: Page 1069 In the last line of text matter, Nq should be 9q Page 1070 In line 8, the first of three signs u should be n. In the last line, suppose K -G and K - B are arc-wise connected. Page 1072 In line 10, replace P by P u K. In lines 11, 12, and 13, delete references to D1. Downloaded by guest on September 23, 2021 10686MA THEMA TICS: MORSE PROC. N. A. S. and E' is the graded module associated with a certain filtration of H(TSK). It follows from a generalization of the Eilenberg-Zilber theorem" by P. Cartier that Hq( T[k]K) = 0 for q < kn if (An): Kq = 0 for q < n. Using Dold,' Section 3, one can replace the hypothesis (An) by (Ba): K is a projective FD-module over a hereditary ring and Hq(K) = 0 for q < n. By standard arguments on spectral se- quences one then obtains from 3.8 3.9. COROLLARY. If K satisfies (An) or (Ba) then there is an exact sequence a 1* a go al* Hq(T[2]K) Hq(TK) Hq+i(TSK) Hl-(T[2]K) Hqi(TK) (3.10) for q < 3n; in particular, o-: H5(TK) --' Hq+i(TSK) is an isomorphism for q < 2n and an epimorphism for q = 2n. It can be shown that the homomorphism ,3 in (3.10) is the composition of d1*: Hq+l(TSK) Hq+l(T[p]SK) and an isomorphism Hq+l(T[2]SK) - Hqi,(T[2]K). If T is a quadratic functor, i.e., T(M1IM`2M3) = 0 for all M1, M2, M3, there is an exact sequence (3.10) without any assumption on K and without restriction on q. All results on the suspension have obvious applications to the left derived func- tors of T. Corollary 3.9 gives, in particular, v; LJ(n)TM _ L(n+'l) TM for q < 2n. (3.11) There is a "dual" to the construction (3.6) which resembles the cobar construc- tion.' Together with the details about the preceding results, it will appear else- where. 1 J. F. Adams, "On the Cobar Construction," these PROCEEDINGS, 42, 409-412, 1956. 2 H. Cartan and S. Eilenberg, Homological Algebra (Princeton, N. J., Princeton University Press, 1956). 3 A. Dold, "Homology of Symmetric Products and Other Functors of Complexes," to appear in Ann. Math., 68, 1958. 4 S. Eilenberg and S. Mac Lane, "On the Groups H(7r, n). I," Ann. Math., 58, 55-106, 1953. 5 S. Eilenberg and S. Mac Lane, "On the Groups H(7r, 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 University, to the topological theory of the Schoenflies problem lead naturally to questions of fundamental importance in the theory of differentiable mappings. In particular, if the hypothesis of the Schoenflies Theorem is stated in terms of regular differentiable mappings of class Cm, m > 0, can the conclusion be stated in terms of regular differentiable mappings of the same class? This paper presents an outline of an answer to this question. It shows how to develop a differential theory corresponding to Mazur's process.' An abstract r-manifold 1, r > 1 of class Cm is understood in the usual sense.23 VOL. 44, 1958 MA THEMA TICS: MORSE 1069 We do not require that 2r be connected. Given manifolds 2r and 2r' with Cm- differentiable structures, a homeomorphism O of 2T onto 2r' which is also a regular differentiable mapping of class Cm, will be called a Cm-diffeomorphism (abbr. Cm- diff.). Com-diffeomorphisms: Let 2r and t' be r-manifolds of class cm, m > 0. Let p and p' be points, respectively, of Zr and 2r'. Then r- p and 2r' - p are r- manifolds which possess Cm-structures derived from those of Tr and 2r'. A homeo- morphism A of 2T onto 'r' will be called a Com"diff., if for some p e 2r and p' e Arts A (p) = p', and if the restriction of A to r- p is a Cm-diff. of r - p onto 2r' - p. In case A is not a Cm-diff. of 2r onto 2r', we term p exceptional under A. Let n > 1 be a fixed integer. Let E and e be two Euclidean n-spaces. We shall be concerned with an (n -1)-sphere Sn-1 of unit radius in E, and a regular, com- pact (n - 1)-manifold Uln-I in e with a Cm-structure, m > 0, derived from that of (. Let 9n- be the image of S,- under a Cm-diff. 4. Let J Sn- and J 9,n-j be respectively the closures of the interiors of Sn-1 and En-1 in E and (. In referring presently to a Co"n-diff. of a neighborhood of J S,,1, we mean a Com-diff. in which the exceptional point (if any exists) is on the interior of S,,-. The principal theorem follows. THEOREM 1.1. Given a C"t-diff.44, m > O of S.