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On a Differential Structure for the Group of Diffeomorphisms View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TO~O~OQYVol. 6. pp. 263-271. Pergamon Press, 1967. Printedin Oreat Britain ON A DIFFERENTIAL STRUCTURE FOR THE GROUP OF DIFFEOMORPHISMS J. A. LBsLnzt (Received 9 May 1966) INTRODUCTION THE PURWSE of this note is to exhibit a differential structure on the group of diffeomorphisms of a compact connected manifold M. The structure in question turns out to be locally FrCchet. In order to obtain some of the classical theorems of analysis in the context of Fr6chet spaces we utilise the notion of quasi-topological spaces of which one can find a detailed exposition in [l]. The local Fr6chet structure is introduced by techniques analogous to those used by Eells in [2] to introduce a locally Banach structure in the space of continuous maps 9(S, M) where S is a compact topological space and M a differentiable manifold. I would like to take this occasion to thank J. Eells without whose encouragement this note would probably not have been written. All topological vector spaces appearing in this paper are considered to be Hausdorff locally convex topological vector spaces over the reals, R; continuous functions will be called Co functions when convenient. Definition. Let U be an open subset of a topological vector space E, and F a topological vector space. A function f : U + F is a C” function (n 2 1) iff is a C”-’ function and for each x E U there exists a continuous symmetric n-multilinear function D “f(x): E x . x E + F such that Y F(u) -j-(x+ v) -f(x) - Df(x)(u) - . - l/n! D”f(x)(u, . , u) satisfies the property that G(t, V) = F(tu)/t”, t#O = 0, t=O is continuous on R x V at (0, v), v~ E and such that D'f : U x E x . x E+ F is continuous. (Convention : D”)f(x) = f(x)). Y PROPOSRION 1. Suppose E, F, and G are topological vector spaces. If U c E, V c F are open andf : U -+ V andg : V -+ G are C” functions then g 0 f : U --, G is u C” function. t Supported by National Science Foundation grant GP-2440. 263 264 .J. A. LESLIE PROPOSITION2. Let f : U + V and g: V + F be C” functions, U c E, V c F open where E, F, and G are topological vector spaces. If h : U -+ G is a C1 function such that Dh(x)(v) = (Dg(f(x)) o Df(x))(v), then h is a C” function. Dejinition 2. Let U c E, V c F be open sets in topological vector spaces E and F, and suppose that G is a topological vector space. A function f :U x V--f G is a C n function in the first (resp. second) variable if f is a C”- ’ function in the first (resp. second) variable and if for each (t, r) E U x V there exists a continuous symmetric n-multilinear function a "faxn(t,q): %xE --f G n-times (resp. anfjlayn(& q): F x . x F+ G n-times such that F(V)= f(< + V,q) - f(5,q) - aflax(t,q)(v)- ... - I/n!a"flax"(r, r)(v,..., U) (resp. G(u) = f(c, q f v) - f(& q) - . - I/n! anfldy”(t, q)(v, . , v)) satisfies the property that rp(t, 4 = F(tu)lt”, t#O N, u) = 0, t=O (resp. y(t, v) = G(tv)/t”, t # 0; r(t, v) = 0, t = 0) is continuous on R x E (resp. R x F) at (0, v), v E E (resp. v EF) and such that anf/lax: U x V x E x . x E+ G (resp. a”f/lay: UX VxFx... x F-t G) is continuous. PROPOSITION 3. Let f : U x V --, G be a function such that for x, x’, x + x’ E U f(x + X’,Y) =f (x9 Y) +f (x’, Y). f is C” if and only if f is C” in both the first and the second variables. The proofs of propositions depend on the notion of quasi-topology developed by Bastiani [l]. We shall now give a short exposition of her theory. Most of the theorems stated here are either proved in [l] or their proofs are simple deductions from what appears in [l]. Let S be a set for x E S designate by %X the ultra filter of all sets containing x. Given two filters % and %’ on S, % < %’ will mean given A’ E %’ there exists A E % such that A c A’. Dejnition 3. A quasi-topological space is a set S together with a relation “+“between filters on S and