# Classification of differentiable structures on the non-Hausdorff line with
two origins

Mykola Lysynskyi  Department of Algebra and Topology, Institute of Mathematics
of NAS of Ukraine, Tereshchenkivska str. 3, Kyiv, 01601, Ukraine
<EMAIL_ADDRESS>and  Sergiy Maksymenko  Department of Algebra and
Topology, Institute of Mathematics of NAS of Ukraine, Tereshchenkivska str. 3,
Kyiv, 01601, Ukraine<EMAIL_ADDRESS>
###### Abstract.

We classify differentiable structures on a line $\mathbb{L}$ with two origins
being a non-Hausdorff but $T_{1}$ one-dimensional manifold obtained by
‘‘doubling’’ $0$. For $k\in\mathbb{N}\cup\\{\infty\\}$ let $H$ be the group of
homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of
$h$ to $\mathbb{R}\setminus 0$ is a $\mathcal{C}^{k}$-diffeomorphism. Let also
$D$ be the subgroup of $H$ consisting of $\mathcal{C}^{k}$-diffeomorphisms of
$\mathbb{R}$ also fixing $0$. It is shown that there is a natural bijection
between $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a
$\mathcal{C}^{k}$-diffeomorphism fixing both origins) and double $D$-coset
classes $D\setminus H/D=\\{DhD\mid h\in H\\}$.

Moreover, the set of all $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a
$\mathcal{C}^{k}$-diffeomorphism which may also exchange origins) are in one-
to-one correspondence with the set of double $(D,\pm)$-coset classes
$D\setminus H^{\pm}/D=\\{DhD\cup Dh^{-1}D\mid h\in H\\}$.

In particular, in contrast with the real line, the line with two origins
$\mathbb{L}$ admits uncountably many pair-wise non-diffeomorphic
$\mathcal{C}^{k}$-structures for each $k=1,2,\ldots,\infty$.

###### Key words and phrases:

Diffeomorphism, smooth structure, $1$-manifold, non-Hausdorff space, line with
two origins, double cosets

###### 2000 Mathematics Subject Classification:

58A05, 57R30

##  1\. Introduction

The present paper is devoted to some steps toward study and classification of
smooth structures on non-Hausdorff manifolds of dimension $1$.

It is well known that each Hausdorff manifold $M$ of dimension $\dim M\leq 3$
has a unique smooth structure up to a diffeomorphism. For $\dim M=1$ this is
classical result, and for $\dim M=2,3$ see Munkres [24, Theorem 6.3] and
Whitehead [38, Corollary 1.18]. Also, Stallings [33] shown that
$\mathbb{R}^{n}$ has a unique $\mathcal{C}^{\infty}$ structure for $n\geq 5$,
see also [3, Section 10]. On the other hand, $\mathbb{R}^{4}$ has uncountably
many pair-wise non-diffeomorphic $\mathcal{C}^{\infty}$ structures, see
Freedman [6], Donaldson [2, Theorem 1], Freed and Uhlenbeck [5], Gompf [8] and
Taubes [37]. It is also worth noting a result by Milnor [23] on existence of
distinct smooth structures on $7$-sphere, and also papers by Kervaire & Milnor
[18] on the group of homotopy spheres, and Kervaire [17] on $10$-manifold
which does not admit any differentible structures.

On the other hand, differentiable structures of non-Hausdorff manifolds seem
to be not essentially and systematically studied in the literature, see e.g.
the book by D. Gauld [7] and references therein. For instance, Haefliger and
Reeb [9] proven existence of smooth structures on non-Hausdorff letter $Y$.
Nyikos [25] studied smooth structures on the long line $X$, and shown that it
admits non-diffeomorphic $\mathcal{C}^{\omega}$ (real analytic) structures,
however it is not clear whether there are non-diffeomorphic
$\mathcal{C}^{\infty}$-structures on $X$. The long line and the Prüfer
manifold are also considered in [32, Appendix A]. Applications of non-
Hausdorff manifold to physics are discussed in [10, 14]. Also, in a recent
papers [26, 27] by D. O’Connel the structure of vector bundles over non-
Hausdorff manifolds are also investigated.

In the present paper we will completely classify $\mathcal{C}^{k}$-structures,
$1\leq k\leq\infty$, on another well-known non-Hausdorff one-dimensional
manifold, called line with two origins $\mathbb{L}$, and which can be
described in (at least) the following three equivalent ways, see Figure 1.1:

  1. (1)

Let $R=(\mathbb{R}\times 0)\sqcup(\mathbb{R}\times 1)$ be a disjoint union of
two copies of $\mathbb{R}$. Then $\mathbb{L}$ is a quotient space of $R$
obtained by identifying points $(t,0)$ with $(t,1)$ for $t\neq 0$.

  2. (2)

Let $\tau$ be the standard topology on $\mathbb{R}$. Then
$\mathbb{L}=\mathbb{R}\sqcup\bar{0}$ is a disjoint union of $\mathbb{R}$ with
some point $\bar{0}$ endowed with the following topology:

$\eta=\tau\cup\\{(W\setminus 0)\cup\\{\bar{0}\\}\mid 0\in W\in\tau\\},$

whose elements are elements of $\tau$ and also open neighborhoods of $0$ in
which $0$ is replaced with $\bar{0}$.

  3. (3)

Let $\mathcal{F}$ be the partition of $\mathbb{R}^{2}\setminus 0$ into
connected components of the intersections of $\mathbb{R}^{2}\setminus 0$ with
vertical lines $\\{x=c\\}_{c\in\mathbb{R}}$. Then $\mathbb{L}$ is the space of
leaves $(\mathbb{R}^{2}\setminus 0)/\mathcal{F}$ of that foliation (with the
corresponding quotient topology), so that the leaves corresponding to non-
distinguishable points $0$ and $\bar{0}$ are the open intervals
$0\times(-\infty;0)$ and $0\times(0;+\infty)$.

Figure 1.1. Line with two origins $\mathbb{L}$

It will be convenient to exploit the second model for $\mathbb{L}$. As in
Figure 1.1 denote

$\displaystyle U:=\mathbb{R}=\mathbb{L}\setminus\bar{0},$ $\displaystyle
V:=\mathbb{L}\setminus 0=(\mathbb{R}\setminus 0)\cup\bar{0}.$ (1.1)

######  Lemma 1.0.1.

$\mathbb{L}$ is locally Euclidean, one-dimensional, second countable,
satisfies $T_{1}$ separation axiom, but does not satisfy $T_{2}$.

###### Proof.

This lemma is trivial and we will only explain the non-$T_{2}$ property. Note
that $U$ is an open neighborhood of $0$ which does not contain $\bar{0}$,
while $V$ is an open neighborhood of $0$ which does not contain $0$. On the
other hand, by the definition of topology $\eta$, any pair of neighborhoods of
$0$ and $\bar{0}$ intersects. Thus, $\mathbb{L}$ is $T_{1}$ but not $T_{2}$. ∎

Our main result is an explicit description of

  * •

the set of differentiable $\mathcal{C}^{k}$-structures on $\mathbb{L}$ up to a
$\mathcal{C}^{k}$-diffeomorphism and

  * •

the sets of $\mathcal{C}^{k}$-diffeomorphisms between
$\mathcal{C}^{k}$-structures, see Theorem 1.2.1.

As a consequence we will show that, in contrast with the real line
$\mathbb{R}$, on $\mathbb{L}$ there exists uncountably many pair-wise non-
diffeomorphic structures, see Theorem 1.2.3. The exposition is intended to be
elementary and self-contained and is aimed to be available for a large
audience of readers interesting in these topics.

###  1.1. Notation

For each $k\in\mathbb{N}\cup\\{\infty\\}$ define the following groups:

  * •

$\mathcal{H}(\mathbb{R},0)$ is the group of homeomorphisms of $\mathbb{R}$
fixing $0$;

  * •

$\mathcal{H}^{+}(\mathbb{R},0)$ be its normal subgroup (of index $2$)
consisting of orientation preserving homeomorphisms, and
$\mathcal{H}^{-}(\mathbb{R},0)$ be the subset consisting of homeomorphisms
reversing orientation;

  * •

$\mathcal{W}^{k}(\mathbb{R},0)$ is the group of homeomorphisms $h$ of
$\mathbb{R}$ such that $h(0)=0$ and the restriction $h|_{\mathbb{R}\setminus
0}\colon\mathbb{R}\setminus 0\to\mathbb{R}\setminus 0$ is a
$\mathcal{C}^{k}$-diffeomorphism of $\mathbb{R}\setminus 0$;

  * •

$\mathcal{D}^{k}(\mathbb{R},0)$ is the subgroup of
$\mathcal{W}^{k}(\mathbb{R},0)$ consisting of
$\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{R}$;

  * •

$\mathcal{E}^{k}(\mathbb{R},0)$ is the subgroup of
$\mathcal{D}^{k}(\mathbb{R},0)$ consisting of diffeomorphisms $h$ such that
$h^{(i)}(0)=0$ for $i=2,3,\ldots,k$;

  * •

$\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$ is the group of
$\mathcal{C}^{k}$-diffeomorphisms fixed on some neighborhood of $0$;

  * •

also for $*\in\\{\pm\\}$ and
$\mathcal{X}\in\\{\mathcal{W},\mathcal{D},\mathcal{E}\\}$ put
$\mathcal{X}^{k,*}(\mathbb{R},0):=\mathcal{H}^{*}(\mathbb{R},0)\cap\mathcal{X}^{k}(\mathbb{R},0)$.

Notice that for all $k^{\prime}>k$ we have the following inclusions

$\textstyle{\mathcal{E}^{k}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{k}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{W}^{k}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}^{k^{\prime}}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{D}^{k^{\prime}}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{W}^{k^{\prime}}(\mathbb{R},0)}$

Also, if $k$ is finite, then every $h\in\mathcal{E}^{k}(\mathbb{R},0)$ is
given by the formula $h(x)=\tau x+x^{k}g(x)$ for some $\tau\neq 0$ and a
continuous function $g\colon\mathbb{R}\to\mathbb{R}$ being $\mathcal{C}^{k}$
on $\mathbb{R}\setminus 0$. On the other hand, if $k=\infty$, then $h(x)=\tau
x+g(x)$ for some $\tau\neq 0$ and a $\mathcal{C}^{\infty}$ function being flat
at $0$, i.e. $g^{(i)}(0)=0$ for all $i\in\mathbb{N}$. In particular,
$\mathcal{E}^{1}(\mathbb{R},0)=\mathcal{D}^{1}(\mathbb{R},0)$.

Finally note that
$\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)\subset\mathcal{E}^{k,+}(\mathbb{R},0)$.

######  Example 1.1.1.

For any real numbers $\alpha,\beta\neq 0$ having the same sign, and $s,t>0$
the following homeomorphism

$h\colon\mathbb{R}\to\mathbb{R},\qquad
h(x)=\begin{cases}\alpha(-x)^{s},&x<0,\\\ \beta x^{t},&x\geq 0.\end{cases}$

belongs to $\mathcal{W}^{k}(\mathbb{R},0)$. Evidently,
$h\in\mathcal{D}^{k}(\mathbb{R},0)$ iff $\alpha=\beta$ and $t=s=1$, i.e. when
$h$ is linear, and in that case it belongs even to
$\mathcal{E}^{\infty}(\mathbb{R},0)$.

Consider the following left action of the product
$\mathcal{D}^{k}(\mathbb{R},0)\times\mathcal{D}^{k}(\mathbb{R},0)$ on
$\mathcal{W}^{k}(\mathbb{R},0)$ defined by

$\mu\colon\mathcal{D}^{k}(\mathbb{R},0)\times\mathcal{D}^{k}(\mathbb{R},0)\times\mathcal{W}^{k}(\mathbb{R},0)\to\mathcal{W}^{k}(\mathbb{R},0),\qquad\mu(a,b,h)=a\circ
h\circ b^{-1}.$

The corresponding orbit space of $\mathcal{W}^{k}(\mathbb{R},0)$ is called the
set of $\mathcal{D}^{k}(\mathbb{R},0)$-double cosets and denoted by

$\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)\,/\,\mathcal{D}^{k}(\mathbb{R},0).$

One easily checks, see Lemma 2.3.3 below, that
$g,h\in\mathcal{W}^{k}(\mathbb{R},0)$ belong to the same orbit of the above
action iff
$\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,h\mathcal{D}^{k}(\mathbb{R},0)g^{-1}\neq\varnothing$.

