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

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Typical Topologies on Homeomorphism Groups
M n : Separable Metrizable Connected n-Manifold possibly with boundaryH(M )⌧ : Homeomorphism Group of M with Topology ⌧Hc(M )⌧ : Subgroup of H(M )⌧ of compactly supported homeomorphisms
H(M )⌧[H(M )⌧0Hc(M )⌧[Hc(M )⌧0

G0 : Identity Connected Component of G

Typical Topologies :[1] Whitney Topologyw[2] Uniform Topologyu[3] Compact - Open Topology

co[4] Direct Limit Topology

dirH(M )w

H(M, d)uH(M )coHc(M )dirHu(M, d)u

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

f 2 H(M )⌧C ⇢ M,

" : M ! (0, 1),d : a metric on M

Nd(f, C, ") = {g 2 H(M ) | d(f (x), g(x))  "(x) (8 x 2 C)}A Fundamental Nbd System of f in H(M )⌧[1] Whitney Topology :Nd(f, M, ") (" : M ! (0, 1))

N (f, U) = {g 2 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, ") (" 2 (0, 1))d(f, g) = sup{d(f (x), g(x)) | x 2 M }[3] Compact - Open Topology :Nd(f, K, ") (K : Compact, " 2 (0, 1))

SubBase : hK, U i = {g 2 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 di↵erent in nature & each topology has some defect.
Whitney Topology : Too Strong
Compact - Open Topology : Too Weak
Uniform Topology : depends on the metric d on M

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Whitney Topology(T. Banakh, K. Mine, K. Sakai and T. Yagasaki)

4

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 ) ⇠= (⇤! `2,!

`2 ⇡ `2 ⇥ R1!

`2)

R1 = dir limn Rn) Hc(R)w ⇡ `2 ⇥ R1
(5) n = 2

(M : a noncompact connected surface)(i) (H(M )w , Hc(M )w ) ⇡loc (⇤! `2,(ii) H(M )w0 ⇡ `2 ⇥ R1(6) n

3 : Conjecture.

!

`2)

at idMHc(M )w ⇡loc `2 ⇥ R1

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 ⇡ `15

6[3] T. Yagasaki(1) ⇡ : (M, d) ! (N, %) : Metric covering projectionN : 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. ⇡ : (X, d) ! (Y, %) : a metric covering projection()(i) 9 U : an open cover of Y s.t. 8 U 2 U⇡ 1(U ) = [i Ui : the disjoint union of open subsets of X
each Ui is mapped isometrically onto U by ⇡(ii) 8 y 2 Y⇡ 1(y) : uniformly discrete in X(iii) %(⇡(x), ⇡(x0))  d(x, x0)(x, x0 2 X)Idea :Metric covering projection ⇡ : (M, d) ! (N, %)7Local deformation lemma for uniform embeddings in (M, d)~w
w Lift by ⇡ : (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)((`2)! , (`2)!f ⇥ S1 (Exceptional case)cocco ⇠(ii) (H(M )0 , H (M )0 ) =(`2)! , (`2)!f(Other Cases)(4) n3:

M = Int N :N : Compact Connected n-manifold=) H(M )co : Locally Contractible

Thank you very muchfor your attention !