Diffeomorphism Groups of Non-Compact Manifolds Endowed

Diﬀeomorphism groups of non-compact manifolds endowed with the Whitney C∞-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 §1. Main Results M : Non-compact C∞ n-manifold (without boundary) D(M) : Group of diﬀeomorphisms of M (Whitney C∞-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 ∞ (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

∞ Main Theorem. Dc(M) ≈ an open subset of l2 × R .

∞ 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, ∞ then D0(M) is homeomorphic to N × R .

Corollary 2. M : connected n = dim M ∞ (1) D0(M) ≈ l2 × R if n = 1, 2 or n = 3 and M : orientable and irreducible ∞ (2) D0(M) ≈ D0(N; ∂N) × R if N : Compact Connected C∞ 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 §2. Local top types of ∞-dim top groups [1] Hilbert space case Characterization. [Dobrowolski - Toruńczyk,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 Fréchet spaces in the category of top vector spaces Top classification of LF spaces [P. Mankiewicz 1974] ◦ ∞-dim separable LF spaces (up to homeo.)

(a) l2 (metrizable) (b) R∞ ≡ dir lim {R1 ⊂ R2 ⊂ R3 ⊂ · · · } ≈ ⊡ω R

∞ ω (c) l2 × R ≈ ⊡ l2 4 Box product · Small box product

(Xn, ∗n)(n ∈ ω) ω = {0, 1, 2, ···} ∏ (1) Box product : □ ∈ X = X n ω ∏n n∈ω n ⊂ Box Topology : n Un (Un Xn : Open subset)

(2) Small box product : ⊡n∈ωXn ⊂ □n∈ωXn

Subspace of ﬁnite sequences : (x0, x1, . . . , xk, ∗k+1, ∗k+2,... ) ◦ The countable box / small box product of (X, ∗) are denoted by □ωX and ⊡ω(X, ∗) R∞ ≈ ⊡ω R

∞ ω l2 × R ≈ ⊡ l2 Proposition. ω ω (1) (D(R), Dc(R)) ≈ (□ l2, ⊡ l2) [BY, 2010]

ω ω (2) (D(M), Dc(M)) ≈ (□ l2, ⊡ l2) [BMSY, 2011] locally ω ∞ ◦ Dc(M) ≈ ⊡ l2 ≈ l2 × R locally 5 G : Top group (e : the unit element of G)

Gn (n ∈ ω) : 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] ∞ ∞ G ≈ an open subset of R or l2 ×R if G satisﬁes the following conditions (1) G : non-metrizable ∃ (2) (Gn)n∈ω : a tower of closed subgroups of G such that

(i) G carries the strong topology with respect to (Gn)n∈ω (ii) for each n ∈ ω

(a) Gn is a separable Hilbert manifold

(b) Gn is LTC in Gn+1

(1) G carries the strong topology with respect to (Gn)n∈ω ⇐⇒ for any neighborhood U of e in G , n ∈ ω ∪ n n ··· ⊂ n∈ω U0U1 Un G is a neighborhood of e in G.

⇐⇒ p : ⊡nGn → G : open at (e)n ∈ ⊡nGn ◦ In this case

the topology of G = the direct limit topology of the tower (Gn)n∈ω in the category of top groups. (i.e., G carries the strongest group topology such that

the inclusion maps Gn → G (n ∈ ω) 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) ⇐⇒ ∀ any open cover U of X ∃ 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 ∞-dim Hilbert manifold, then every point of X is a strong Z-point. 8 ∞ Open subspace of l2 × R Triangulation Theorem. [Mine - Sakai, 2008] ∞ (1) ∀ any open subspace X of l2 × R ∞ X ≈ K × l2 × R for a locally ﬁnite simplicial complex K. ∞ (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.

∞ Main Theorem. Dc(M) ≈ an open subset of l2 × R .