Also, consider a partition of $\mathcal{W}^{k}(\mathbb{R},0)$ into larger
sets. Say that $g,h\in\mathcal{W}^{k}(\mathbb{R},0)$ are
$(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-equivalent, if either $g$ or $g^{-1}$
belong to the orbit of $h$ with respect to the above action $\mu$. One easily
checks that this relation is an equivalence, see Section 2.4. The
corresponding equivalence classes will be called
$(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-double cosets, and the set of all
classes will be denoted by

$\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)^{\pm
1}\,/\,\mathcal{D}^{k}(\mathbb{R},0).$

It can also be defined as the orbit set of a certain natural action of the
wreath product $\mathcal{D}^{k}(\mathbb{R},0)\wr\mathbb{Z}_{2}$ exstending the
action $\mu$, see Section 2.4.

###  1.2. Main results

Let $\mathcal{H}(\mathbb{L})$ be the group of all homeomorphisms of
$\mathbb{L}$. It is easy to show that every homemorphism $h$ of $\mathbb{L}$
either fixes or exchanges points $0$ and $\bar{0}$, see Lemma 2.5.1 below. In
particular, the restriction of $h$ to the complement
$\mathbb{L}\setminus\\{0,\bar{0}\\}=\mathbb{R}\setminus 0$ is also invariant
under $h$. Moreover, $h|_{\mathbb{R}\setminus 0}\colon\mathbb{R}\setminus
0\to\mathbb{R}\setminus 0$ is either strictly increasing or strictlty
decreasing homeomorphism. Therefore we will also say that $h$ preserves
orientation of $\mathbb{L}$ if $h|_{\mathbb{R}\setminus 0}$ is increasing, and
$h$ reverses orientation of $\mathbb{L}$ otherwise.

For $*\in\\{+,-,\mathrm{fix},\mathrm{ex}\\}$ denote by
$\mathcal{H}^{*}(\mathbb{L})$ the subset of $\mathcal{H}(\mathbb{L})$
consisting of homeomorphisms respectively preserving or reversing orientation
of $\mathbb{L}$, or fixing or exchanging $0$ and $\bar{0}$. Also for
$*\in\\{+,-\\}$ and $\bullet\in\\{\mathrm{fix},\mathrm{ex}\\}$ we denote
$\mathcal{H}^{*,\bullet}(\mathbb{L})=\mathcal{H}^{*}(\mathbb{L})\cap\mathcal{H}^{\bullet}(\mathbb{L})$.
Evidently, $\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$ and
$\mathcal{H}^{+}(\mathbb{L})$ are normal subgroups of
$\mathcal{H}(\mathbb{L})$ of index $2$, while their intersection
$\mathcal{H}^{+,\mathrm{fix}}(\mathbb{L})$ has index $4$.

Fix once and for all some $k\in\mathbb{N}\cup\\{\infty\\}$. Given two
$\mathcal{C}^{k}$-structures $\mathfrak{A}$ and $\mathfrak{B}$ on $\mathbb{L}$
let:

  * •

$\mathcal{D}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ be the set of all
$\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{L}$ from $\mathfrak{A}$ to
$\mathfrak{B}$;

  * •

$\mathcal{D}^{*}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})=\mathcal{D}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\cap\mathcal{H}^{*}(\mathbb{L})$,
where $*\in\\{-,+,\mathrm{fix},\mathrm{ex}\\}$;

  * •

$\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})=\mathcal{D}^{*}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\cap\mathcal{D}^{\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})=\mathcal{D}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\cap\mathcal{H}^{*,\bullet}(\mathbb{L})$,
where $*\in\\{-,+\\}$, and $\bullet\in\\{\mathrm{fix},\mathrm{ex}\\}$.

Notice that though we omitted $k$ from notations, it is implicitly determined
by the notations for $\mathcal{C}^{k}$-structures $\mathfrak{A}$ and
$\mathfrak{B}$.

If $\mathfrak{A}=\mathfrak{B}$, then we will abbreviate
$\mathcal{D}^{*}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ and
$\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ to
$\mathcal{D}^{*}_{\mathfrak{A}}(\mathbb{L})$ and
$\mathcal{D}^{*,\bullet}_{\mathfrak{A}}(\mathbb{L})$ respectively. For
instance, $\mathcal{D}^{+}_{\mathfrak{A}}(\mathbb{L})$, resp.
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$, is the group of
$\mathcal{C}^{k}$-diffeomorphisms of $\mathfrak{A}$ preserving orientation,
resp. fixing $0$ and $\bar{0}$, while
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ is their
intersection.

Evidently, we have the following partition

$\displaystyle\mathcal{D}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\ =\
\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\ \sqcup\
\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\ \sqcup\
\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\ \sqcup\
\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L}),$

however depending on $\mathfrak{A}$ and $\mathfrak{B}$ some of
$\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ may be
empty.

The following theorem classifies $\mathcal{C}^{k}$-structures on $\mathbb{L}$.
In fact, we will prove more detailed and explicit statements, describing also
sets of diffeomorphisms between $\mathcal{C}^{k}$-structures fixing or
exchanging $0$ and $\bar{0}$, see Theorem 5.4.1.

######  Theorem 1.2.1.

Let $k\in\mathbb{N}\cup\\{\infty\\}$. Then

  * •

$\mathcal{C}^{k}$-structures on $\mathbb{L}$ up to a
$\mathcal{C}^{k}$-diffeomorphism are in one-to-one correspondence with the set
$\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)^{\pm
1}\,/\,\mathcal{D}^{k}(\mathbb{R},0)$ of
$(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-double coset classes;

  * •

while $\mathcal{C}^{k}$-structures on $\mathbb{L}$ up to a
$\mathcal{C}^{k}$-diffeomorphism fixing $0$ and $\bar{0}$ are in one-to-one
correspondence with the set
$\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)\,/\,\mathcal{D}^{k}(\mathbb{R},0)$
of $\mathcal{D}^{k}(\mathbb{R},0)$-double coset classes.

The following lemma coincides with statements 3) and 4) of Theorem 5.3.2
below.

######  Lemma 1.2.2.

For any $\mathcal{C}^{k}$-structure,
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ contains a normal
subgroup isomorphic to the group $\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$
of diffeomorphisms of $\mathbb{R}$ fixed near $0$. In particular,
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ is always
uncountable.

More generally, let $\mathfrak{A}$ and $\mathfrak{B}$ be
$\mathcal{C}^{k}$-structures on $\mathbb{L}$ and
$*\in\\{\mathrm{fix},\mathrm{ex}\\}$. If
$\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ is non-
empty, then it is uncountable as well.

Our last statement presents examples of $\mathcal{C}^{k}$-structures from
which either of sets
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ and
$\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ is empty
or non-empty.

A $\mathcal{C}^{k}$-atlas $\mathsf{A}$ on $\mathbb{L}$ will be called minimal
if it consisting of only two charts $\varphi\colon U\to\mathbb{R}$ and
$\psi\colon V\to\mathbb{R}$ being homeomorphisms onto and such that
$\varphi(0)=\psi(\bar{0})=0$. We will show in Lemma 5.3.1 that every
$\mathcal{C}^{k}$-structure on $\mathbb{L}$ admits a minimal atlas.

In particular, let $\mathrm{id}_{u}\colon U\to\mathbb{R}$ be the identity map,
and $\mathrm{id}_{v}\colon V\to\mathbb{R}$ be the homeomorphism given by
$\mathrm{id}_{v}=\mathrm{id}_{u}$ on $V\setminus\\{0\\}$ and
$\mathrm{id}_{v}(\bar{0})=0$. Then for each
$g\in\mathcal{W}^{k}(\mathbb{R},0)$ one can define the following minimal atlas
$\mathsf{S}(g)=\\{(U,\mathrm{id}_{u}),(V,g\circ\mathrm{id}_{v})\\}$. Such an
atlas will be called special minimal of $g$, see Section 5.5. Evidently, its
transition map is

$g\circ\mathrm{id}_{v}\circ\mathrm{id}_{u}^{-1}|_{\mathbb{R}\setminus
0}=g|_{\mathbb{R}\setminus 0},$

whence $\mathsf{S}(g)$ is $\mathcal{C}^{k}$.

Further, for each $a>0$ define the following homeomorphism
$w_{a}\in\mathcal{W}^{k}(\mathbb{R},0)$ by

$w_{a}(x)=\begin{cases}x,&x\leq 0,\\\ ax,&x>0,\end{cases}$ (1.2)

see Figure 1.2.

Figure 1.2. Homeomorphism $w_{a}$

It is a particular case of homeomorphism $h$ from Example 1.1.1 for
$\alpha=1$, $\beta=a$, $s=t=1$. Let $\mathfrak{S}(a)$ be the corresponding
$\mathcal{C}^{k}$-structure defined by the minimal atlas $\mathsf{S}(w_{a})$.

######  Theorem 1.2.3.

  1. (1)

Suppose $a=1$, so $w_{a}=\mathrm{id}_{\mathbb{R}}$. Then there is an
isomorphism

$\eta\colon\mathcal{D}_{\mathfrak{S}(1)}(\mathbb{L})\cong\mathcal{D}^{k}(\mathbb{R},0)\times\mathbb{Z}_{2},$

such that for $*\in\\{\pm\\}$

$\displaystyle\eta\bigl{(}\mathcal{D}^{*,\mathrm{fix}}_{\mathfrak{S}(1)}(\mathbb{L})\bigr{)}=\mathcal{D}^{k,*}(\mathbb{R},0)\times\\{1\\},$
$\displaystyle\eta\bigl{(}\mathcal{D}^{*,\mathrm{ex}}_{\mathfrak{S}(1)}(\mathbb{L})\bigr{)}=\mathcal{D}^{k,*}(\mathbb{R},0)\times\\{-1\\}.$
(1.3)

  2. (2)

Let $a\neq 1$. Then
$\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L}){}=\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}=\varnothing$,
and
$\mathcal{D}_{\mathfrak{S}(a)}(\mathbb{L})=\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}\sqcup\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L}){}$,
so every self-diffeomorphism of $\mathfrak{S}(a)$ either fixes $0$ and
$\bar{0}$ and preserves orientation, or exchanges $0$ and $\bar{0}$ and
reverses orientation. Moreover, there exists an isomorphism

$\eta\colon\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}\cong\mathcal{E}^{k,+}(\mathbb{R},0),$

and an element
$p\in\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L}){}$ of order
$2$. Hence, $\mathcal{D}_{\mathfrak{S}(a)}(\mathbb{L})$ splits into a
semidirect product
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}\rtimes\langle
p\rangle$, and thus it is isomorphic to a certain111Notice that
$\mathcal{E}^{k,-}(\mathbb{R},0)$ also contains the diffeomorphism
$q\colon\mathbb{R}\to\mathbb{R}$, $q(x)=-x$, of order $2$, and therefore
$\mathcal{E}^{k}(\mathbb{R},0)$ also splits into the semidirect product
$\mathcal{E}^{k,+}(\mathbb{R},0)\rtimes\langle q\rangle$. However, it is not
clear whether there is an isomorphism between
$\mathcal{D}_{\mathfrak{S}(a)}(\mathbb{L})\cong\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}\rtimes\langle
p\rangle$ and
$\mathcal{E}^{k}(\mathbb{R},0)\cong\mathcal{E}^{k,+}(\mathbb{R},0)\rtimes\langle
q\rangle$ extending $\eta$. semidirect product
$\mathcal{E}^{k,+}(\mathbb{R},0)\rtimes\mathbb{Z}_{2}$.

