Prl:E X R E; Pr,(X,P) = X Pr2:E X Rfv R; Pr2(X,P) = P
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1036 MATHEMATICS: HUEBSCH AND MORSE PROC. N. A. S. ever, it is easy to show that this theory is not generally acceptable. If, for instance, a dilute toluidine blue solution is added gradually to a relatively concentrated solution of a few per cent of polyethylsulfonate, the first trace of dye added, detectable by the naked eye, is already meta- chromatic. It seems unreasonable to suppose that this minute quantity of dye would form stacks and not distribute itself among the great excess of SO, groups. Similarly, if a solution of poly- ethylsulfonate is gradually added to a solution of toluidine blue, the maximal metachromatic shift is obtained when there is one SO, group for every dye molecule. Thus, there being no generally acceptable theory for metachromasia, one may sum up the situation by saying that while the weak interaction between two dye molecules in a dimer causes a small shift and a small de- crease in absorbancy, the stronger interaction between the basic dye and strongly negative SO3, P03, or COOH groups causes a greater shift with decrease and broadening of the absorption. Though most proteins contain COOH groups they are not usually metachromatic, the number of COOH groups being low and their electrostatic action being, possibly, also compensated by the NH2 groups present. 4Bailey, K., Biochem. J., 43, 271, (1948). 6 White, J. I., H. B. Bensusan, S. Himmelfarb, B. E. Blankenhorn, and W. R. Amberson, Am. J. Physiol., 188, 212, (1957). 6 Other metachromatic dyes, like acridine orange, can equally be used. THE DEPENDENCE OF THE SCHOENFLIES EXTENSION ON AN ACCESSORY PARAMETER (THE TOPOLOGICAL CASE) BY WILLIAM HUEBSCH AND MARSTON MORSE INSTITUTE FOR ADVANCED STUDY Communicated October 28, 1963 The theorem stated below has been established in detail in a paper published by the authors in 19601 and referred to here as (I). The theorem is here restated, with appropriate explanations, because a recent review of this paper in Mathematical Reviews refers only to the "differentiable case," whereas in paper (I) the authors explicitly state and prove the theorem both in the "topological case" and the "differentiable case." As a result of this omission in Mathematical Reviews, mathematicians have started work on the problem in the "topological case" under the assumption that (I) covers only the differentiable case. The parameter p in the problem in the topological case has for domain an arbi- trary paracompact space r. Let E be a Euclidean n-space n > 1 of points x with rectangular coordinates (xi, . ., xn). Let S be an (n-1)-sphere in E, and B the closed n-ball bounded by S. The Product Space E X r.-Let x and p be points in E and r, respectively, and X a subset of E X r. One introduces the projections prl:E X r E; pr,(x,p) = x pr2:E X rFv r; pr2(x,p) = p. For p fixed in r, the p-section of X is by definition the set Xv = {x E EI(x,p) E X}. If (xp) -- F(x,p) is a mapping of X into E X r set, prF(x,p) = Fi(x,p), pr2F(x,p) = F2(x,p), Downloaded by guest on September 29, 2021 VOL. 50, 1963 MATHEMATICS: HUEBSCH AND MORSE 1037 Thus, (x,p) -- Fl(x,p) is a mapping of X into E, and (x,p) -0 F2(x,p) is a mapping of X into r. The mapping F will be termed p-invariant if for each (x,p) E X, F2(x,p) = p. For (x,p) E X one sets F1(x,p) = FP(x), and for fixed p E pr2X terms the mapping x -- FP(x) of XP into E the p-section of F. The Data.-Let L be an open neighborhood of S X r relative to E X F, and let there be given a p-invariant homeomorphism b:L E x r; (x,p) -) t(x,p) such that for each p E r, (DP maps points of LP which are interior (exterior) to S in E into points which are interior (exterior) to the topological (n -1)-sphere cP(S) in E. Set Le =L - (B x r). THEOREM. Corresponding to 1, L, and r, conditioned as above, and to any suffi- ciently small open neighborhood L* C L of S X r relative to E X r, there exists a p-invariant homeomorphism A:L U (B X r) -- E X r which extends bI(Le U L*). For fixed p c r, AP is a solution of the classical Schoenflies problem in which 4P :LP E is the given homeomorphism. Moreover, AP varies continuously with p, since A is continuous. The problem with parameter is made difficult by the fact that the domain LP of definition of (P in E varies with p. In fact, it is not excluded that the subset nfLP (pFr) of E fails to be a neighborhood of S in E. A solution of the "problem with parameter" is made possible by the explicit nature of the authors' earlier solution of the classical Schoenflies problem without parameter.2 Formulas in reference 2 in the classical problem which give an exten- sion of a homeomorphism so explicitly involve a restriction of so. These formulas are of such a nature that they apply whether (p is a homeomorphism, a diffeomorphism, or even an analytic diffeomorphism.A They apply in the case at hand when so is replaced by a mapping b which depends upon a parameter p, provided a preliminary transformation of the r-problem into a "uniform" F- problem is made (cf. ref. 1). In the differentiable case r is assumed to be a connected differentiable r-manifold, r > 0 of class C', 0< m < c, with a countable base. For statement of the theorem in the differentiable case see (I) or Mathematical Reviews. 1 Huebsch, W., and M. Morse, "The dependence of the Schoenflies extension on an accessory parameter," Jour. d'Analyse Math., 8, 209-271 (1960-61). 2 Huebsch, W., and M. Morse, "An explicit solution of the Schoenflies extension problem," J. Math. Soc. Japan, 12, 271-289 (1960). 3 Huebsch, W., and M. Morse, "Schoenflies extensions of analytic families of diffeomorphisms," Math, Annalen, 144, 162-174 (1961). Downloaded by guest on September 29, 2021.