∞ 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, ∞ then D0(M) is homeomorphic to N × R . 9 §3. Spaces of embeddings and Bundle Theorem M : a C∞ n-manifold without boundary D(M) (Whitney C∞-topology) (1) K ⊂ M

D(M; K) = {h ∈ D(M): h|K = idK} < D(M) : a closed subgroup

D0(M,K) : the identity connected components of D(M,K) (2) L : a C∞ submanifold of MK ⊂ L ∞ EK(L, M) : the space of C -embeddings f : L → M with f|K = idK. (the compact-open C∞-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) = h|L : a natural restriction map

r0 : D0(M; K) → EK(L, M)0 : the restriction of r E⋆ { | ∈ D } K(L, M) := Im r = h L : h (M,K)

(4) D(M,K) ↷ 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 C∞ n-manifold without boundary K ⊂ L : C∞ n-submanifolds of M s.t .

K, L are closed subsets of M, K ⊂ Int L, clM (L − K) is compact.

Theorem 1. ∀ any closed subset C of M with C ∩ L = ∅

∃ a neighborhood U of the inclusion iL in EK(L, M)

∃ a map s : U → D0(M; K ∪ C) s.t. s(g)|L = g (g ∈ U) and s(iL) = idM .

Theorem 2.

∀ any f ∈ EK(L, M)

∃ an open neighborhood Vf of f in EK(L, M)

∃ a map ηf : Vf → D0(M,K) s.t. ηf (g)f = g (g ∈ Vf ) and ηf (f) = idM . 11 Theorem 2.

∀ any f ∈ EK(L, M)

∃ an open neighborhood Vf of f in EK(L, M)

∃ a map ηf : Vf → D0(M,K) s.t. ηf (g)f = g (g ∈ Vf ) and ηf (f) = idM .

(1) r : D(M; K) → EK(L, M) V ∩ E⋆ ̸ ∅ ⇒ 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) ∀ ∈ 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 ≠ ∅ =⇒ Vf ⊂ Im r0

the map r0 has a section on Vf .

(ii) Im r0 = EK(L, M)0.

(iii) ∀ f ∈ 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 ≠ ∅.

(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 ∞-dim separable Fr´echet manifolds ∴ top l2-manifolds 13 §4. Proof of Main Theorem M : a non-compact (σ-compact) smooth n-manifold without boundary

∞ Main Theorem. Dc(M) ≈ an open subset of l2 × R .

Outline of Proof. (1) ∃ Sequence of compact n-submanifolds of M :

∅= ̸ M0 ⊂ M1 ⊂ M2 ⊂ M3 ⊂ · · · ∪ ⫋ ∈ s.t. Mi Int Mi+1 (i ω), M = i∈ω Mi

Ki := M − Int Mi : an n-submanifold of M (i ∈ ω) (M−1 = ∅)

◦ M ⊃ K1 ⊃ K2 ⊃ K3 ⊃ · · ·

(2) G = Dc(M)

Gi = D(M; Ki)(i ∈ ω) : a tower of closed subgroups of G

p : ⊡i∈ωGi → G : the multiplication map

∞ We apply Criterion to show that G ≈ an open subset of l2 × R 13′

∞ Criterion for l2 × R [Banakh - Repovˇs,2012], [BMRSY 2013] ∞ a top group G ≈ an open subset of l2 × R if G satisﬁes the following conditions (1) G : non-metrizable

(2) ∃ (Gi)i∈ω : a tower of closed subgroups of G such that (i) for each i ∈ ω

(a) Gi is a separable l2-manifold

(b) Gi is LTC in Gi+1 (i.e., Gi+1 → Gi+1/Gi has a local section)

(ii) G carries the strong topology with respect to (Gi)i∈ω 13′′

∞ Criterion for l2 × R [Banakh - Repovˇs,2012], [BMRSY 2013] ∞ G ≈ an open subset of l2 × R if G satisﬁes the following conditions

equiv (1) G : non-metrizable ( ⇐⇒ G : not 1st countable) ∃ (2) (Gi)i∈ω : a tower of closed subgroups of G such that (i) for each i ∈ ω