  3. (3)

Also, for each $a\neq 1$ the structures $\mathfrak{S}(a)$ and
$\mathfrak{S}(1/a)$ are $\mathcal{C}^{k}$-diffeomorphic. In fact,
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\varnothing$
and

$\mathcal{D}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})\sqcup\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L}),$

so every $\mathcal{C}^{k}$-diffeomorphism from $\mathfrak{S}(a)$ to
$\mathfrak{S}(1/a)$ either fixes $0$ and $\bar{0}$ and reverses orientation,
or exchanges $0$ and $\bar{0}$ and preserves orientation. Moreover, for every
$h\in\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})$
and
$g\in\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})$
we have the following bijections:

$\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})\xrightarrow{\leavevmode\nobreak\
\leavevmode\nobreak\ k\,\mapsto\,g^{-1}\circ k\leavevmode\nobreak\
\leavevmode\nobreak\
}\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L})\xrightarrow{\leavevmode\nobreak\
\leavevmode\nobreak\ k\,\mapsto\,h\circ k\leavevmode\nobreak\
\leavevmode\nobreak\
}\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L}).$

  4. (4)

Finally, if $a\neq b,1/b$, then
$\mathcal{D}_{\mathfrak{S}(a),\mathfrak{S}(b)}(\mathbb{L})=\varnothing$, i.e.
these $\mathcal{C}^{k}$-structures are not diffeomorphic. In particular,
$\mathbb{L}$ has uncountably many pair-wise non-diffeomorphic
$\mathcal{C}^{k}$-structures of each $k\in\mathbb{N}\cup\infty$.

######  Remark 1.2.4.

Notice that we have a surjective homomorphism
$j^{1}\colon\mathcal{E}^{k,+}(\mathbb{R},0)\to\mathbb{R}\setminus 0$,
$j^{1}(h)=h^{\prime}(0)$. Let $G_{k}$ be its kernel. Then $G_{k}$ consists of
diffeomorphisms $k$-tangent to the identity and was studied in [34]. Moreover,
the group $G_{\infty}$ plays an important role in the theory of foliations,
see [35, 31, 16].

### Structure of the paper

Section 2 is preliminary: it contains definitions of double cosets and branch
points of non-Hausdorff spaces. In Section 3 we discuss differentiable
structures on locally Euclidean spaces not assuming them to be manifolds in a
usual sense, i.e. Hausdorff and second countable. In Section 4 we present a
proof of a well-known result that any two $\mathcal{C}^{k}$-structures, $1\leq
k\leq\infty$, on the real line are diffeomorphic, see Corollary 4.4. In fact
we need a more explicit statement claiming that if $M$ is a one-dimensional
locally Euclidean space endowed with some $\mathcal{C}^{k}$-structure and
$U\subset M$ is an open subset homeomorphic with $\mathbb{R}$, then there
exists a $\mathcal{C}^{k}$-compatible chart $\varphi\colon U\to\mathbb{R}$
being a homeomorphism onto, see Corollary 4.4.1. In Section 5 we classify
$\mathcal{C}^{k}$-structures on $\mathbb{L}$, and, in particular, prove
Theorem 1.2.1, see Theorem 5.4.1. Finally in Section 6 we prove Theorem 1.2.3.

##  2\. Preliminaries

Throughout the paper the arrows $\lhook\joinrel\rightarrow$ and
$\rightarrow\\!\\!\\!\\!\\!\to$ mean monomorphism and epimorphism
respectively. We will also regard the cyclic group $\mathbb{Z}_{2}$ of order
$2$ as the multiplicative group $\\{\pm 1\\}$.

###  2.1. Semidirect products

Let $H$ be a group with a unit $e$ and $A$ be some subgroup of $H$. A
homomorphism $s\colon H\to A$ is called retraction if $s(a)=a$ for all $a\in
A$.

Suppose $s\colon H\to A$ is a homomorphism-retraction, and let $K$ be its
kernel. Then $A$ acts on $K$ by conjugations, and this allows to define the
following group $K\rtimes_{s}A$ called the semidirect product of $K$ and $A$
corresponding to the retraction $s$. Namely, $K\rtimes_{s}A$ is a cartesian
product of sets $K\times A$ with the following multiplication:

$(k,a)(l,b):=(kala^{-1},ab),\qquad(k,a),(l,b)\in K\times A.$

Moreover, the map $\gamma\colon K\rtimes_{s}A\to H$, $\gamma(k,a)=ka$, is an
isomorphism of groups making commutative the following diagram:

$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k\,\mapsto\,(k,e)}$$\textstyle{K\rtimes_{s}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(k,a)\,\mapsto\,a}$$\scriptstyle{\gamma}$$\scriptstyle{\cong}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{A}$

We will exploit this construction several times.

###  2.2. Wreath product $D\wr\mathbb{Z}_{2}$

Let $D$ be a group and

$\phi\colon D\times D\to D\times D,\qquad\phi(a,b)=(b,a),$

be the automorphism of exchanging coordinates. It has the order $2$, and
therefore defines a monomorphism
$\bar{\phi}\colon\mathbb{Z}_{2}\to\mathrm{Aut}(D\times D)$, given by
$\bar{\phi}(1)=\mathrm{id}_{D\times D}$ and $\bar{\phi}(-1)=\phi$.

The corresponding semidirect product $(D\times D)\rtimes\mathbb{Z}_{2}$ is
called the wreath product of $D$ and $\mathbb{Z}_{2}$ and denoted by
$D\wr\mathbb{Z}_{2}$, see e.g. [22]. Thus, $D\wr\mathbb{Z}_{2}$ is the direct
product of sets $D\times D\times\mathbb{Z}_{2}$ with the following binary
operation:

$(a,b,\delta)(c,d,\varepsilon)=\begin{cases}(ac,bd,\delta\varepsilon),&\delta=+1,\\\
(ad,bc,\delta\varepsilon),&\delta=-1.\end{cases}$

###  2.3. Double cosets $C\setminus H\,/\,D$

We recall here the definition of double cosets, see e.g. [11, Section 1.7].
Let $H$ be a group and $C$, $D$ be two subgroups. Then we have a natural left
action of the product $C\times D$ on $H$:

$C\times D\times H\to H,\qquad(c,d)\cdot h=chd^{-1}.$

The corresponding orbit of $h\in H$ is called the $(C,D)$-double coset of $h$
and is denoted by $ChD$. If $C=D$, then $DhD$ is also called the $D$-double
coset of $h$.

Notice that

$ChD=\\{chd^{-1}\mid c\in C,d\in D\\}=\\{chd\mid c\in C,d\in
D\\}=\mathop{\cup}\limits_{c\in C}chD=\mathop{\cup}\limits_{d\in D}Chd.$

In particular, it is the union of left cosets of $D$ corresponding to elements
of the right coset $Ch$, as well as the union of right cosets of $C$
corresponding to elements of the left coset $hD$. The partition of $H$ into
$(C,D)$-double cosets is denoted by

$C\setminus H\,/\,D.$

Partitions into double cosets are not so ‘‘regular’’ as partitions into left
or right cosets, and they are often hard to compute, see e.g. [30, 13, 12,
20], and references therein. For instance, the double coset classes may have
distinct sizes. The reason is that the above action of $C\times D$ on $H$ is
not always free.

######  Example 2.3.1.

Let $\mathbb{D}_{3}=\langle r,s\mid r^{3}=s^{2}=(rs)^{2}=1\rangle$ be the
dihedral group consisting of six elements $e,r,r^{2},s,sr,sr^{2}$. Consider
two subgroups $A=\\{e,s\\}$ and $B=\\{e,sr\\}$ of $\mathbb{D}_{3}$. One easily
checks that

$\displaystyle A\setminus\mathbb{D}_{3}\,/\,B=\bigl{\\{}\\{e,s,r,sr\\},\
\\{r^{2},sr^{2}\\}\bigr{\\}},$ $\displaystyle
A\setminus\mathbb{D}_{3}\,/\,A=A\setminus\mathbb{D}_{3}^{\pm
1}\,/\,A=\bigl{\\{}\\{e,s\\},\\{r,r^{2},sr,sr^{2}\\}\bigr{\\}}.$

The following lemma is trivial. We will refer to it several times, and
therefore it will be convenient to put it as a separate statement:

######  Lemma 2.3.2.

If $C=D$ and this group is normal in $H$, then $C\setminus H\,/\,D$ coincides
with the partition $H/D$ into (no matter left or right) cosets of $D$ in $H$.
∎

Let us mention one more description of $C\setminus H\,/\,D$ which will be
useful in this paper.

######  Lemma 2.3.3.

Let $g,h\in H$. Then the following conditions are equivalent:

  1. (1)

$g\in ChD$, which is the same as $CgD=ChD$;

  2. (2)

$C\cap gDh^{-1}\neq\varnothing$.

###### Proof.

(1)$\Rightarrow$(2) Suppose $g\in ChD$, so $g=chd^{-1}$ for some $c\in C$ and
$d\in D$. Then $c=gdh^{-1}\in C\cap gDh^{-1}\neq\varnothing$.

(2)$\Rightarrow$(1) Conversely, if $C\cap gDh^{-1}$ contains some element $c$,
i.e. $c=gdh^{-1}$ for some $d\in D$, then $g=chd^{-1}\in ChD$. ∎

###  2.4. Double cosets $D\setminus H^{\pm 1}\,/\,D$

Suppose $C=D$. Then for each $h\in H$, we have that $(DhD)^{-1}=Dh^{-1}D$ is
the $D$-double coset of $h^{-1}$, see e.g. the discussion in [4, §3]. The
following set

$Dh^{\pm 1}D:=DhD\cup Dh^{-1}D=\\{chd,ch^{-1}d\mid c,d\in D\\}$

will be called $(D,\pm)$-double coset of $h$. Thus, if $DhD$ mutually contains
some $g\in H$ and its inverse, then $Dh^{\pm 1}D=DhD=Dh^{-1}D$. Otherwise,
$Dh^{\pm 1}D=DhD\sqcup Dh^{-1}D$.

It follows that $(D,\pm)$-double cosets yield a partition on $H$ which will be
denoted by

$D\setminus H^{\pm 1}\,/\,D.$

One can give another description of $D\setminus H^{\pm 1}\,/\,D$. Note that we
have a natural left action

$(D\wr\mathbb{Z}_{2})\times H\to H,\qquad(a,b,\delta)\cdot
h:=ah^{\delta}b^{-1}.$

Indeed, if $a,b,c,d\in D$ and $h\in H$, then

$\displaystyle(a,b,1)\bigl{(}(c,d,1)\cdot h\bigr{)}$
$\displaystyle=a(chd^{-1})b^{-1}=(ac,bd,1)\cdot
h=\bigl{(}(a,b,1)(c,d,1)\bigr{)}\cdot h,$
$\displaystyle(a,b,-1)\bigl{(}(c,d,1)\cdot h\bigr{)}$
$\displaystyle=a(chd^{-1})^{-1}b^{-1}=(ad,bc,-1)\cdot
h=\bigl{(}(a,b,-1)(c,d,1)\bigr{)}\cdot h,$
$\displaystyle(a,b,1)\bigl{(}(c,d,-1)\cdot h\bigr{)}$
$\displaystyle=ach^{-1}d^{-1}b^{-1}=(ac,bd,-1)\cdot
h=\bigl{(}(a,b,1)(c,d,-1)\bigr{)}\cdot h,$
$\displaystyle(a,b,-1)\bigl{(}(c,d,-1)\cdot h\bigr{)}$
$\displaystyle=a(ch^{-1}d^{-1})^{-1}b^{-1}=(ad,bc,1)\cdot
h=\bigl{(}(a,b,-1)(c,d,-1)\bigr{)}\cdot h.$

It is easy to see that $D\setminus H^{\pm 1}\,/\,D$ coincides with the
partition into orbits of that action of $D\wr\mathbb{Z}_{2}$.

