SUBMERSIONS AND FOLIATIONS OF TOPOLOGICAL MANIFOLDS
David B . Gauld
(received 25 September, 1970; revised December, 1970)
In this paper a proposed definition for a foliation on a topologi cal manifold is given, and sane results concerning submersions and foliations are proven. See [3] for a treatment of differentiable foliations.
Notation. Rn denotes Euclidean n-space. Rn is included as a subspace of Rn+1 by identifying ( x x ) € Rn with (x.,...,x ,0) € Rn+ . For i n i n each pair of non-negative integers m and n, Yn; Rm -*■ Rn denotes either projection on the first n factors or the natural inclusion (according as m > n or m 5 n). By an m-manifold ^ , we mean a paraccmpact Hausdorff space in which each point has a neighbourhood homecmorphic to Bm , the unit ball of Rm . A local chart at x in M will mean an em bedding of either Bm or r” as a neighbourhood of x in M. 3M, the boundary of M, consists of all points of M which do not have neighbour hoods haneaitorphic to Rn .
The following definition appears in [l]. Definition 1. Let and Nn be manifolds with m > n and suppose 3M = 0. A map Lemma 1. Let (p:Mn -*■ Nn be a submersion. Suppose x,y € M satisfy g-1 Proof: Let f and g be charts with f (0) = x and g”*(pf = ym . ------x x x n Let f and g be charts with f(0) = y and g-1 Math. Chronicle 1^ (1971), 139-146. 139 restricting and g to small cubes centred at 0 if necessary and then observing that these cubes are themselves hcmecmorphs of Rm and Rn , we may assume that g(Rn) c g(Rn). Define the remaining recruired chart f by fy = f[(g_1g) 1 ] x . ■ For further similar results refer to [l], Lemmas 3 and 4. Theorem 2. Let cprM01 -► Nn be a submersion. Then Q = {(x,y) € M * M : cp (x) = cp(y)} is a (2m - n) - submanifold of M x M. ------Proof: Let (x,y) € 0. Let xf ,f y :Rm ->Mand q:Rn -► N be local charts given to us by Lemma 1. Consider the embedding eiR2111 = Rm x Rm -v m x m given by e(Xj,...,x2m) . (f (x,,...,x ), f (x + x^_ ....,X + X , . X , . • . /X )). x 1 m y 1 2m-n+l n 2m m+1 2m-n This gives us a local chart for M * M at (x,y). It remains to shew that m-n /- m. n _ e(R 2 ) = e(R 2 ) fl Q. (i) Let (x x__ ) C R^ n. Then l 2m-n ^ x ' 35:.... xm> = .... V = g (x ,... ,x ) 1 n = g g_1 = (ii) Let (x. ,... ,x,_) € R2™ be such that e (x,,... ,x^) € Q. i 2m l 2m Then (x ,... ,x ) = g ! (xl + X 2m-n+l' * * * ,Xn + X 2m) * 140 Thus x_ = ... = x =0, 2m-n+l 2m i.e. (x ,... ,x ) € R2111 n , so that 1 2m e(R ) n Q c e(R ) . This completes the proof. ■ Definition 2. Let Mm be a manifold, 3M = J3. A foliation F of dimen sion p on M is a collection , Since each cp is a submersion, we have that for each x in U , a a (f>a~ 1 ( Uj = {z € M : Real (z) > U 2 - {z € M : |Real (z)| < 2} U 3 = {z 6 M : Real (z) < - Define maps (a) if z = 2 + re*^, where 9 € [- > set cp, (z) = 6 (b) if Real (z) < 2 and Imag (z) > 0, set (p^z) = — + 2 - Real (z) 141 (c) if Real (z) 5 2 and Imag (z) < 0, set One can readily check that {lK,cp^} gives a foliation of dimension 1 on M, the leaves being as shown in Figure 1. In [5], Phillips gives infinitely many distinct foliations of dimension 2 on the manifold S2 x R. Theorem 3. Let M be a manifold, 3M = 0, and F be a foliation of dimension p on M. Then * Q = { (x ,y) £ m x ft : 3a with Proof: The proof is as for Theorem 2, with f^ and f being chosen to satisfy f (Rm ) U f (Rm ) c u for seme a for which cp (x) = cp (y). x y a a a The only other essential difficulty is in translating (ii) of the above proof to the case at hand, since we would only know that V x (xl ...... V ■ W (X1 + X2m-n+l...... X2m-n> for seme 3 € A. The overlap condition in the definition of a folia tion allows us to replace 6 by a in this equation and then proceed as before. ■ We new look at the relationship between foliations and microbundles, See Milnor [4] for definitions and results of microbundles. Theorem 4. Let yirf11 + 5 m be a