On Groups of Diffeomorphisms Author(s): T. E. Stewart Source: Proceedings of the American Mathematical Society, Vol. 11, No. 4 (Aug., 1960), pp. 559-563 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2034711 . Accessed: 05/05/2011 17:16

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http://www.jstor.org ON GROUPS OF DIFFEOMORPHISMS

T. E. STEWART I. We consider here the groups of homeomorphisms on Euclidean n-space and the n-sphere Sn. Chiefly we will be concerned with the question of whether or not these groups reduce in an homotopy sense to the ordinary orthogonal group acting on these spaces. Such ques- tions are intimately connected with the theory of fibre bundles in which these spaces occur as fibres. We will restrict ourselves to the case where the homeomorphisms are of class C' and will topologize the various groups taking account of the differentiability. We first consider Euclidean n-space En. We denote by K the group of all homeomorphisms f of En such that f and f'l are of class C'. K becomes a topological group by demanding uniform convergence of f and its derivatives on compact sets,' i.e. a typical neighborhood of the identity function is given by

Vre = g Ez K I ||g(x) -xll <,e, |- (x)- k |< e,|x|

i, k =1...,n}

Clearly the orthogonal group, On, of En is imbedded in K and the topology induced on 0n is the usual topology. In fact any locally compact or complete metric group which acts as a transformation group of En so that each motion is of class C' is imbedded in K (see [1, p. 197]). With this topology on K we now show THEOREM I. On is a deformation retract of K. First we will demonstrate a number of lemmas which lead to the proof of Theorem I.

LEMMA 1. Let Ko= {gCKlg(O)=0}. Then Ko is a closed subgroup of K and K decomposes topologically into En XKo. Hence if On is a deformation retract of Ko, then On is a deformation retract of K.

PROOF. With each xCEn associate oGK, a2(y) =y+x. Clearly xa-+o is topological. If fEK, consider f(O) =xo. Then aoz4f is in Ko Received by the editors October 26, 1959. 1 For details on this topology, see R. Thom, Les singularites des applications diff6rentiables,Ann. Inst. Fourier, Grenoblevol. 6 (1956) pp. 43-87. 559 560 T. E. STEWART [August and f= ag, g Ko. Furthermore, the decomposition is unique. Indeed if aog=a,h then gh-'=oa, is a translation. Since gh-'EKo, y=x, g =h. That the topology on K is the topology of the product space is easily established. Now let

H= gEKo - =k

H is then a closed, invariant subgroup and

LEMMA 2. Ko decomposes topologically into Gl(n, R) XH where Gl(n, R) is the group of non-singular matrices on En.

PROOF. Consider the homomorphism 4: Ko-+Gl(n, R) defined by +(f) = ((dfi/dxk) (0)). 4 is continuous on Ko since we have demanded close partial derivatives. If aEGl(n, R) then q(a) =a. If f Ko con- sider the element k(f-)f under 4. We have 4W(4f)-'f)= o(OW-9(f) = W(f)-'(f) = I the identity matrix, and ck(f->f is in the kernel of X which is clearly H, and f=O(f)g, geH. If ag=3h, a, 3C-Gl(n, R), g, hCH then #-Ta=hg-' and since hg-'GH, #-'a=I and consequently a=fl and g = h. Again the product topology of Ko can be shown to be the product topology on Gl(n, R) and H.

LEMMA 3. The group H is contractible to a point.

PROOF. For 0

0 c(f, t) = { L/gfLt, < t _ 1,

We have 41(f, 1) =f and 41(f, 0) =I, thus if 4Dis continuous, 4Dcon- tracts H to I. First we see that 1 is continuous on HX [0, 1] since on this set 1 amounts to a continuous system of inner automorphisms and these are continuous since the group operations are continuous. Now let f H, we will show that 1 is continuous at (f, 0). Suppose we are given a neighborhood V7,E of I as given by (1), (we suppose r> 1). We must show that there exists a neighborhood, U, off in H and a a > 0 such that for g U, t <6, N(g, t) C V7.. Now if hCH then by the theorem of the mean we have

h,(x) = xi + E c,(h; X)Xk k=1 I960] ON GROUPS OF DIFFEOMORPHISMS 561

and c4(h; x) tends to 0 with x, (in fact c4(h; x) = (dhi/lxk) (6Ix) -8,O< 0 <0 <1). Furthermore, denoting by h' the transformation Nh, t), t >0 we have (h'/dxj) (x) = (dhi/dxj) (tx). Then we first select a neighborhood Ui of f so that for

g C Ui, ||x||

Since (dfi/dxk)(0) =8 , we can choose ti>0 so that l|x|| <71 implies | (dfi/dxk) (X)-t I < e/2rn. Choose 61> 0 so that for 0

