On the reconstruction of topological spaces from their groups of homeomorphisms
Authors: Rubin Matatyahu 
Published: June 2010

For various classes

K
K

of topological spaces we prove that if

X
1

,

X
2

∈
K

{X_1},{X_2} \in K

and

X
1

,

X
2

{X_1},{X_2}

have isomorphic homeomorphism groups, then

X
1

{X_1}

and

X
2

{X_2}

are homeomorphic. Let

G
G

denote a subgroup of the group of homeomorphisms

H
(
X
)

H(X)

of a topological space

X
X

. A class

K
K

of

⟨
X
,
G
⟩

\langle X,G\rangle

’s is faithful if for every

⟨

X
1

,

G
1

⟩
,
⟨

X
2

,

G
2

⟩
∈
K

\langle {X_1},{G_1}\rangle ,\langle {X_2},{G_2}\rangle \in K

, if

φ
:

G
1

→

G
2

\varphi :{G_1} \to {G_2}

is a group isomorphism, then there is a homeomorphism

h
h

between

X
1

{X_1}

and

X
2

{X_2}

such that for every

g
∈

G
1

φ
(
g
)
=
h
g

h

−
1

g \in {G_1}\;\varphi (g) = hg{h^{ - 1}}

. Theorem 1: The following class is faithful:

{
⟨
X
,
H
(
X
)
⟩

|

(
X

\{ \langle X,H(X)\rangle |(X

is a locally finite-dimensional polyhedron in the metric or coherent topology or

X
X

is a Euclidean manifold with boundary) and for every

x
∈
X

x

x \in X\;x

is an accumulation point of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet g left-parenthesis x right-parenthesis vertical-bar g element-of upper H left-parenthesis upper X right-parenthesis EndSet right-brace union left-brace mathematical left-angle upper X comma upper G mathematical right-angle vertical-bar upper X">

{
g
(
x
)

|

g
∈
H
(
X
)
}
}
∪
{
⟨
X
,
G
⟩

|

X

\{ g(x)|g \in H(X)\} \} \cup \{ \langle X,G\rangle |X

is a differentiable or a

P
L

PL

-manifold and

G
G

contains the group of differentiable or piecewise linear homeomorphisms

}
\}

∪
{
⟨
X
,
H
(
X
)
⟩

|

X

\cup \{ \langle X,H(X)\rangle |X

is a manifold over a normed vector space over an ordered field

}
\}

. This answers a question of Whittaker

[

W

]

[{\text {W}}]

, who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful:

{
⟨
X
,
G
⟩

|

X

\{ \langle X,G\rangle |X

is a locally compact Hausdorff space and for every open

T
⊆
X

T \subseteq X

and

x
∈
T

{
g
(
x
)

|

g
∈
H
(
X
)

x \in T\;\{ g(x)|g \in H(X)

and

g
↾
(
X
−
T
)
=
Id
}

g \upharpoonright (X - T) = \operatorname {Id}\}

is somewhere dense

}
\}

. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful:

{
⟨
X
,
H
(
X
)
⟩

|

\{ \langle X,H(X)\rangle |

(1)

X
X

is a

0
0

-dimensional Hausdorff space; (2) for every

x
∈
X

x \in X

there is a regular open set whose boundary is

{
x
}

\{ x\}

; (3) for every

x
∈
X

x \in X

there are

g

1
,

g
2

∈
G

{g_{1,}}{g_2} \in G

such that

x
≠

g
1

(
x
)
≠

g
2

(
x
)
≠
x

x \ne {g_1}(x) \ne {g_2}(x) \ne x

, and (4) for every nonempty open

V
⊆
X

V \subseteq X

there is

g
∈
H
(
X
)
−
{
Id
}

g \in H(X) - \{ \operatorname {Id}\}

such that

g
↾
(
X
−
V
)
=
Id
}

g \upharpoonright (X - V) = \operatorname {Id}\}

. Note that (2) is satisfied by

0
0

-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from

R

{\mathbf {R}}

. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class

K
T

{K^T}

of trees
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet mathematical left-angle upper A u t left-parenthesis upper T right-parenthesis comma upper T semicolon less-than-or-equal-to comma ring comma upper O p mathematical right-angle vertical-bar upper T element-of upper K Superscript upper T Baseline EndSet">

{
⟨
Aut
⁡
(
T
)
,
T
;
≤
,
∘
,
Op
⟩

|

T
∈

K
T

}

\{ \langle \operatorname {Aut}(T),T; \leq , \circ ,\operatorname {Op}\rangle |T \in {K^T}\}

is first-order interpretable in

{
Aut
⁡
(
T
)

|

T
∈

K
T

}

\{ \operatorname {Aut}(T)|T \in {K^T}\}

.