-I onto 9)fn-I there exists a Co"-diff. A,, of some open neighborhood of J S.-,, relative to E, onto some open neighborhood of J 9,n-l, relative to (A, such that A, sn-1 = 4). There are indications that this theorem would be false if Com were replaced by Cm in its statement, at least false for some integers n.4 2. Composite n-Manifolds.-Let M and 9Y1 be two n-manifolds abstractly given, without points in common. Let W and Q be fixed open subsets of M and 91, re- spectively, and ,t a homeomorphism of W onto W. Let each point p e W be identi- fied with its image i(p) to form an identified point [p:y(p)]. Subject to this identi- fication, let 2 be the ensemble of all points of M and 9W. The -mappings ir, 71, r2: To each point p e M corresponds a point ir(p) e represented by p if p f W, and by [p :,(p)]if p e W. To each point q E corresponds a point 7r(q) E 2, represented by q if q f i, and by [jc-I(q) q], if q E U3. Set 71 = TIM, 72 = r[9. The mappings 7r, and 7r2 are biunique, but not in general onto 2; the mapping 7r of M u 9R (points unidentified) is onto A, but not in general bi- unique. We write 2 = [Ml, 9), u, W, Q], and term this a canonical representation of Z. Let z be topologized as follows. Let X and OC be arbitrary open sets of M and 911, respectively. Let 4P be the ensemble of subsets of 2 which contains all subsets 7ri(X) and 72(l) and is closed with respect to finite intersections of its sets. An open set of 2: shall be the union of an arbitrary collection of sets of 4. So topolo- gized, E is not necessarily a Hausdorf space or a manifold. It is, however, if Condi- tion (a) is satisfied (as we assume). Let 1MW denote the boundary of W relative to M, and 13.1,3 the boundary of Q3, relative to 9). Condition (a). For arbitrary points p E AmW and q e flO there shall exist neighbor- hoods N, and N,, relative to M and 9), respectively, such that 7r,(Np n W) n 7r2(9q n JU) = 0. 1070 MA THEMA TICS: MORSE PROC. N. A. S. M-Represented subsets of 2: Let A and 21 be subsets, respectively, of M and 9), and set 7ri(A) U 7r2(21) = [A, 21, 2;]. This subset of E will be said to be u-represented if (A n W) = 21 n G. Two n-represented subsets of I, [A, 2, ] and [B, i, ], have a M-represented inter- section and union [A n B, 2 u 3,2], [A u B, 21 u Q, ]. This extends to infinite unions. The complement of a jA-represented subset [A, 21, z ] of 2 is a n-represented subset [1- A, )1 - 21, ;]. The algebra of these subsets is basic. A Cm-structure on 2;: Suppose M and 9) have Cm-structures and that the mapping 1A is a Cm-diff. Let [F] and [H] be, respectively, sets of local representations of Mkl and 9), where F e [F] maps an open Euclidean region U onto a domain of coordi- nates X c M, and a e [H] maps an open Euclidean region U onto a domain of coordinates X c 91. We suppose that a Cm-structure is thereby defined on M and 9). To each F E [F] corresponds a mapping 7r1F of U onto 7r,(X) c 2, and to each 0 e [0] corresponds a mapping T2 of U onto 7r2(T) C Z. We show that the set of these mappings defines a Cm-structure on I. 3. Differential Transformation of the problem.-We establish a fundamental lemma. Let (xi ...., x,) and (y', . , Yn) be Euclidean coordinates in E and (i, re- spectively. Let I be the mapping of E onto d under which yi = Xi, i = 1, .