Whitney, Uniform and Compact-Open topologies
on homeomorphism groups of noncompact manifolds

Tatsuhiko Yagasaki(Kyoto Institute of Technology)“ General Topology Symposium 2012”
Kobe University Dec. 14, 2012

1
Typical Topologies on Homeomorphism Groups
M n : Separable Metrizable Connected n-Manifold possibly with boundaryH(M )τ : Homeomorphism Group of M with Topology τHc(M )τ : Subgroup of H(M )τ of compactly supported homeomorphisms
H(M )τ ⊃ Hc(M )τ
∪∪H(M )τ0 ⊃ Hc(M )τ0G0 : Identity Connected Component of G

Typical Topologies :[1] Whitney TopologywH(M )w[2] Uniform TopologyuH(M, d)u ⊃ Hu(M, d)u[3] Compact - Open Topology

coH(M )co[4] Direct Limit Topology

dirHc(M )dir

2Fundamental Nbd Systems for Topologies τ = w, u, co
Notation.

f ∈ H(M )τ

C ⊂ M,

ε : M → (0, ∞),d : a metric on M

Nd(f, C, ε) = {g ∈ H(M ) | d(f (x), g(x)) ≤ ε(x) (∀ x ∈ C)}
A Fundamental Nbd System of f in H(M )τ
[1] Whitney Topology :Nd(f, M, ε) (ε : M → (0, ∞))N (f, U) = {g ∈ H(M ) | (f, g) ≤ U} (U : an open cover of M )[2] Uniform Topology w.r.t. a metric d on M :the sup-metricNd(f, M, ε) (ε ∈ (0, ∞))d(f, g) = sup{d(f (x), g(x)) | x ∈ M }[3] Compact - Open Topology :Nd(f, K, ε) (K : Compact, ε ∈ (0, ∞))SubBase : hK, U i = {g ∈ H(M ) | g(K) ⊂ U }(K : Compact, U : Open in M )Compact case.

n = dim M

When M is compact, w = u = co andthis topology has the following properties:(1) H(M ) : Separable Completely metrizable, Locally Contractible(2) n = 1, 2 :(i) H(M ) : Top. `2-manifold(ii) Homotopy Type & Topological Type of H(M )0 : Well-Known(3) n ≥ 3 :Conjecture.

H(M ) : Top. `2-manifold

Non - Compact case.
w, u, co : strictly different in nature & each topology has some defect.
Whitney Topology : Too Strong
Compact - Open Topology : Too Weak
Uniform Topology : depends on the metric d on M34
Whitney Topology(T. Banakh, K. Mine, K. Sakai and T. Yagasaki)

M n : Non-Compact Connected n-Manifold possibly with boundary(1) H(M )w : not Locally Connected (not Normal)(2) Hc(M )w : Paracompact (but Not Metrizable)
Locally Contractible(i) K ⊂ Hc(M )w : Compact =⇒ K has Common Compact Support(3) H(M )w0 = Hc(M )w0(4) n = 1 :(H+(R)w , Hc(R)w ) ∼= (ω `2,ωω`2 )R∞ = dir limn Rn`2 ≈ `2 × R∞

∴ Hc(R)w ≈ `2 × R∞
(5) n = 2

(M : a noncompact connected surface)(i) (H(M )w , Hc(M )w ) ≈loc (ω `2,ω`2 )at idM(ii) H(M )w0 ≈ `2 × R∞
(6) n ≥ 3 : Conjecture.

Hc(M )w ≈loc `2 × R∞

Uniform Topology(M n, d) : Non-Compact Connected n-Manifold possibly with boundarywith a metric d[1] A.V. Černavskiı̆ (1969)
M = Int N :N : Compact n-manifold

d = Restriction of some metric on N=⇒ Hu(M, d)u : Locally Contractible[2] K. Mine, K. Sakai, T. Yagasaki and A. Yamashita (2011)
Hu(R)u0 ≈ `∞56[3] T. Yagasaki(1) π : (M, d) → (N, %) : Metric covering projection
N : Compact n-manifold possibly with boundary=⇒ Hu(M, d)u : Locally Contractible
Metric covering projections
C 0 version of Riemannian covering spaces over compact Riem. manifolds
Def. π : (X, d) → (Y, %) : a metric covering projection⇐⇒(i) ∃ U : an open cover of Y s.t. ∀ U ∈ U
π −1(U ) = ∪i Ui : the disjoint union of open subsets of X
each Ui is mapped isometrically onto U by π(ii) ∀ y ∈ Y

π −1(y) : uniformly discrete in X(iii) %(π(x), π(x0)) ≤ d(x, x0)(x, x0 ∈ X)

Idea :Metric covering projection π : (M, d) → (N, %)7Local deformation lemma for uniform embeddings in (M, d)~w
w Lift by π : (M, d) → (N, %) + Arzela-Ascoli theorem
Edwards – Kirby’s Local deformation Lemma for
top. embeddings of compact subsets in N
(2) Hu(Rn)u0 ' ∗(3) N : Compact connected 2-manifold with ∂N 6= ∅M = Int N
d : a metric on M s.t. each end of M is a (bi-Lipschitz) Euclidean end=⇒ Hu(M, d)u0 ' ∗Idea : Local deformation for uniform embeddingsww Similarity transformations on Rn or Euclidean ends
Global deformation for uniform homeomorphisms

8Compact-Open Topology
M n : Non-Compact Connected n-Manifold possibly with boundary(1) H(M )co : Separable Completely Metrizable(2) n = 1 :(3) n = 2 :∼H(R)co0 = `2(M : a noncompact connected surface)(T. Yagasaki)

Exceptional case : M = R2, (Half) Open Annulus, Open Möbius Band`2 × S1 (Exceptional case)co ∼(i) H(M )0 =`2(Other Cases)(ωω1(`),(`)(Exceptional case)2
2 f ×S
cocco ∼(ii) (H(M )0 , H (M )0 ) =(`2)ω , (`2)ωf(Other Cases)(4) n ≥ 3 :M = Int N :N : Compact Connected n-manifold=⇒ H(M )co : Locally Contractible

Thank you very muchfor your attention !