Diffeomorphism groups of non-compact manifolds endowed with the Whitney C1-topology Taras Banakh (Ivan Franko National University of Lviv, Ukraine) Tatsuhiko Yagasaki (Kyoto Institute of Technology, Japan) \ Foliations and Diffeomorphism Groups 2015" Tambara Seminar House Oct. 27, 2015 1 x1. Main Results M : Non-compact C1 n-manifold (without boundary) D(M) : Group of diffeomorphisms of M (Whitney C1-topology) ◦ D(M) is a topological group. D0(M) : Identity connected component of D(M) Dc(M) ⊂ D(M) : Subgroup of diffeomorphisms with compact support Problem. Global Top Type of Dc(M) and D0(M). Basic Properties [BMSY, 2011] (1) Dc(M) is paracompact, but not metrizable (not first countable). (2) D0(M) ⊂ Dc(M) : an open, normal subgroup 1 (3) Whitney C -topology on Dc(M) = Direct limit top in the category of Top Groups ◦ Direct limit top in the category of Top spaces is not a group topology. (≈ : homeomorphic) 2 1 Main Theorem. Dc(M) ≈ an open subset of l2 × R . 1 Corollary 1. D0(M) ≈ N × R for some l2-manifold N. (i) The topological type of N is uniquely determined by the homotopy type of D0(M). (ii) If D0(M) is homotopy equivalent to an l2-manifold N, 1 then D0(M) is homeomorphic to N × R . Corollary 2. M : connected n = dim M 1 (1) D0(M) ≈ l2 × R if n = 1; 2 or n = 3 and M : orientable and irreducible 1 (2) D0(M) ≈ D0(N; @N) × R if N : Compact Connected C1 n-manifold with Boundary M is diffeomorphic to Int N (Int N ≡ N − @N) ◦ D0(N; @N) is an l2-manifold. ◦ for example, if D0(N; @N) is contractible, then D0(N; @N) ≈ l2 3 x2. Local top types of 1-dim top groups [1] Hilbert space case Characterization. [Dobrowolski - Toru´nczyk,1981] A top group G is a separable Hilbert manifold () G is a Polish ANR. ◦ a Polish space = a separable completely metrizable space ◦ an ANR = an absolute neighborhood retract for metric spaces [2] LF-space case LF spaces = Direct Limits of Fr´echet spaces in the category of top vector spaces Top classification of LF spaces [P. Mankiewicz 1974] ◦ 1-dim separable LF spaces (up to homeo.) (a) l2 (metrizable) (b) R1 ≡ dir lim fR1 ⊂ R2 ⊂ R3 ⊂ · · · g ≈ ⊡! R 1 ! (c) l2 × R ≈ ⊡ l2 4 Box product · Small box product (Xn; ∗n)(n 2 !) ! = f0; 1; 2; · · · g Q (1) Box product : □ 2 X = X n ! Qn n2! n ⊂ Box Topology : n Un (Un Xn : Open subset) (2) Small box product : ⊡n2!Xn ⊂ □n2!Xn Subspace of finite sequences : (x0; x1; : : : ; xk; ∗k+1; ∗k+2;::: ) ◦ The countable box / small box product of (X; ∗) are denoted by □!X and ⊡!(X; ∗) R1 ≈ ⊡! R 1 ! l2 × R ≈ ⊡ l2 Proposition. ! ! (1) (D(R); Dc(R)) ≈ (□ l2; ⊡ l2) [BY, 2010] ! ! (2) (D(M); Dc(M)) ≈ (□ l2; ⊡ l2) [BMSY, 2011] locally ! 1 ◦ Dc(M) ≈ ⊡ l2 ≈ l2 × R locally 5 G : Top group (e : the unit element of G) Gn (n 2 !) : a tower of closed subgroups of G (G0 ⊂ G1 ⊂ G2 ⊂ · · · , G = [nGn) p : ⊡n(Gn; e) −! G : p(x0; x1; : : : ; xk; e; e; : : : ) = x0 x1 ··· xk ◦ p : continuous, surjective Criterion. [Banakh - Repovˇs,2012], [BMRSY 2013] 1 1 G ≈ an open subset of R or l2 ×R if G satisfies the following conditions (1) G : non-metrizable 9 (2) (Gn)n2! : a tower of closed subgroups of G such that (i) G carries the strong topology with respect to (Gn)n2! (ii) for each n 2 ! (a) Gn is a separable Hilbert manifold (b) Gn is LTC in Gn+1 (c) each Z-point of Gn+1=Gn is a strong Z-point. 6 Terminology (1) G carries the strong topology with respect to (Gn)n2! () for any neighborhood U of e in G , n 2 ! S n n ··· ⊂ n2! U0U1 Un G is a neighborhood of e in G. () p : ⊡nGn ! G : open at (e)n 2 ⊡nGn ◦ In this case the topology of G = the direct limit topology of the tower (Gn)n2! in the category of top groups. (i.e., G carries the strongest group topology such that the inclusion maps Gn ! G (n 2 !) are continuous.) (2) A subgroup H of a top group G is called locally topologically complemented (LTC) in G if H is closed in G and the quotient map q : G ! G=H has a local section at some point of G=H ◦ In this case the map q : G ! G=H is a principal H-bundle. 