###  2.5. Branch points

Let $M$ be a topological space with topology $\eta$, and $x\in M$ be a point.
Denote by $\mathsf{hcl}(x):=\mathop{\cap}\limits_{x\in W\in\eta}\overline{W}$
the intersection of closures of all open neighborhoods of $x$, we will call it
the Hausdorff closure of $x$. Evidently, $x\in\mathsf{hcl}(x)$. Also note that
$\mathsf{hcl}(x)$ consists of $y\in M$ such that $x$ and $y$ have no disjoint
neighborhoods.

Say that $x$ is a branch point of $M$. It is well known and is easy to see
that $M$ is Hausdorff iff $\\{x\\}=\mathsf{hcl}(x)$ for all $x\in M$, i.e. $M$
has no branch points. There are not standard terms for such points. For
instance in [27] they are called Hausdorff-violating.

The following simple lemma is left for the reader.

######  Lemma 2.5.1.

Let $h\colon M\to M$ be a homeomorphism of $M$. Then
$h(\mathsf{hcl}(x))=\mathsf{hcl}(h(x))$ for all $x\in M$. In particular, the
set of branch points of $M$ is invariant under homeomorphisms.

##  3\. Differentiable structures on locally euclidean spaces

This section briefly recalls definition of differentiable structures on
locally Euclidean spaces. We refer the reader to the book by J. Lee [21,
Chapters 1,2], and also to the paper of F. Takens [36] for an equivalent
approach.

Usually a manifold is a Hausdorff second countable locally Euclidean
topological space. The first two conditions are necessary and sufficient for
metrizability of manifolds and for the possibility to embed them into some
Euclidean spaces. However, for defining differentiable structures only the
property of being locally Euclidean is enough. Therefore we will assume here
only that property. For simplicity we also do not consider manifolds with
boundary.

For each $m\geq 0$ the real $m$-dimensional linear space $\mathbb{R}^{m}$ with
the usual topology is called Euclidean. Note that $\mathbb{R}^{0}=\\{0\\}$ is
a one-point space being a $0$-dimensional linear space. Also recall that a
homeomorphism $h\colon U\to V$ between open subset of $\mathbb{R}^{m}$ is
called a $\mathcal{C}^{k}$-diffeomorphism for some
$k\in\mathbb{N}\cup\\{\infty\\}$, if both $h$ and $h^{-1}$ are
$\mathcal{C}^{k}$-maps.

###  3.1. Locally Euclidean spaces

Let $M$ be a topological space. A chart on $M$ is an open embedding (i.e. a
homeomorphism onto open subset) $\varphi\colon U\to\mathbb{R}^{m}$ of some
open subset $U\subset M$ for some $m\geq 0$. In this case we will say that
this chart is $m$-dimensional and $U$ is its domain. It will often be
convenient to regard such a chart as a pair $\mathsf{u}=(U,\varphi)$.

Notice that then for every open subset $U^{\prime}\subset U$ the restriction
$\varphi|_{U^{\prime}}\colon U^{\prime}\to\mathbb{R}^{m}$ is also an
$m$-dimensional chart on $M$.

Any collection $\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ of charts
on $M$ will be called a partial atlas. If their domains cover $M$, i.e.
$M=\mathop{\cup}\limits_{i\in\Lambda}U_{i}$, then $\mathsf{A}$ is called an
atlas. Evidently, the union of any collection of (partial) atlases is a
(partial) atlas. It will also be convenient to regard each chart $\mathsf{u}$
as a partial atlas $\\{\mathsf{u}\\}$ consisting of that chart only. This will
never lead to confusion.

Say that $M$ is locally Euclidean if either of the following equivalent
conditions holds:

  1. (LE1)

every point $x\in M$ has an open neighborhood homeomorphic to some open subset
of some Euclidean space (whose dimension may depend on $x$);

  2. (LE2)

$M$ admits an atlas.

###  3.2. Differentiable structures

Let $\mathsf{u}=(U,\varphi)$ and $\mathsf{v}=(V,\psi)$ be two overlapping
charts on $M$ with values in $\mathbb{R}^{m}$, i.e. $U\cap V\neq\varnothing$.
Then the following well-defined homeomorphism between open subsets of
$\mathbb{R}^{m}$

$\psi\circ\varphi^{-1}\colon\varphi(U\cap V)\to\psi(U\cap V)$ (3.1)

is called the transition map from $\mathsf{u}$ to $\mathsf{v}$. Note that its
inverse, $\varphi\circ\psi^{-1}\colon\psi(U\cap V)\to\varphi(U\cap V)$, is
then the transition map from $\mathsf{v}$ to $\mathsf{u}$.

In what follows fix once and for all some $k\in\mathbb{N}\cup\\{\infty\\}$.
Then the charts $\mathsf{u}$ and $\mathsf{v}$ are $\mathcal{C}^{k}$-compatible
if either $U\cap V=\varnothing$ or the transition map $\psi\circ\varphi^{-1}$
is a $\mathcal{C}^{k}$-diffeomorphism.

A partial atlas $\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ on $M$ is
$\mathcal{C}^{k}$ if all its charts are pair-wise
$\mathcal{C}^{k}$-compatible.

Further, two partial $\mathcal{C}^{k}$-atlases $\mathsf{A}$ and $\mathsf{B}$
are $\mathcal{C}^{k}$-compatible if their union $\mathsf{A}\cup\mathsf{B}$ is
still a partial $\mathcal{C}^{k}$-atlas, i.e. the charts from $\mathsf{A}$ and
$\mathsf{B}$ are pair-wise $\mathcal{C}^{k}$-compatible as well.

Notice that the relation ‘‘to be $\mathcal{C}^{k}$-compatible’’ on the set of
all partial atlases on $M$ is not an equivalence: it is reflexive and
symmetric, but not transitive. However, it is easy to show that it is
transitive on the set of all atlases:

######  Lemma 3.2.1.

The relation ‘‘to be $\mathcal{C}^{k}$-compatible’’ on the set of all atlases
on $M$ is an equivalence relation. In particular, the union of any collection
of pair-wise $\mathcal{C}^{k}$-compatible $\mathcal{C}^{k}$-atlases is still a
$\mathcal{C}^{k}$-atlas.

The corresponding equivalence classes of pair-wise
$\mathcal{C}^{k}$-compatible atlases on $M$ are called
$\mathcal{C}^{k}$-structures. It will be convenient to say that an atlas
$\mathsf{A}$ on $M$ represents a $\mathcal{C}^{k}$-structure $\mathfrak{A}$ if
$\mathsf{A}\in\mathfrak{A}$. Also, a partial atlas $\mathsf{A}$ on $M$, e.g. a
chart, is $\mathcal{C}^{k}$-compatible with $\mathfrak{A}$ if it is
$\mathcal{C}^{k}$-compatible with some atlas from $\mathfrak{A}$.

Note also that taking the union of all atlases from the same
$\mathcal{C}^{k}$-structure $\mathfrak{A}$, we will get a maximal atlas of
$\mathfrak{A}$, in the sense that it contains any other
$\mathcal{C}^{k}$-compatible atlas. Hence, each $\mathcal{C}^{k}$-structure is
represented by a unique maximal atlas. However, such a description is not very
useful from the ‘‘practical’’ point of view, since the maximal
$\mathcal{C}^{k}$-atlases are too large to be ‘‘observable’’. In many cases it
is more convenient to work with atlases having as small number of charts as
possible. See also Lemma 3.4.1 below for another characterization of
differentiable structures.

Moreover, the number of $\mathcal{C}^{k}$-structures on $M$ is still too
large, and in order to get some reasonable classification one should consider
those structures up to a homeomorphism being $\mathcal{C}^{k}$-diffeomorphism
between them.

###  3.3. Orientations

The transition map (3.1) preserves orientation if the determinant of the
Jacobi matrix of $\psi\circ\varphi^{-1}$ is positive at each point
$x\in\varphi(U\cap V)$. A $\mathcal{C}^{k}$-atlas
$\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ on $M$ is called
orientable if all its transition maps preserve orientation. A
$\mathcal{C}^{k}$-structure on $M$ is orientable, if it contains an orientable
atlas.

###  3.4. Differentiable maps between locally Euclidean spaces

Till the end of this section we assume that $M$ and $N$ are locally Euclidean
spaces endowed with some $\mathcal{C}^{k}$-structures $\mathfrak{A}$ and
$\mathfrak{B}$ respectively.

Let $h\colon M\to N$ be a continuous map and $\mathsf{u}=(U,\varphi)$ and
$\mathsf{v}=(V,\psi)$ be charts in $M$ and $N$ respectively such that
$h(U)\subset V$. Then the induced map between open subsets of Euclidean
spaces:

$\psi\circ h\circ\varphi^{-1}\colon\varphi(U)\to\psi(V)$ (3.2)

is called a (local) presentation of $h$ in the charts $\mathsf{u}$ and
$\mathsf{v}$.

Say that $h$ is $\mathcal{C}^{k}$ (with respect to $\mathfrak{A}$ and
$\mathfrak{B}$), if for each $x\in M$ there exist local charts
$\mathsf{u}=(U,\varphi)$ on $M$ and $\mathsf{v}=(V,\psi)$ on $N$ such that

  * •

$x\in U$ and $\mathsf{u}$ is $\mathcal{C}^{k}$-compatible with $\mathfrak{A}$;

  * •

$h(x)\in V$ and $\mathsf{v}$ is $\mathcal{C}^{k}$-compatible with
$\mathfrak{B}$;

  * •

$h(U)\subset V$ and the presentation $\psi\circ
h\circ\varphi^{-1}\colon\varphi(U)\to\psi(V)$ of $h$ in $\mathsf{u}$ and
$\mathsf{v}$ is $\mathcal{C}^{k}$ at the point $\varphi(x)$.

In this case we will also say use the notation
$h\colon(M,\mathfrak{A})\to(N,\mathfrak{B})$ to indicate that $h$ is a
$\mathcal{C}^{k}$-map with respect to $\mathfrak{A}$ and $\mathfrak{B}$, and
denote by $\mathcal{C}^{k}_{\mathfrak{A},\mathfrak{B}}(M,N)$ the set of all
such maps.

A homeomorphism $h\colon M\to N$ is a $\mathcal{C}^{k}$-diffeomorphism (with
respect to $\mathfrak{A}$ and $\mathfrak{B}$) or between $(M,\mathfrak{A})$
and $(N,\mathfrak{B})$, if
$h\in\mathcal{C}^{k}_{\mathfrak{A},\mathfrak{B}}(M,N)$ and
$h^{-1}\in\mathcal{C}^{k}_{\mathfrak{B},\mathfrak{A}}(N,M)$.

The set of all $\mathcal{C}^{k}$-diffeomorphisms
$h\colon(M,\mathfrak{A})\to(N,\mathfrak{B})$ will be denoted by
$\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(M,N)$. In particular, if $M=N$,
and $\mathfrak{A}=\mathfrak{B}$, then
$\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{A}}(M,M)$ will also be abbreviated to
$\mathcal{D}^{k}_{\mathfrak{A}}(M)$ or even to $\mathcal{D}^{k}(M)$ if the
corresponding $\mathcal{C}^{k}$-structure is assumed from the context.

The following easy lemma gives another characterization of
$\mathcal{C}^{k}$-structures.

######  Lemma 3.4.1.

Two $\mathcal{C}^{k}$-structures $\mathfrak{A}$ and $\mathfrak{B}$ on $M$
coincide iff
$\mathrm{id}_{M}\in\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(M,M)$, i.e. the
identity map of $M$ is a $\mathcal{C}^{k}$-diffeomorphism with respect to
them.