microbundle. Then there is a neighbourhood W of i (M) in Q such that v |w is a submersion. Thus tt determines a natural foliation on W , called the foliation by fibres of p. 142 Proof: For each x € M, there are neighbourhoods Ux of x in M and of i(x) in Q and a hcmecmorphism -> V x making the following diagram commute (c.f. Milnor [4,p.54]). u Ux x ? X Let W = . Then W is a neighbourhood of i (M) in Q and is a submanifold of Q. Let y € W and let x * ir(y). We may assune that is hcmecmorphic to Rm , say by a hcmecmorphism satisfying g(0) = x. Then g is a chart for M at x. Define the chart f :Rq -► W by f(z) « h^ (g x 1) [z + (g-1 x l)hx"1(y)], z € R9. Then f(0) = y, so f is a chart at y. Using ccnmutativity of the above diagram, one readily sees that g”1irf = . Thus tt|w is a submersion. ■ Theorem 5. Let M,F and Q be as in Theorem 3. Then t F:M + Q + M is a microbundle, where i(x) = (x,x) and ir(x,y) = x. Proof; Since iri(x) =* x for all x CM, it remains to find for each x in M, neighbourhoods U of x in M and V of i(x) in Q and a hcmecmorphism h:U x r P -► v such that the diagram U U commutes. 143 Suppose given x € M. Choose a such that x € U^, and choose local charts fiF M, g:I^ ^ -► F ^ such that f (0) = x , f(Fm ) c u and ?1 ?1 a - m g 1 Let U = f(Rm ). Then U is a neighbourhood of x in M. Define h:U * rP- q as follows: let (y,(x ,...,x )) be a typical m+i m+p element of U * F? , and let (x,,...,x ) * f *1 (y). Set 1 m h(y» (XmM.mX J ) - (y ,f (x ,... ,x .x + x + x in))« m +1 m + p 1 m-p m+l m—p+l m+p in Then V = h (U x Tp) is a neighbourhood of i(x) and h is a haneanor- phism onto V. With such a choice of h, the diagram obviously commutes. ■ Remark. A microbundle £ :B + e ^ b is called a subraicroburdle of a j microbundle n:B -*■ F -+■ B if there is a neighbourhood E' of i(B) in E satisfying (i) E' c f (ii) i (b) = j (b) V b € B (iii) 7T (e) = p(e) V e € E'. The microbundle t F of Theorem 5 is a submicrobundle of the tangent A P1 microbundle M -► M x M 4 M of H, where A(x) = (x,x) and p ^ X / y ) * x. Definition 3. Let if1 and Nn be manifolds without boundary and let F be a foliationof dimension p on N. Then a map h:F?* -► M, g:Rn -*■ N satisfying: (i) h (0) = x ,. .. _i , m m n (ii) g (iii) for each y £ g (Rn ) , say y = g(T, , *.. ,T ), i n 9-1 Lp(y) = t(£j»...*Cn) € R j 5 =^ for each i 5 n - p}. Note that if p = 0 and m'i n, then the definition reduces to that of a submersion. Theorem 6. Let M01, Nn , F^ and n. Then f pulls back the foliation F to M. Precisely, if F = {U ,cp }, then 144 Proof: Let V - qT1 (0 ) and let i * Define g : R n “P •> R n “P by g « Then g(Rn”p) - - Thus g(Rn P) * 0af(R°). Next, x € M f P 1) - g is injective, for if g(£,) » g(n,,...,n ), * n-p i n—p then This proves that $ is a submersion, a 145 (II) Hie overlap condition is satisfied, for if x € V fl then a $ {j/ w (x)i n v * * Remark. The foliation on M determined by a submersion cp:M -*• N is merely the pull-back by cp of the foliation of N by points. Definition 4. Let M*" and Nn be manifolds without boundary and let T be a foliation of dimension p on N. A bundle map : tM -+• t N is fib rewise transverse to tF if for each x € M, the map $ : n”1 (x) -*■ N given by 4> (xfx ‘) = p Note that df is fibrewise transverse to t F. In [2] there is given a characterisation of maps which are trans verse to a foliation F in terms of bundle maps fibrewise transverse to tF. This result generalises Theorem 2 of [l]. REFERENCES 1. D. Gauld, Mersions of Topological Manifolds, Trans. Amer. Math. Soc. 149 (1970), 539-560. 2. D. Gauld, Foliations on Topological Manifolds, Math. Chronicle, to appear. 3. A. Haefliger, Vari£t£s Feuillet€es, Ann. Scuola Norm. Sup. Pisa (3) 16 (1962), 367-397. 4. J. Milnor, Microbundles I, Topology supp. 1 (1964), 53-80. 5. A. Phillips, Foliations of Open Manifolds, I, Comment. Math. Helv. 43 (1968), 204-211. University of Auckland 146