7 Lemma. Suppose G is a top group and K ⊂ H are closed subgroups of G. (i) K is LTC in H and H is LTC in G =) K is LTC in G. (ii) H is LTC in G =) π : G=K ! G=H : a locally trivial bundle with the fiber H=K. (3) X : a top space A ⊂ X : a closed subset A is a Z-set in X ( a strong Z-set in X) () 8 any open cover U of X 9 a continuous map f : X ! X such that f is U-close to idX : X ! X and f(X) \ A = ; ( f(X) \ A = ;) ◦ If X is an 1-dim Hilbert manifold, then every point of X is a strong Z-point. 8 1 Open subspace of l2 × R Triangulation Theorem. [Mine - Sakai, 2008] 1 (1) 8 any open subspace X of l2 × R 1 X ≈ K × l2 × R for a locally finite simplicial complex K. 1 (2) X, Y : open subspaces of l2 × R X ≈ Y () X ' Y (homotopically equivalent) ◦ N = K × l2 : an l2-manifold the top type is determined by its homotopy type. 1 Main Theorem. Dc(M) ≈ an open subset of l2 × R . 1 Corollary 1. D0(M) ≈ N × R for some l2-manifold N. (i) The topological type of N is uniquely determined by the homotopy type of D0(M). (ii) If D0(M) is homotopy equivalent to an l2-manifold N, 1 then D0(M) is homeomorphic to N × R . 9 x3. Spaces of embeddings and Bundle Theorem M : a C1 n-manifold without boundary D(M) (Whitney C1-topology) (1) K ⊂ M D(M; K) = fh 2 D(M): hjK = idKg < D(M) : a closed subgroup D0(M; K) : the identity connected components of D(M; K) (2) L : a C1 submanifold of MK ⊂ L 1 EK(L; M) : the space of C -embeddings f : L ! M with fjK = idK. (the compact-open C1-topology) EK(L; M)0 : the connected component of the inclusion iL in EK(L; M) (3) r : D(M; K) !EK(L; M); r(h) = hjL : a natural restriction map r0 : D0(M; K) !EK(L; M)0 : the restriction of r E? f j 2 D g K(L; M) := Im r = h L : h (M; K) (4) D(M; K) y EK(L; M) : the continuous action by the left composition ◦ r is the orbit map at the inclusion iL : L ⊂ M ◦ E? K(L; M) is the orbit of iL. 10 Bundle Theorem M : a C1 n-manifold without boundary K ⊂ L : C1 n-submanifolds of M s.t . K, L are closed subsets of M, K ⊂ Int L, clM (L − K) is compact. Theorem 1. 8 any closed subset C of M with C \ L = ; 9 a neighborhood U of the inclusion iL in EK(L; M) 9 a map s : U!D0(M; K [ C) s.t. s(g)jL = g (g 2 U) and s(iL) = idM . Theorem 2. 8 any f 2 EK(L; M) 9 an open neighborhood Vf of f in EK(L; M) 9 a map ηf : Vf !D0(M; K) s.t. ηf (g)f = g (g 2 Vf ) and ηf (f) = idM . 11 Theorem 2. 8 any f 2 EK(L; M) 9 an open neighborhood Vf of f in EK(L; M) 9 a map ηf : Vf !D0(M; K) s.t. ηf (g)f = g (g 2 Vf ) and ηf (f) = idM . (1) r : D(M; K) !EK(L; M) V \E? 6 ; )V ⊂ E? (i) f K(L; M) = = f K(L; M) the map r has a section on Vf E? E (ii) K(L; M) is a clopen subset of K(L; M) 8 2 E? (iii) f K(L; M) the map r has a local section at f. (2) r0 : D0(M; K) !EK(L; M)0 (i) Vf \ Im r0 =6 ; =)Vf ⊂ Im r0 the map r0 has a section on Vf . (ii) Im r0 = EK(L; M)0. (iii) 8 f 2 Im r0 the map r0 has a local section at f. 12 Corollary. D !E ? (1) r : (M; K) K(L; M) is a top principal bundle E? E (i) K(L; M) is a clopen subset of K(L; M) (ii) the structure group D(M; L) D D D D ≈ E ? (iii) (M; L) is LTC in (M; K) (M; K)= (M; L) K(L; M) (2) r0 : D0(M; K) !EK(L; M)0 is a top principal bundle (i) the structure group G = D(M; L) \D0(M; K). (ii) G is LTC in D0(M; K) D0(M; K)=G ≈ EK(L; M)0. Proposition. Suppose L − K =6 ;. (i) D(M; K [ (M − L)) ⊃ D0(M; K [ (M − L)) E ⊃ E ? ⊃ E (ii) K(L; M) K(L; M) K(L; M)0 These are 1-dim separable Fr´echet manifolds ) top l2-manifolds 13 x4. Proof of Main Theorem M : a non-compact (σ-compact) smooth n-manifold without boundary 1 Main Theorem.