######  Example 3.4.2.

Let $U\subset\mathbb{R}^{m}$ be an open subset. Then $U$ admits an atlas
$\mathsf{Id}_{U}:=\\{(U,\mathrm{id}_{U})\\}$ consisting of a unique chart
being just the inclusion map $\mathrm{id}_{U}\colon U\subset\mathbb{R}^{m}$.
This atlas is $\mathcal{C}^{k}$ for all $k\in\mathbb{N}\cup\\{\infty\\}$. The
corresponding $\mathcal{C}^{k}$-structure on $U$ will be called canonical and
denoted by $\mathcal{I}_{U}$.

Also, if $V\subset\mathbb{R}^{n}$ is another open subset, then a map $h\colon
U\to V$ belongs to $\mathcal{C}^{k}_{\mathcal{I}_{U},\mathcal{I}_{V}}(U,V)$
iff $h$ is $\mathcal{C}^{k}$ as a map between open subsets of Euclidean spaces
in the usual sense.

######  Example 3.4.3.

Let $\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ be a
$\mathcal{C}^{k}$-atlas on $M$ representing a $\mathcal{C}^{k}$-structure
$\mathfrak{A}$, and $X\subset M$ be an open subset. Then the following
statements hold.

  1. (1)

$X$ is locally Euclidean as well;

  2. (2)

$\mathsf{A}|_{X}:=\\{(X\cap U_{i},\varphi_{i}|_{X\cap U_{i}})\mid X\cap
U_{i}\neq\varnothing,i\in\Lambda\\}$ is a $\mathcal{C}^{k}$-atlas on $X$;

  3. (3)

Denote by $\mathfrak{A}|_{X}$ the $\mathcal{C}^{k}$-structure on $X$
represented by $\mathsf{A}|_{X}$. Then the inclusion map $i\colon X\subset M$
is $\mathcal{C}^{k}$ with respect to $\mathfrak{A}|_{X}$ and $\mathfrak{A}$.

Moreover, let $\mathsf{v}=(V,\psi)$ be any $m$-dimensional chart
$\mathcal{C}^{k}$-compatible with $\mathfrak{A}$. Thus $\psi(V)$ is an open
subset of $\mathbb{R}^{m}$, and so it has a canonical
$\mathcal{C}^{k}$-structure $\mathcal{I}_{\psi(V)}$ given by the atlas
$\mathsf{Id}_{\psi(V)}$. Then the corresponding homeomorphism $\psi\colon
V\to\psi(V)$ is a $\mathcal{C}^{k}$-difeomorphism with respect to the
$\mathcal{C}^{k}$-structures $\mathfrak{A}|_{V}$ and $\mathcal{I}_{\psi(V)}$.

######  Lemma 3.4.4.

Let $h\colon M\to N$ be a homeomorphism, $0\leq k\leq\infty$, and
$\mathsf{u}=(U,\varphi)$ and $\mathsf{v}=(V,\psi)$ be charts on $M$. Then the
following statements hold.

  1. (1)

$h^{*}(\mathsf{u}):=(h(U),\varphi\circ h^{-1})$ is a chart on $M$ with the
same image $\varphi(U)$, and the presentation of $h$ in the charts
$\mathsf{u}$ and $h^{*}(\mathsf{u})$ is the identity map of $\varphi(U)$:

$(\varphi\circ h^{-1})\circ h\circ\varphi^{-1}=\mathrm{id}_{\varphi(U)};$

  2. (2)

The transition maps from $\mathsf{u}$ to $\mathsf{v}$ and from
$h^{*}(\mathsf{u})$ to $h^{*}(\mathsf{v})$ coincide, i.e.

$(\psi\circ h^{-1})\circ(\varphi\circ h^{-1})^{-1}=\psi\circ\varphi^{-1};$

  3. (3)

Let $\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ be a
$\mathcal{C}^{k}$-atlas on $M$ representing a $\mathcal{C}^{k}$-structure
$\mathfrak{A}$. Then

$h^{*}\mathsf{A}=\\{(h(U_{i}),\varphi_{i}\circ h^{-1})\\}_{i\in\Lambda}$

is an atlas on $N$ with the same transition maps between the charts with the
corresponding indices. In particular, the following statements hold.

    1. (a)

$h^{*}\mathsf{A}$ is of class $\mathcal{C}^{k}$ as well, and we will denote
its $\mathcal{C}^{k}$-structure by $h^{*}\mathfrak{A}$;

    2. (b)

$\mathsf{A}$ is a maximal $\mathcal{C}^{k}$-atlas iff so is $h^{*}\mathsf{A}$;

    3. (c)

If $k\geq 1$, then $h$ is a $\mathcal{C}^{k}$-diffeomorphism with respect to
$\mathfrak{A}$ and $h^{*}\mathfrak{A}$, i.e.
$h\in\mathcal{D}^{k}_{\mathfrak{A},h^{*}\mathfrak{A}}(M,M)$.

Thus, due to Lemma 3.4.4(3)(3b), for each $k\in\mathbb{N}\cup\\{\infty\\}$ the
group $\mathcal{H}(M)$ of homeomorphisms of $M$ naturally acts on the set of
$\mathcal{C}^{k}$-structures, and these homeomorphisms are diffeomorphisms
between the corresponding $\mathcal{C}^{k}$-structures. Therefore the right
formulation of the problem of classification of $\mathcal{C}^{k}$-structures
on $M$ reduces to the following one:

######  Problem 3.5.

Describe the orbits of the action of the group $\mathcal{H}(M)$ on the set of
$\mathcal{C}^{k}$-structures on $M$.

In particular, the statement that each manifold $M$ of dimension $\leq 3$ has
a unique (up to a diffeomorphism) $\mathcal{C}^{k}$-structure means that the
above action is transitive. In next section we will present an elementary
proof of that fact for $\dim M=1$. Such a statement is usually referred as a
classical result, though the authors does not find its proof in the available
literature. In fact, we prove some technical statement which will be useful
for studying non-Hausdorff manifolds.

Let us mention few simple statements which will be useful lated in the paper
for the study of differentiable structures on non-Hausdorff $1$-manifolds.

######  Lemma 3.5.1.

Let $X$ be an open subset of $\mathbb{R}^{n}$ for some $n\geq 0$, and
$\mathfrak{A}$ and $\mathfrak{B}$ be two $\mathcal{C}^{k}$-structures on $X$
represented by atlases $\\{(X,\varphi)\\}\in\mathfrak{A}$ and
$\\{(X,\psi)\\}\in\mathfrak{B}$ consisting of single charts and such that
$\varphi(X)=\psi(X)\subset\mathbb{R}^{n}$. Then the homeomorphism

$\psi^{-1}\circ\varphi\colon
X\xrightarrow{\varphi}\varphi(X)=\psi(X)\xrightarrow{\psi^{-1}}X$

is a $\mathcal{C}^{k}$-diffeomorphism between $(X,\mathfrak{A})$ and
$(X,\mathfrak{B})$, i.e.
$\psi^{-1}\circ\varphi\in\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(X,X)$.

###### Proof.

Consider the following commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{-1}\circ\varphi}$$\scriptstyle{\varphi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\varphi(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\textstyle{\psi(X)}$

It means that the local presentation of $\psi^{-1}\circ\varphi$ in the charts
$\mathsf{u}$ and $\mathsf{v}$ is the identity map
$\mathrm{id}\colon\varphi(X)=\psi(X)$ which is a diffeomorphism of class
$\mathcal{C}^{k}$. Hence $\psi^{-1}\circ\varphi$ is a
$\mathcal{C}^{k}$-diffeomorphism. ∎

######  Lemma 3.5.2.

Let $M$ and $N$ be two locally Euclidean spaces endowed with
$\mathcal{C}^{k}$-structres $\mathfrak{A}$ and $\mathfrak{B}$ respectively.
Suppose there exists a $\mathcal{C}^{k}$-diffeomorphism
$h\in\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(M,N)$. Then we have the
following commutative diagram of bijections:

$\textstyle{\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(M,N)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
k\,\mapsto\,k\circ
h^{-1}}$$\textstyle{\mathcal{D}^{k}_{\mathfrak{A}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\,\mapsto\,h\circ
g\ }$$\scriptstyle{g\,\mapsto\,h\circ g\circ
h^{-1}}$$\textstyle{\mathcal{D}^{k}_{\mathfrak{B}}(N)}$

in which the lower horizontal arrow is an isomorphism of groups. In
particular, if the groups $\mathcal{D}^{k}_{\mathfrak{A}}(M)$ and
$\mathcal{D}^{k}_{\mathfrak{B}}(N)$ are not isomorphic, then
$(M,\mathfrak{A})$ and $(N,\mathfrak{B})$ are not
$\mathcal{C}^{k}$-diffeomorphic.

The inverse statement, i.e. that every isomorphism
$\alpha\colon\mathcal{D}^{k}_{\mathfrak{A}}(M)\to\mathcal{D}^{k}_{\mathfrak{B}}(N)$
is induced by some diffeomorphism between $M$ and $N$, is called a
reconstrucion theorem. It holds for Hausdorff second countable manifolds $M$
and $N$, see [39, 28, 19, 29, 1] for precise statements and references
therein.

##  4\. Uniqueness of $\mathcal{C}^{k}$-structures on $\mathbb{R}$ up to a
diffeomorphism

In this section $M$ is a one-dimensional locally Euclidean space and
$k\in\mathbb{N}\cup\\{\infty\\}$. We will describe a way of joining charts on
$M$. This technique is mentioned in [15].

######  Definition 4.1.

Let $\mathsf{u}=(U,\varphi)$ and $\mathsf{v}=(V,\psi)$ be two charts on $M$.
Say that $\mathsf{u}$ and $\mathsf{v}$ are $\mathcal{C}^{k}$-joinable if they
are $\mathcal{C}^{k}$-compatible and

$\displaystyle\varphi(U)=(a;c),$ $\displaystyle\varphi(U\cap V)=\psi(U\cap
V)=(b;c),$ $\displaystyle\psi(V)=(b;d)$

for some $a<b<c<d\in\mathbb{R}$.

In this case every open embedding $\sigma\colon U\cup V\to\mathbb{R}$,
regarded as a chart $\mathsf{w}=(U\cup V,\sigma)$ on $M$, will be called a
$\mathcal{C}^{k}$-join of $\mathsf{u}$ and $\mathsf{v}$ if, see Figure 4.1,

  1. (1)

$\sigma(U)=\varphi(U)=(a;c)$ and $\sigma(V)=\psi(V)=(b;d)$;

  2. (2)

$\mathsf{w}$ is $\mathcal{C}^{k}$-compatible with $\mathsf{u}$ and
$\mathsf{v}$;

  3. (3)

$\sigma|_{U\setminus V}=\varphi|_{U\setminus V}\colon U\setminus V\to(a;c]$
and $\sigma|_{V\setminus U}=\psi|_{V\setminus U}\colon V\setminus U\to[b;d)$.

Figure 4.1. $\mathcal{C}^{k}$-joinable charts

The first condition implies that $\sigma(U\cap V)=(c;b)$, while the latter
condition can be formulated so that $\sigma$ differs from $\varphi$ and $\psi$
only on the intersection $U\cap V$, or equivalently, that the transition maps
$\sigma\circ\varphi^{-1}$ and $\sigma\circ\psi^{-1}$ are identity on
$U\setminus V$ and $V\setminus U$ respectively.

Our aim is to establish the following theorem

######  Theorem 4.2.

  1. (1)

Every pair of $\mathcal{C}^{k}$-joinable charts $\mathsf{u}$ and $\mathsf{v}$
has a $\mathcal{C}^{k}$-join.

  2. (2)

Let $\mathsf{A}$ be a $\mathcal{C}^{k}$-atlas on $M$ having a pair of
$\mathcal{C}^{k}$-joinable charts $\mathsf{u}$ and $\mathsf{v}$ such that for
every other chart $\mathsf{y}=(Y,\xi)$ from $\mathsf{A}$ we have that

$Y\cap U\cap V=\varnothing.$

Then for any $\mathcal{C}^{k}$-join $\mathsf{w}$ of $\mathsf{u}$ and
$\mathsf{v}$ the following collection of charts

$\mathsf{B}=\bigl{(}\mathsf{A}\setminus\\{\mathsf{u},\mathsf{v}\\}\bigr{)}\cup\\{\mathsf{w}\\},$

obtained by replacing $\mathsf{u}$ and $\mathsf{v}$ with $\mathsf{w}$, is
still a $\mathcal{C}^{k}$-atlas on $M$ being also $\mathcal{C}^{k}$-compatible
with $\mathsf{A}$.

