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)

2015 Math. Society of Japan
Annual Meeting, Topology Section
March. 22, 2015, Meiji Univ.

∞§1. Diﬀeomorphism Groups with Whitney C -topology

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M : Non-compact C ∞ n-manifold (without boundary)D(M ) : Diﬀeo Group of M (Whitney C ∞-topology)— Top group

D0(M ) : the identity connected component of D(M )Dc(M ) ⊂ D(M ) : Subgroup of Diﬀeo’s with compact support
Basic Property.

(BMSY, 2011)(1) Dc(M ) : Paracompact

D0(M ) ⊂ Dc(M ) : Open normal subgroup(2) (Mi)i∈N : Compact n-submfds of M s.t. Mi ⊂ IntM Mi+1, M = ∪iMi
G(Mi) := {h ∈ Dc(M ) | supp h ⊂ Mi} < Dc(M ) : Closed subgroup
G(M1) ⊂ G(M2) ⊂ · · ·Dc(M ) = ∪i G(Mi)(i) Dc(M ) = g-lim−→i G(Mi) (Direct limit in Category of Top Groups)(ii) G(Mi) : Top manifold modeled on Infinite-dim separable Fréchet space≈ l2
Problem. Global Top Type of Dc(M ) ?

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LF spaces = Direct Limits of Fréchet spaces in Category of Top vector spaces
Top classification of LF spaces(P. Mankiewicz 1974)

Infinite-dim separable LF spaces (up to homeo.)(1) R∞ ≡ dir lim {R1 ⊂ R2 ⊂ R3 ⊂ · · · } ≈ ⊡ω R(2) l2(3) l2 × R∞ ≈ ⊡ω l2
Box product · Small box productDefinition.

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

Subspace of finite sequences :(x0, x1, . . . , xi, ∗, ∗, . . . )∏Box product
Small box product ⊡ωi (Xi, ∗i)i Xi ,3

Top. Type
1-dim case

(D(R), Dc(R)) ≈ (□ω l2, ⊡ω l2)(BY, 2010)

n-dim case

(D(M ), Dc(M )) ≈ℓ (□ω l2, ⊡ω l2)(BMSY, 2011)* Dc(M ) ≈ℓ ⊡ω l2 ≈ l2 × R∞
Thm. Dc(M ) ≈ Open subset of l2 × R∞
Cor.

Dc(M ) ≈ L × R∞ for some l2-manifold L(BY 2015)( L ≃ Dc(M ) )(1) M : Connected
D0(M ) ≈ l2 × R∞

if n = 1, 2 orn = 3 and M : orientable, irreducible(2) D0(M ) ≈ D0(N ; ∂N ) × R∞
if N : Compact Connected C ∞ n-manifold with Boundary
M = Int N

4§4. Results on Top Groups and Towers of Subgroups
G : Top group(e : the identity element of G)

Gn (n ∈ ω) : Tower of Closed subgroups of G
(G0 ⊂ G1 ⊂ G2 ⊂ · · · , G = ∪nGn)
p : ⊡n(Gn, e) −→ G :p(x0, x1, . . . , xk , e, e, . . . ) = xk · · · x1 x0∗ p : continuous, surjective[1] (BMSY 2011, cf. BMSY [arXiv:0802.0337v1])
p : open =⇒ G = g-lim Gn[2] (2007 - 2008)(i) p : open=⇒

(in Category of Top Groups)(BMSY [arXiv:0802.0337v1])(ii) Gn → Gn/Gn−1 admits a global section sn.
o7n noooooo
ooooos=⊡ s⊡n(Gn/Gn−1)⊡nGnps
≈

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EE"/pG5[3] (2009 - )T. Banakh - D. Repovš — Series of papers

Study of Top LF-manifolds and Direct limit of Uniform spaces
Criterion for open subsets of ℓ2 × R∞ (BMRSY, 2013)(i) G : Non-metrizable(ii) Gn : separable ℓ2-manifold(iii) p : ⊡nGn → G : open(iv) Gn → Gn/Gn−1 has a local section(v) Gn/Gn−1 is an ℓ2-manifold(more generally, each Z-point of Gn/Gn−1 is a strong Z-point).
=⇒ G ≈ Open subset of ℓ2 × R∞

6References.
BMRSY = T. Banakh, K. Mine, D. Repovš, K. Sakai, T. Yagasaki[0] BMSY, arXiv:0802.0337v1 (2008)−→ [2], [3], [4][1] BMS, Classifying homeomorphism groups of infinite graphs, Topology Appl., 157(2009) 108–122.
[2] BMSY, Homeomorphism and diﬀeomorphism groups of non-compact manifolds withthe Whitney topology, Topology Proceedings, 37 (2011) 61 – 93.
[3] BMSY, On homeomorphism groups of non-compact surfaces, endowed with theWhitney topology, Topology Appl., 164 (2014) 170 – 181.
[4] BY, Diﬀeomorphism groups of non-compact manifolds endowed with the Whitney
C ∞ -topology, Topology Appl., 179 (2015), 51 – 61 (online: 23 Oct. 2014).
[5] MS, Open subsets of LF-spaces, Bull. Pol. Acad. Sci. Math., 56 (2008) 25–37.
[6] BR, A topological characterization of LF-spaces, Topology Appl., 159 (2012) 1475 –1488.
[7] BMRSY, Detecting topological groups which are (locally) homeomorphic to LFspaces, Topology Appl., 160 (2013) 2272 – 2284

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Thank you very muchfor your attention !