(a) Gi is a separable l2-manifold

(b) Gi is LTC in Gi+1 (i.e., Gi+1 → Gi+1/Gi has a local section)

(for example, Gi+1/Gi is a separable l2-manifold)

(ii) G carries the strong topology with respect to (Gi)i∈ω ( equiv ) ⇐⇒ p : ⊡i∈ωGi → G is an open map at (e)i

(for example, p has a local section at idM ) 14 G = Dc(M)

Gi = D(M; Ki)(i ∈ ω) : a tower of closed subgroups of G p : ⊡i∈ωGi → G : the multiplication map

Proposition 1. (1) G is not 1st countable (2) For each i ∈ ω

(i) Gi is a separable l2-manifold,

(ii) Gi is LTC in Gi+1,

(iii) Gi+1/Gi is a separable l2-manifold.

(3) p : ⊡i∈ωGi → G admits a local section at idM .

Proof. (1) follows from the non-compactness of M and the diagonal argument (2) follows from Results in §3. (Spaces of embeddings and Bundle Theorem)

◦ clM (M − Ki) = Mi, clM (Ki − Ki+1) : compact (3) follows from Bundle Theorem 1. 15 §5. Proof of Corollary 2 for Main Theorem

Properties of D0(M)

H = D0(M) (the identity connected component of G = Dc(M))

Hi = D0(M; Ki) (the identity connected component of Gi = D(M; Ki)) (i ∈ ω) Proposition 2. (1) H is an open subgroup of G

(Hi)i∈ω is a tower of of closed subgroups of H

◦ Any compact subset of H is included in some Hi.

◦ H and Hi (i ∈ ω) are path-connected and have the homotopy type of CW-complexes.

(2) The group H and the tower (Hi)i∈ω satisfy

the same properties (1) ∼ (3) in Proposition 1 (for G and (Gi)i∈ω). ◦ H = g-lim−→ Hi 15′ Proposition 2. (1) H is an open subgroup of G

(Hi)i∈ω is a tower of of closed subgroups of H

◦ Any compact subset of H is included in some Hi.

◦ H and Hi (i ∈ ω) are path-connected and have the homotopy type of CW-complexes.

(2) The group H and the tower (Hi)i∈ω satisfy

the same properties (1) ∼ (3) in Proposition 1 (for G and (Gi)i∈ω). ◦ ≈ × R∞ H = g-lim−→ Hi H L for some l2-manifold L (3) (i) For any pair of compact spaces (L, K) the inclusion inducies a bijection ∼ −→= ι : lim−→ [L, K; Hi, idM ] [L, K; H, idM ] (ii) For m = 0, 1, ··· , ∞,

if each Hi ⊂ Hi+1 is an m-equivalence, then so is H1 ⊂ H. ∞ ◦ For example, if each Hi is contractible, then so is H and H ≈ l2×R . 16 Corollary 2 ∞ (1) D0(M) ≈ l2 × R if n = 1, 2 or n = 3 and M : orientable and irreducible ∞ (2) D0(M) ≈ D0(N; ∂N) × R if N : Compact Connected C∞ n-manifold with Boundary M is diﬀeomorphic to Int N (Int N ≡ N − ∂N)

Proof. We keep the notations Mi, Ki, Hi (i ∈ ω) and H. (1) (i) Since M is connected, we may assume that for each i ∈ ω

(a) Mi is connected and

(b) each connected component of Ki = M − Int Mi is non-compact.

(ii) The restriction map Hi → D0(Mi, ∂Mi) is a homotopy equivalence.