###### Proof.

First we establish the following simple lemma.

######  Lemma 4.2.1.

Let $b<c\in\mathbb{R}$ and $g\colon(b;c)\to(b;c)$ be a
$\mathcal{C}^{k}$-diffeomorphism preserving orientation. Then there exist
$\varepsilon>0$ and a diffeomorphism $p\colon(b;c)\to(b;c)$ such that

$\displaystyle p(x)$ $\displaystyle=x\ \text{for}\ x\in(b;b+\varepsilon],$
$\displaystyle p(x)$ $\displaystyle=g(x)\ \text{for}\ x\in[c-\varepsilon;c).$

###### Proof.

It suffices to find $\varepsilon>0$ and a strictly positive function
$\gamma\colon[b;c)\to(0;+\infty)$ such that

$\displaystyle\gamma(x)=1\ \text{for}\ x\in[b;b+\varepsilon],$
$\displaystyle\gamma(x)=g^{\prime}(x)\ \text{for}\ x\in[c-\varepsilon;c),$
$\displaystyle\smallint_{b}^{c}\gamma(s)ds=c-b,$

and then put $p(x)=b+\int_{b}^{x}\gamma(s)ds$.

As $g$ is a preserving orientation homeomorphism of $(b;c)$, we have that

$b=\lim\limits_{x\to b+0}g(x)<\lim\limits_{x\to c-0}g(x)=c.$

Then one can easily find $\varepsilon>0$ and two $\mathcal{C}^{\infty}$
functions $\alpha,\beta\colon[b;c]\to[0;+\infty)$ such that

$\displaystyle\alpha=1\ \text{on}\ [b;b+\varepsilon],\qquad\alpha=0\
\text{on}\ [b+2\varepsilon;c-2\varepsilon],\qquad\alpha=g^{\prime}\ \text{on}\
[c-\varepsilon;c],$ $\displaystyle
A:=c-b-\smallint_{b}^{c}\alpha(s)ds>0,\qquad\beta^{-1}(0)=[b;b+\varepsilon]\cup[c-\varepsilon;c].$

Denote $B=\smallint_{b}^{c}\beta(s)ds$. Then the function
$\gamma=\alpha+(A/B)\beta$ has the above properties. ∎

Now we can prove Theorem 4.2.

(1) Let $\mathsf{u}=(U,\varphi)$ and $\mathsf{v}=(V,\psi)$ be two
$\mathcal{C}^{k}$-joinable charts on $M$. Not loosing generality, assume that

$\displaystyle\varphi(U)=(0;2),$ $\displaystyle\varphi(U\cap V)=\psi(U\cap
V)=(1;2),$ $\displaystyle\psi(V)=(1;3).$

Denote $W=U\cup V$. We need to construct a $\mathcal{C}^{k}$-join
$\mathsf{w}=(W,\sigma)$ for these charts.

Let $g=\psi\circ\varphi^{-1}\colon(1;2)\to(1;2)$ be the transition map from
$\mathsf{u}$ to $\mathsf{v}$. Then by Lemma 4.2.1 one can find $\varepsilon>0$
and a $C^{k}$-diffeomorphism $p\colon(0;2)\to(0;2)$ such that

$\displaystyle p(x)$ $\displaystyle=x\ \text{for}\ x\in(0;1+\varepsilon),$
(4.1) $\displaystyle p(x)$ $\displaystyle=g(x)\ \text{for}\
x\in(2-\varepsilon;2).$ (4.2)

Consider the homeomorphism $\bar{\varphi}:=p\circ\varphi\colon U\to(0;2)$
regarded as a chart $\bar{\mathsf{u}}=(U,\bar{\varphi})$. Then by (4.1),
$\bar{\varphi}=\varphi$ on $(0;1+\varepsilon)$, whence

$\bar{\varphi}(U\cap V)=\varphi(U\cap V)=\psi(U\cap V)=(1;2).$

Moreover, by (4.2), the transition map from $\mathsf{v}$ to
$\bar{\mathsf{u}}$, being a $\mathcal{C}^{k}$-diffeomorphism

$q:=\bar{\varphi}\circ\psi^{-1}=p\circ\varphi\circ\psi^{-1}=p\circ
g^{-1}\colon(1;2)\to(1;2),$

is fixed on $(2-\varepsilon^{\prime};2)$ for a sufficiently small
$\varepsilon^{\prime}>0$. Hence $q$ extends by the identity on $(2;3)$ to a
self-diffeomorphism of $(1;3)$, which we will denote by the same letter $q$.

Since $\bar{\varphi}|_{U\cap V}=q\circ\psi|_{U\cap V}$, we get a wel-defined
homeomorphism

$\sigma:W=U\cup V\to(0;3),\qquad\sigma(x)=\begin{cases}\bar{\varphi}(x),&x\in
U,\\\ q\circ\psi(x),&x\in V,\end{cases}$

i.e. the chart $\mathsf{w}=(W,\sigma)$.

Note that $p$ and $q$ are the transition maps from $\mathsf{u}$ and
$\mathsf{v}$ to $\mathsf{w}$ respectively:

$\displaystyle\sigma\circ\varphi^{-1}$
$\displaystyle=\bar{\varphi}\circ\varphi^{-1}=p\circ\varphi\circ\varphi^{-1}=p\colon(0;2)\to(0;2),$
$\displaystyle\sigma\circ\psi^{-1}$
$\displaystyle=q\circ\psi\circ\psi^{-1}=q\colon(1;3)\to(1;3).$

Since they are $\mathcal{C}^{k}$-diffeomorphisms, $p$ is fixed on $U\setminus
V$, and $q$ is fixed on $V\setminus U$, we see that $\mathsf{w}$ is a
$\mathcal{C}^{k}$-join of $\mathsf{u}$ and $\mathsf{v}$.

(2) Let $\mathsf{u}=(U,\varphi)$ and $\mathsf{v}=(V,\psi)$ be two
$\mathcal{C}^{k}$-joinable charts in the atlas $\mathsf{A}$ such that for
every other chart $\mathsf{y}=(Y,\xi)$ from $\mathsf{A}$ we have that $Y\cap
U\cap V=\varnothing$. Let also $\mathsf{w}$ be any $\mathcal{C}^{k}$-join of
$\mathsf{u}$ and $\mathsf{v}$. We need to prove that

$\mathsf{B}=\bigl{(}\mathsf{A}\setminus\\{\mathsf{u},\mathsf{v}\\}\bigr{)}\cup\\{\mathsf{w}\\},$

is a $\mathcal{C}^{k}$-atlas on $M$ being also $\mathcal{C}^{k}$-compatible
with $\mathsf{A}$.

Since $W=U\cup V$, the domains of charts of $\mathsf{B}$ cover $M$.

Let us show that $\mathsf{w}$ is $\mathcal{C}^{k}$-compatible with all other
charts $\mathsf{y}=(Y,\xi)\in\mathsf{B}$ distinct from $\mathsf{w}$, i.e. the
transition map $\sigma\circ\xi^{-1}\colon\xi(Y\cap W)\to\sigma(Y\cap W)$ is a
$\mathcal{C}^{k}$-diffeomorphism.

Indeed, since $Y\cap U\cap V=\varnothing$, we have that $Y\cap W$ is a union
of disjoint open subsets $Y\cap U$ and $Y\cap V$, i.e. $Y\cap W=(Y\cap
U)\sqcup(Y\cap V)$. Since $\sigma$ coincides with $\varphi$ on $U\setminus
V\supset Y\cap U$ and with $\psi$ on $V\setminus U\supset Y\cap V$, we see
that

$\displaystyle\sigma\circ\xi^{-1}|_{\xi(Y\cap
U)}=\varphi\circ\xi^{-1}|_{\xi(Y\cap U)},$
$\displaystyle\sigma\circ\xi^{-1}|_{\xi(Y\cap
V)}=\psi\circ\xi^{-1}|_{\xi(Y\cap V)}.$ (4.3)

But $\mathsf{A}$ is a $\mathcal{C}^{k}$-atlas, i.e. $\mathsf{y}$ is
$\mathcal{C}^{k}$-compatible with $\mathsf{u}$ and $\mathsf{v}$, whence both
restrictions (4.3) are $\mathcal{C}^{k}$, and therefore $\sigma\circ\xi^{-1}$
is $\mathcal{C}^{k}$ as well. ∎

###  4.3. Chain-like atlases

Let $\mathsf{B}=\\{\mathsf{v}_{j}=(V_{j},\psi_{j})\\}_{j\in\mathbb{Z}}$ be a
countable $\mathcal{C}^{k}$-atlas on the real line $\mathbb{R}$. Say that
$\mathsf{B}$ is chain-like if for each $j\in\mathbb{Z}$ the following
conditions hold:

  1. (G1)

the charts $\mathsf{v}_{j}$ and $\mathsf{v}_{j+1}$ are
$\mathcal{C}^{k}$-joinable;

  2. (G2)

$V_{i}\cap V_{j}\cap V_{j+1}=\varnothing$ for all $i\neq j,j+1$.

Equivalently, suppose that $\psi_{j}(V_{j})=(a_{j};b_{j})$ for some
$a_{j}<b_{j}\in\mathbb{R}\cup\\{\pm\infty\\}$, $j\in\mathbb{Z}$. Then
$\mathsf{B}$ is chain-like if for every $j\in\mathbb{Z}$ we have that

  * •

all $a_{j},b_{j}$ are finite,

  * •

$a_{j}<b_{j-1}<a_{j+1}<b_{j}$,

  * •

$\psi_{j}(V_{j}\cap V_{j+1})=\psi_{j+1}(V_{j}\cap V_{j+1})=(a_{j+1};b_{j})$.

Notice that for any pairs of strictly increasing two-sided sequences
$\\{x_{i}\\}_{i\in\mathbb{Z}}$ and $\\{y_{i}\\}_{i\in\mathbb{Z}}$ in
$\mathbb{R}$ there exists a $\mathcal{C}^{\infty}$-diffeomorphism
$q\colon\mathbb{R}\to\mathbb{R}$ such that $q(x_{i})=y_{i}$ for
$i\in\mathbb{Z}$. Hence one can always assume that for such a chain-like atlas
we have that

$\displaystyle\lim_{j\to-\infty}a_{j}=\lim_{j\to-\infty}b_{j}=-\infty,$
$\displaystyle\lim_{j\to+\infty}a_{j}=\lim_{j\to+\infty}b_{j}=+\infty,$ (4.4)

which is equivalent to the assumption that

  1. (G3)

$\mathop{\cup}\limits_{j\in\mathbb{Z}}\psi_{j}(V_{j})=\mathbb{R}$.

######  Theorem 4.3.1.

  1. (1)

For every $\mathcal{C}^{k}$-atlas
$\mathsf{A}=\\{(U_{i},\varphi_{i})\\}_{i\in\Lambda}$ on $\mathbb{R}$ there
exists a chain-like $\mathcal{C}^{k}$-atlas $\mathsf{B}$ being
$\mathcal{C}^{k}$-compatible with $\mathsf{B}$.

  2. (2)

Let $\mathsf{B}$ be a chain-like $\mathcal{C}^{k}$-atlas on $\mathbb{R}$. Then
there exists a homeomorphism $\sigma\colon\mathbb{R}\to\mathbb{R}$ such that
the chart $(\mathbb{R},\sigma)$ is $\mathcal{C}^{k}$-compatible with
$\mathsf{B}$.