(x)X | (X) k | IOXk O9Xk

'agi 'Ofi-~~Of < (tx) - (ix) + -(tx)-&k OXk OXk SXk

E rn

Since the inequality holds for all x for which ||x||

I g(X) - I C(g; Xi= j-1 tX);Xi n _E I c(g; tx) I I xj I

Hence 1 maps U1X [0, 61) into V7,Eand 1 is continuous at (f, 0) and the lemma is proved. PROOF OF THEOREMI. It follows from Lemmas 1, 2, 3 that K = En X GI(n, R) X H as a topological space. Now GI (n, R) can be topologically decomposed as the product of the orthogonal group and a euclidean space. Then K is topologically the product OnXA where A is a contractible space, and Theorem I follows readily.

COROLLARY I. Let G= {B, p, X, En, K} be a fibre bundle in the sense of Steenrod, [2, p. 8], with fibre E , group K and base space X a locally finite polyhedra. Then MBis equivalent in K to a bundle with group On. 562 T. E. STEWART [August

PROOF. According to [2, p. 36] it suffices to show that the bundle 63' associated to (B with fibre K/On has a cross-section. It follows from Theorem I that lis(K/O.) =0, i_ O. Since X is a locally finite polyhedra it follows in the usual way that we can define a cross- section inductively on the k-dimensional skeleton of X and obtain the corollary. II. We consider now the sphere, Sn, and the group K of homeomor- phisms f of Sn such that f and fP are of class C' in the usual differ- entiable structure of Sn. We suppose Sn covered by two coordinate neighborhoods, U1 and U2, U1=Sn\{xo}, U2=Sn\{ -XO}. If (x) are the coordinates on U1 and (y) the coordinates on U2 we take as coordinate transformations

Xi YS =

E Xk k-1 K is topologized with the usual Cl-topology (cf. the reference in footnote 1).

THEOREM II. If On+iis the orthgonal group then i*: Hkk(On+i)-+Hk(K) is an isomorphism into where i* is induced by the injection i: On+,--K. The proof follows LEMMA4. Let Ko denote the subgroup of K which holds x0 fixed. Then i: On-+Ko induces an isomorphism i* of Hk(On) into Hk(Ko) for each k.2

PROOF. We suppose xO to have coordinates (0, * * *, 0) in Ul. De- fine the continuous homomorphism 4: Ko-+Gl(n, R) by

+(g) = ( (O)).

For each g, +/(g) can be expressed uniquely and continuously as the product of an orthogonal matrix i4(g) and a triangular matrix 7r(g), + (g) = 4(g)r(g). Then i/: KO-+On and for h iOn, 4(h) =h so that i/ is a retraction of Ko onto On. The lemma then follows immediately. PROOF OF THEOREM II. It is easily seen that the map f: K--+S, f(gj -=g(xo) is open (since f is open on On+i) and consequently that K/Ko is canonically homeomorphic to Sn. Furthermore, the local cross-section of On+1over Sn provides a local cross-section for K over

2 Qn can be imbedded in Ko so that its action is that of On,on the coordinate neighborhood U1. 1960] GAUSS' FIRST PROOF OF THE QUADRATIC RECIPROCITY THEOREM 563

Sn, and K is then a fibre bundle over Sn. Consequently we have the following commutative diagram of exact sequences

* * * -+ Ik(On) -+ Hk(On+1) -- Hk(Sn) -) Hkkl(On) -+ Ik-1(O+) - * *

wherej* is the identity. It is easily seen that Ker(i*: Hk(On+1)-+I1k(K)) is the image under Hk(On))-+Hk(On+l) of Ker(i*: Hk(On)-+Hk(Ko)). Since this last kernel is 0 the theorem follows.

REFERENCES 1. D. Montgomery and L. Zippin, Topologicaltransformation groups, New York, IntersciencePublishers, 1955. 2. N. Steenrod, The topologyof fibre bundles, Princeton, Princeton University Press, 1951.

UNIVERSITY OF NOTRE DAME

A NOTE ON GAUSS' FIRST PROOF OF THE QUADRATIC RECIPROCITY THEOREM

L. CARLITZ We assume that the reader is familiar with Mathews' exposition [1, pp. 45-50] of the inductive proof of the reciprocity theorem. There are three main cases: I. pRq, II. pNq, q-3 (mod 4), III. pNq, q-1 (mod 4). In I we have e2-p=qf, in II we have e2+p=qf. In III we have first the lemma which asserts the existence of a prime p'