points in S satisfying the following axioms: (1) x~Simplies%~--,x (2) .T’ < I and 9” + x implies 97”-+ x (3) $” + x and 95 + x‘ implies x = x’ (4) 3 + x and 3”’ + x implies 35 n 97”’+ X. Definition 4. Let {A’, -} and { Y, -t} be quasi-topological spaces. A function f : X-+ Y is called quasi-continuous if S + x E S implies f(s) + f (x) E Y for all x E 3’. (By abuse of notation f(X) is the filter generated by f (S), S E 9-j. The relation .!Z+ x will be described by saying X quasi-converges to x. ON A DIFFERENTIAL STRUCTURE FOR THE GROUP OF DIFFEOMORPHISMS 265 The quasi-topological spaces and the quasi-continuous maps form a category that we shall designate by fi. Composition in +. is ordinary composition of maps of sets. Let Y be the category of Hausdorff topological spaces; define a functor T: F --) +i as follows: for X an object of Ysuppose that the underlying set of T(X) is the same as the underlying set X with quasi-topology given by 3 + x if and only if I converges to x for the topology on X E Y. As a function of sets set T(f) = 6 T(f) is clearly quasi-continuous for f a morphism in Y. Now define a functor S: p -tY as follows: for X E +z, suppose that the underlying set of S(X) is the same as the underlying set of X. The topology on S(X) is the strongest topology on S(X) such that S 4 x in XE /Z implies d converges to xinS(X)Er. Forf :X+ Y~fisetS(f)=f as a function of sets. With these definitions S: b +Y is a functor (see [l], pp. 24-25). S o T: F -P F is clearly the identity functor. The identity mapping on X defines a transformation of functors /3(X) : X + T 0 S(X). It is easy to verify that T is the adjoint of S(S -! T); that is, a(f) = T(f) 0 b(B) defines an isomorphism of functors from Horn, (S(B), A) to Horn,, (B, T(A)) having b(g) = S(g) as an inverse. Given a collection of quasi-topological spaces Xi, i E I, there exists on n Xi a quasi- id topology having the universal property for products defined by % + x E n Xi if and only id if pi(%) + Pi(x) for all i E I where pi: I-J Xj -P Xi is the canonical projection. jd Definition 5. A quasi-topological vector space is a vector space E over R together with a quasi-topology + such that, (1) the sum and scalar multiplication define quasi-continuous functions E x E + E and T(R) x E + E; (2) there exists a locally convex topology on E such that if I + x then ?Z converges to x for this locally convex topology on E; (3) if %+O then -%+O. The quasi-topological vector spaces form an additive category where the morphisms are the linear quasi-continuous functions between quasi-topological vector spaces; compo- sition is ordinary set theoretic composition. Designate the category of quasi-topological vector spaces by QV and the category of locally convex topological vector spaces by V. Let T : r --f Qr be the functor that associates to V E V the quasi-topological vector space whose underlying vector space structure is the same as that of V and whose under- lying quasi-topology is given by considering V E F and considering T(V) E #Z(see p. 3). Definition 6. Let E, F, E Qy and U c E open in S(E) where S(E) is the finest locally convex topology on E so that filters which quasi-converge converge, and let f : U + F be a function. f is called a C1 function if for each x E U there exists a quasi-continuous linear transformation D f (x) : E -+ F such that F(a) = f (x + 4 - f(x) - Df (x)(4 266 J.A. LESLIE satisfies the property that cp(t, 0) = F(tv)/t, f#O = 0, t=O is a quasi-continuous function and Df : U-+LZ(E, F) is a quasi-continuous function. Set D”f = D(D”-‘f). Define a functor Y : QY x QY -P QY as follows: 3’(E, F) as a vector space is the vector space of all quasi-continuous linear transformations E -+ F; the quasi-topology on S(E, F) is given by .F + f if and only if d --f x implies F”(3) + f (x). With the obvious definition of 8 on morphisms one verifies readily that 49: QV x QV -_) QV is a functor.
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