###### Proof.

(1) Since $\mathbb{R}$ is one-dimensional and paracompact, and the domains of
$\mathsf{A}$ constitute an open cover of $\mathbb{R}$, one can find a
countable $\\{V_{j}\\}_{j\in\mathbb{Z}}$ cover of $\mathbb{R}$ such that for
every $j\in\mathbb{Z}$

  1. (a)

$V_{j}=(a_{j};b_{j})$ for some $a_{j}<b_{j}\in\mathbb{R}$ such that
$a_{j}<b_{j-1}<a_{j+1}<b_{j}$, i.e. $V_{j}$ intersects only $V_{j-1}$ and
$V_{j+1}$ and those intersections are proper subsets of $V_{j}$, so in
particular,

$V_{i}\cap V_{j}\cap V_{j+1}=\varnothing,\qquad i\neq j,j+1;$ (4.5)

  2. (b)

there exists a chart $\mathsf{u}_{j}=(U_{j},\varphi_{j})\in\mathsf{A}$ with
$V_{j}\subset U_{j}$;

Note that for any $\mathcal{C}^{k}$-embdding
$p_{j}\colon\varphi_{j}(V_{j})\to\mathbb{R}$, the chart
$\mathsf{v}_{j}:=(V_{j},\psi_{j}:=p_{j}\circ\varphi_{j}|_{V_{j}})$ is still
$\mathcal{C}^{k}$-compatible with $\mathsf{A}$. In particular, one can choose
$p_{j}$ so that

$\displaystyle\psi_{j}(V_{j})=(a_{j};b_{j}),$
$\displaystyle\psi_{j}(V_{j-1}\cap V_{j})=(a_{j};b_{j-1})$
$\displaystyle\psi_{j}(V_{j}\cap V_{j+1})=(a_{j+1};b_{j}),$ (4.6)

which will imply that $\mathsf{v}_{j}$ and $\mathsf{v}_{j+1}$ are
$\mathcal{C}^{k}$-compatible for all $j\in\mathbb{Z}$.

Indeed, note $\varphi_{j}(V_{j})=(c_{j};d_{j})$ for some
$c_{j}<d_{j}\in\mathbb{R}$. Take $p_{j}\colon(c_{j};d_{j})\to(a_{j};b_{j})$ to
be any $\mathcal{C}^{k}$-diffeomorphism such that
$p_{j}\circ\varphi_{j}(b_{j-1})=b_{j-1}$ and
$p_{j}\circ\varphi_{j}(a_{j+1})=a_{j+1}$. Then the composite map

$\psi_{j}:=p_{j}\circ\varphi_{j}|_{V_{j}}\colon
V_{j}=(a_{j};b_{j})\xrightarrow{\leavevmode\nobreak\
\varphi_{j}\leavevmode\nobreak\
}(c_{j};d_{j})\xrightarrow{\leavevmode\nobreak\ p_{j}\leavevmode\nobreak\
}(a_{j};b_{j})\subset\mathbb{R}$

has the desired property (4.6).

Then (4.6) and (4.5) mean respectively conditions (G1) and (G2), so
$\mathsf{B}:=\\{(V_{j},\psi_{j})\\}_{j\in\mathbb{Z}}$ is a chain-like
$\mathcal{C}^{k}$-atlas on $\mathbb{R}$.

(2) Suppose now that $\mathbb{R}$ has a chain-like atlas
$\mathsf{B}=\\{\mathsf{v}_{j}=(V_{j},\psi_{j})\\}_{j\in\mathbb{Z}}$. As noted
above, one can assume that condition (4.4) is also satisfied.

Then by Theorem 4.2 there exists a $\mathcal{C}^{k}$-join
$\mathsf{w}_{0,1}=(V_{0}\cup V_{1},\sigma_{0,1})$ of $\mathsf{v}_{0}$ and
$\mathsf{v}_{1}$, such that the collection of charts

$\mathsf{B}_{0,1}:=\bigl{(}\mathsf{B}\setminus\\{\mathsf{v}_{0},\mathsf{v}_{1}\\}\bigr{)}\cup\\{\mathsf{w}_{0,1}\\}$

is a $\mathcal{C}^{k}$-atlas being $\mathcal{C}^{k}$-compatible with
$\mathsf{B}$. By definition, $\mathsf{w}_{0,1}$ differs from $\psi_{0}$ and
$\psi_{1}$ only on the intersection $V_{0}\cap V_{1}$ which is disjoint from
other chars from $\mathsf{B}$. This easily implies that $\mathsf{B}_{0,1}$ is
still chain-like.

In particular, the charts $\mathsf{v}_{-1}$ and $\mathsf{w}_{0,1}$ are
$\mathcal{C}^{k}$-joinable, and again by Theorem 4.2 for any of their
$\mathcal{C}^{k}$-join

$\mathsf{w}_{-1,1}=\bigl{(}V_{-1}\cup W_{0,1}=V_{-1}\cup V_{0}\cup V_{1},\
\sigma_{-1,1}\bigr{)}$

the collection of charts

$\mathsf{B}_{-1,1}:=\bigl{(}\mathsf{B}\setminus\\{\mathsf{v}_{-1},\mathsf{w}_{0,1}\\}\bigr{)}\cup\\{\mathsf{w}_{-1,1}\\}=\bigl{(}\mathsf{B}\setminus\\{\mathsf{v}_{-1},\mathsf{v}_{0},\mathsf{v}_{1}\\}\bigr{)}\cup\\{\mathsf{w}_{-1,1}\\}$

is a chain-like $\mathcal{C}^{k}$-atlas being $\mathcal{C}^{k}$-compatible
with $\mathsf{B}_{0,1}$ and therefore with $\mathsf{B}$. Again notice that
$\sigma_{-1,1}$ differs from with $\psi_{-1}$ and $\sigma_{0,1}$ only on the
intersection

$V_{-1}\cap(V_{0}\cap V_{1})=V_{-1}\cap V_{0}.$

In particular, $\sigma_{-1,1}=\psi_{0}$ on the compact segment

$A_{1}:=V_{0}\setminus(V_{-1}\cup V_{1})=\psi_{0}^{-1}([b_{-1},a_{1}]).$

Thus, by induction for each $n\geq 1$ we can construct a sequence of chain-
like $\mathcal{C}^{k}$-atlases

$\mathsf{B}_{-n,n}:=\bigl{(}\mathsf{B}\setminus\\{\mathsf{v}_{j}\\}_{j=-n}^{n}\bigr{)}\cup\\{\mathsf{w}_{-n,n}\\}$

where
$\mathsf{w}_{-n,n}=\bigl{(}\mathop{\cup}\limits_{j=-n}^{n}V_{j},\sigma_{-n,n}\bigr{)}$
is a chart such that
$\sigma_{-n,n}\bigl{(}\mathop{\cup}\limits_{j=-n}^{n}V_{j}\bigr{)}=(a_{-n};b_{n})$
and $\sigma_{-n,n}=\sigma_{-n+1,n-1}$ on the compact segment

$A_{n}:=\bigl{(}\mathop{\cup}\limits_{j=-n}^{n}V_{j}\bigr{)}\setminus\bigl{(}V_{-n}\cup
V_{n}\bigr{)}.$

Since $A_{n}\subset\mathrm{Int}{A_{n+1}}$ and
$\mathbb{R}=\mathop{\cup}\limits_{n=0}^{+\infty}A_{n}$, for each
$x\in\mathbb{R}$ there exists $n>0$ such that

$\sigma_{-i,i}(x)=\sigma_{-j,j}(x)\ \text{for all}\ i,j\geq n.$

Then due to (4.4), we get a well-defined homeomorphism
$\sigma\colon\mathbb{R}\to\mathbb{R}$,
$\sigma(x)=\lim\limits_{n\to\infty}\sigma_{-n,n}(x)$, which is
$\mathcal{C}^{k}$-compatible with all the above atlases and, in particular,
with $\mathsf{B}$. ∎

######  Corollary 4.4.

Every $\mathcal{C}^{k}$-structure $\mathfrak{B}$ on $\mathbb{R}$ is
$\mathcal{C}^{k}$-diffeomorphic with the canonical one given by the atlas
$\mathsf{Id}_{\mathbb{R}}=\\{(\mathbb{R},\mathrm{id}_{\mathbb{R}})\\}$ with a
single chart.

###### Proof.

By Theorem 4.3.1, $\mathfrak{B}$ is represented by an atlas
$\mathsf{B}=\\{(\mathbb{R},\psi)\\}$ consisting of a single chart given by
some homeomorphism $\psi\colon\mathbb{R}\to\mathbb{R}$. On the other hand, the
canonical $\mathcal{C}^{k}$-structure $\mathcal{I}_{\mathbb{R}}$ is also given
by the atlas with a single chart whose coordinate homeomorphism
$\mathrm{id}\colon\mathbb{R}\to\mathbb{R}$ has the same image $\mathbb{R}$ as
$\psi$. Then by Lemma 3.5.1, the map
$\psi=\psi\circ\mathrm{id}_{\mathbb{R}}^{-1}\colon\mathbb{R}\to\mathbb{R}$ is
a $\mathcal{C}^{k}$-diffeomorphism between $(\mathbb{R},\mathfrak{B})$ and
$(\mathbb{R},\mathcal{I}_{\mathbb{R}})$. ∎

######  Corollary 4.4.1.

Let $M$ be a one-dimensional locally Euclidean topological space, $\mathsf{A}$
a $\mathcal{C}^{k}$-atlas on $M$, and $U\subset M$ be an open subset
homeomorphic with $\mathbb{R}$. Then there exists a homeomorphism
$\varphi\colon U\to\mathbb{R}$ being $\mathcal{C}^{k}$-compatible with
$\mathsf{A}$. In other words, $\mathsf{A}\cup\\{(U,U)\\}$ is a
$\mathcal{C}^{k}$-atlas on $M$ being $\mathcal{C}^{k}$-compatible with
$\mathsf{A}$.

We finish this section with the following well-known statement which could be
proved in using the approach of similar to Theorem 4.3.1.

######  Theorem 4.5.

Let $M$ be a connected Hausdorff one-dimensional locally Euclidean topological
space. Then for any two $\mathcal{C}^{k}$-structures $\mathfrak{A}$ and
$\mathfrak{B}$ on $M$ there exists a $\mathcal{C}^{k}$-diffeomorphism
$h\in\mathcal{D}^{k}_{\mathfrak{A},\mathfrak{B}}(M,M)$. In other words, the
action of the group of homeomorphisms $\mathcal{H}(M)$ on the set of all
$\mathcal{C}^{k}$-structures on $M$ is transitive.

##  5\. Differentiable structures on the line with double origin

It this section we will give a classification of $\mathcal{C}^{k}$-structures
on the line with two origins $\mathbb{L}=\mathbb{R}\sqcup\\{\bar{0}\\}$ for
all $k=1,2,\ldots,\infty$. It will be convenient to describe at first the
structure of the groups $\mathcal{H}(\mathbb{L})$ of all homeomorphisms of
$\mathbb{L}$, and then compare it with the subgroups of diffeomorphisms of
distinct differentiable structures. Denote
$U:=\mathbb{R}=\mathbb{L}\setminus\bar{0}$ and $V:=\mathbb{L}\setminus 0$.

###  5.1. Homeomorphisms of $\mathbb{R}$ and $\mathbb{L}$

Consider the diffeomorphism $q\colon\mathbb{R}\to\mathbb{R}$, $q(x)=-x$,
reversing orientation and having order $2$. Evidently,
$q\in\mathcal{E}^{\infty,-}(\mathbb{R},0)$.

Let $\mathcal{G}$ be one of the groups $\mathcal{H}^{+}(\mathbb{R},0)$,
$\mathcal{W}^{k,+}(\mathbb{R},0)$, $\mathcal{D}^{k,+}(\mathbb{R},0)$ or
$\mathcal{E}^{k,+}(\mathbb{R},0)$ defined in Section 1.1, and
$\mathcal{G}^{+}=\mathcal{G}\cap\mathcal{H}^{+}(\mathbb{R},0)$.