D0(Mi, ∂Mi) is contractible. [Earle - Eells, Ivanov, Hatcher, et al] ∞ (iii) Each Hi is contractible =⇒ H is contractible ∴ H ≈ l2 × R Proposition 2 (3) 17 (2) (i) Take a collar ∂N × [0, 1] of ∂N = ∂N × {0} in N − × 1 ∈ Mi = N (∂N [0, i+1)) (i ω) ◦ − × 1 Ki = M Int Mi = ∂N (0, i+1]

(ii) The inclusion Hi ⊂ Hi+1 is a homotopy equivalence for each i ∈ ω. (by the property of collars)

∴ H ≃ H1 by Proposition 2 (3)

H1 ≃ D0(M1; ∂M1) ≈ D0(N; ∂N)

(iii) H ≃ D0(N; ∂N) (an l2-manifold) ∞ ∴ H ≈ D0(N; ∂N) × R by Corollary 1 for Main Theorem 18

Claim. p : ⊡i∈ωGi → G admits a local section at idM . Proof. (1) We use the following notations: K ⊂ M

GK = Dc(M,K) and G(K) = Dc(M,M − K) ◦ Gi = GKi = G(Mi). (2) Consider three discrete families of compact n-submanifolds of M :

(Fi)i∈ω : Fi = M2i − Int M2i−1 F = ∪i∈ωFi

(Li)i∈ω : Li = M2i+1 − Int M2i L = ∪i∈ωLi

(Ni)i∈ω : Ni is a small neighborhood of Li N = ∪i∈ωNi

◦ M = F ∪ L, GL = G(F )

(3) The map λF : ⊡i∈ωG(Fi) −→ G(F )

λF (h0, h1, ··· , hm, idM , ··· ) = h0h1h2 ··· hm

◦ λF is an open embedding.

The map λN : ⊡i∈ωG(Ni) −→ G(N) is deﬁned similarly. 19

(4) The map θ : ⊡i∈ωG(Ni) × G(F ) −→ G

θ((gi)i∈ω, h) = λN ((gi)i∈ω)h

The map θ has a local section at idM .

(∵) (i) For each i ∈ ω (applying Bundle Theorem 1 to Ci = M − Int Ni) ∃ V E⋆ an open neighborhood i of the inclusion iLi in (Li,M) ∃ V → | ∈ V a map si : i G(Ni) s.t. si(f) Li = f (f i) and si(iLi) = idM .

The maps si (i ∈ ω) induce the map

s = ⊡i∈ωsi : ⊡i∈ωVi −→ ⊡i∈ωG(Ni): s((fi)i∈ω) = (si(fi))i∈ω. −→ ⊡ E⋆ | (ii) The map rL : G i∈ω (Li,M): rL(g) = (g Li)i∈ω. −1 ◦ The inverse image V = rL (⊡i∈ωVi) is an open nbd of idM in G

(iii) The composition η = λN s rL : V → G(N) −1 ◦ η(g) g ∈ GL = G(F )(g ∈ V)

(iv) The required local section of θ at idM is deﬁned by −1 σ0 : V −→ ⊡i∈ωG(Ni) × G(F ): σ0(g) = (srL(g), η(g) g) 20

(5) The map ϱ = θ(id × λF ): ⊡i∈ωG(Ni) × ⊡i∈ωG(Fi) −→ G

ϱ((gi)i∈ω, (hi)i∈ω) = λN ((gi)i∈ω)λF ((hi)i∈ω)

Since λF is an open embedding, if we replace V by a smaller one, then a local section of ϱ on V is deﬁned by −1 σ = (id × λF )σ0 : U −→ ⊡i∈ωG(Ni) × ⊡i∈ωG(Fi).

(i) For each h ∈ U the image σ(h) = ((fi)i∈ω, (gi)i∈ω) has the properties:

(a) gifj = fjgi (j ≥ i + 1)

(b) h = ϱσ(h) = (f0f1f2 ··· )(g0g1g2 ··· ) = f0g0f1g1f2g2 ··· .

(6) The required local section γ at idM of p : ⊡i∈ωGi → G is deﬁned by ( ) γ : V −→ ⊡i∈ωGi : γ(h) = idM , f0, g0, f1, g1, f2, g2,... . □ This completes the proof of Proposition 1 and Main Theorem.