Then it is easy to see that $q\,\mathcal{G}^{+}q^{-1}=\mathcal{G}^{+}$, so the
conjugation by $q$ is an automorphism of $\mathcal{G}^{+}$ of order $2$. Hence
one can define the semidirect product $\mathcal{G}^{+}\rtimes\mathbb{Z}_{2}$
corresponding to that automorphism. Thus, by definition,
$\mathcal{G}^{+}\rtimes\mathbb{Z}_{2}$ is a cartesian product
$\mathcal{G}^{+}\times\mathbb{Z}_{2}$ with the following multiplication
$(h,\delta)(g,\varepsilon)=(h\circ q^{\delta}\circ g\circ
q^{-\delta},\delta\varepsilon)$. In particular, the inclusion
$\mathcal{G}^{+}\subset\mathcal{H}^{+}(\mathbb{R},0)$ allows to consider
$\mathcal{G}^{+}\rtimes\mathbb{Z}_{2}$ as a subgroup of
$\mathcal{H}^{+}(\mathbb{R},0)\rtimes\mathbb{Z}_{2}$.

The following two lemmas are straightforward, see Section 2.1, and we leave
them for the reader. They describe certain algebraic properties of the groups
of homeomorphisms of $\mathbb{R}$ and $\mathbb{L}$ and also establish
relationships between them.

######  Lemma 5.1.1.

The map
$\eta\colon\mathcal{H}^{+}(\mathbb{R},0)\rtimes\mathbb{Z}_{2}\to\mathcal{H}^{+}(\mathbb{R},0)$,
$\eta(h,\delta)=h\circ q^{\delta}$, is an isomorphism making commutative the
following diagram:

$\textstyle{\mathcal{H}^{+}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\,\mapsto\,(g,0)}$$\textstyle{\mathcal{H}^{+}(\mathbb{R},0)\rtimes_{q}\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g,a)\,\mapsto\,a}$$\scriptstyle{\eta}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}^{+}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\langle
q\rangle\cong\mathcal{H}(\mathbb{R},0)/\mathcal{H}^{+}(\mathbb{R},0)}$

and isomorphically mapping $\mathcal{G}^{+}\times\mathbb{Z}_{2}$ onto
$\mathcal{G}$.

######  Lemma 5.1.2.

  1. (1)

Let $\zeta\colon\mathbb{L}\to\mathbb{L}$ be the (unique) map exchanging $0$
and $\bar{0}$ and fixed on the complement
$\mathbb{L}\setminus\\{0,\bar{0}\\}$. Then

    1. (a)

$\zeta$ is a homeomorphism of $\mathbb{L}$ and
$\zeta^{2}=\mathrm{id}_{\mathbb{L}}$, so it generates the subgroup
$\langle\zeta\rangle\cong\mathbb{Z}_{2}$;

    2. (b)

$\zeta$ commutes with each $h\in\mathcal{H}(\mathbb{L})$ and
$\langle\zeta\rangle$ coincides with the center of $\mathcal{H}(\mathbb{L})$.

  2. (2)

$\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$ is the kernel of the ‘‘restriction to
$\\{0,\bar{0}\\}$’’ homomorphism

$\mathsf{p}\colon\mathcal{H}(\mathbb{L})\to\mathrm{Perm}(\\{0,\bar{0}\\})=\mathbb{Z}_{2},\qquad\mathsf{p}(h)=h|_{\\{0,\bar{0}\\}},$

to the group of permutations of $\\{0,\bar{0}\\}$. Moreover, the ‘‘restriction
to $U$’’ map

$\mathsf{r}_{U}\colon\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\to\mathcal{H}(\mathbb{R},0),\qquad\mathsf{r}_{U}(h)=h|_{U},$

is an isomorphism of groups. The inverse map
$\mathsf{r}_{U}^{-1}\colon\mathcal{H}(\mathbb{R},0)\to\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$
associates to each $h\in\mathcal{H}(\mathbb{R},0)$ its unique extension to a
homeomorphism of all $\mathbb{L}$ fixing $\bar{0}$.

  3. (3)

The normal subgroups $\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$ and
$\langle\zeta\rangle$ generate $\mathcal{H}(\mathbb{L})$, and
$\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\cap\langle\zeta\rangle=\\{\mathrm{id}_{\mathbb{L}}\\}$,
whence $\mathcal{H}(\mathbb{L})$ splits into the direct product
$\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\times\langle\zeta\rangle$. More
explicitly, the map

$\alpha\colon\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\times\langle\zeta\rangle\equiv\mathcal{H}(\mathbb{R},0)\times\mathbb{Z}_{2}\to\mathcal{H}(\mathbb{L}),\qquad\alpha(g,\varepsilon)=g\circ\xi^{\varepsilon},$

is an isomorphism of groups making commutative the following diagram:

$\textstyle{\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{r}_{U}\colon
h\,\mapsto\,h|_{U}}$$\scriptstyle{\cong}$$\textstyle{\mathcal{H}(\mathbb{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{p}\colon
h\,\mapsto\,h|_{\\{0,\bar{0}\\}}}$$\textstyle{\mathrm{Perm}(\\{0,\bar{0}\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}(\mathbb{R},0)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\,\mapsto\,(g,0)}$$\textstyle{\mathcal{H}(\mathbb{R},0)\times\mathbb{Z}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(g,\varepsilon)\,\mapsto\,\varepsilon}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$$\textstyle{\mathbb{Z}_{2}}$

  4. (4)

Let $q\in\mathcal{H}(\mathbb{R},0)$ be any homeomorphism such that
$q^{2}=\mathrm{id}_{\mathbb{R}}$, and
$\hat{q}=\mathsf{r}_{U}^{-1}(q)\in\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$ be
it unique extension to all of $\mathbb{L}$ fixing $\bar{0}$. Then we have an
isomorphism

$\bigl{(}\mathcal{H}^{+}(\mathbb{R},0)\rtimes_{q}\mathbb{Z}_{2}\bigr{)}\times\mathbb{Z}_{2}\,\cong\,\mathcal{H}(\mathbb{L}),\qquad(g,a,\varepsilon)\mapsto
g\circ\hat{q}^{a}\circ\zeta^{\varepsilon}.$

###  5.2. Group $\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$

From now on fix some $k\in\mathbb{N}\cup\\{\infty\\}$, and let
$\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$ be the group of
monotone $\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{R}\setminus 0$ (i.e.
either increasing or decreasing).

######  Example 5.2.1.

The diffeomorphism $h\colon\mathbb{R}\setminus 0\to\mathbb{R}\setminus 0$,
$h(x)=1/x$, is not monotone and therefore does not belong to
$\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$.

Notice that for each $h\in\mathcal{W}^{k}(\mathbb{R},0)$ its restriction
$h|_{\mathbb{R}\setminus 0}\colon\mathbb{R}\setminus 0\to\mathbb{R}\setminus
0$ is a monotone $\mathcal{C}^{k}$-diffeomorphism.

######  Lemma 5.2.2.

The map
$\gamma\colon\mathcal{W}^{k}(\mathbb{R},0)\to\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus
0)$, $\gamma(h)=h|_{\mathbb{R}\setminus 0}$, is an isomorphism of groups.

###### Proof.

Evidently, $\gamma$ is a monomorphism, and we need to show that $\gamma$ is
surjective. Let $g\in\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$.
Since $g$ is monotone, $\lim\limits_{x\to 0}g(x)=0$, whence $g$ extends to a
homeomorphism $\hat{g}\colon\mathbb{R}\to\mathbb{R}$, defined by $\hat{g}=g$
on $\mathbb{R}\setminus 0$ and $\hat{g}(0)=0$. It follows, that
$\hat{g}\in\mathcal{W}^{k}(\mathbb{R},0)$ and $\gamma(\hat{g})=g$, i.e.
$\gamma$ is surjective as well. ∎

For each $g\in\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$ the
homeomorphism $\gamma^{-1}(g)$ will be called the $0$-extension on $g$. Note
also that we have the following inclusion

$\mathcal{D}^{k}(\mathbb{R},0)\ \subset\ \mathcal{W}^{k}(\mathbb{R},0)\
\stackrel{{\scriptstyle\gamma^{-1}}}{{\equiv}}\
\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0).$

Thus, one can regard $\mathcal{D}^{k}(\mathbb{R},0)$ as a subgroup of
$\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$. It is easy to check
that $\mathcal{D}^{k}(\mathbb{R},0)$ is not normal in
$\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$.

######  Lemma 5.2.3.

The group $\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$ is normal in
$\mathcal{W}^{k}(\mathbb{R},0)$.

###### Proof.

Let $h\in\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$ and
$g\in\mathcal{W}^{k}(\mathbb{R},0)$. Then $g\circ h\circ g^{-1}$ is fixed near
$0\in\mathbb{R}$ and its restriction to $\mathbb{R}\setminus 0$ is a
$\mathcal{C}^{k}$-diffeomorphism. Hence $g\circ h\circ g^{-1}$ is a
$\mathcal{C}^{k}$-diffeomorphism of $\mathbb{R}$ fixed near $0$, i.e. it
belongs to $\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$. ∎

###  5.3. Minimal atlases on $\mathbb{L}$

A $\mathcal{C}^{k}$-atlas $\mathsf{A}$ on $\mathbb{L}$ will be called minimal
if it consisting of only two charts $\varphi\colon U\to\mathbb{R}$ and
$\psi\colon V\to\mathbb{R}$ being homeomorphisms onto and such that
$\varphi(0)=\psi(\bar{0})=0$.

Notice that for every such an atlas, its transition map from $(U,\varphi)$ to
$(V,\psi)$

$\psi\circ\varphi^{-1}\colon\mathbb{R}\setminus 0\to\mathbb{R}\setminus 0$

is a monotone $\mathcal{C}^{k}$-diffeomorphism of $\mathbb{R}\setminus 0$,
i.e. it belongs to $\mathcal{D}^{k}_{\mathrm{mon}}(\mathbb{R}\setminus 0)$.
The corresponding $0$-extension of $\psi\circ\varphi^{-1}$ will be denoted by
$g_{\mathsf{A}}$, i.e.

$g_{\mathsf{A}}:=\gamma(\psi\circ\varphi^{-1})\in\mathcal{W}^{k}(\mathbb{R},0).$

Note also that by definition, $\mathsf{A}$ is orientable iff its unique
transition map $\psi\circ\varphi^{-1}$ preserves orientation. The latter is,
evidently, equivalent to the assumption that $g_{\mathsf{A}}$ preserve
orientations.

The following lemma is the reason of our detailed exposition of uniqueness of
$\mathcal{C}^{k}$-structures on $\mathbb{R}$.

######  Lemma 5.3.1.

Every $\mathcal{C}^{k}$-structure $\mathfrak{A}$ on $\mathbb{L}$ is
represented by some orientable minimal atlas.

###### Proof.

Since $U$ and $V$ are open subsets homeomorphic with $\mathbb{R}$, it follows
from Corollary 4.4.1 that there exist two homeomorphisms onto $\varphi\colon
U\to\mathbb{R}$ and $\psi\colon V\to\mathbb{R}$ being
$\mathcal{C}^{k}$-compatible with $\mathfrak{A}$. Replacing $\varphi$ and
$\psi$ with $\varphi-\varphi(0)$ and $\psi-\psi(\bar{0})$ respectively, one
can achieve that $\varphi(0)=\psi(\bar{0})=0$. Moreover, replacing (if
necessary) $\psi$ with $-\psi$, we can assume that the transition map
$\psi^{-1}\circ\varphi$ preserves orientation. As $\mathbb{L}=U\cup V$, we
have that $\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}$ is the desired orientable
minimal $\mathcal{C}^{k}$-atlas belonging to $\mathfrak{A}$. ∎

######  Theorem 5.3.2.

Let $\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}$ and
$\mathsf{B}=\\{(U,\bar{\varphi}),(V,\bar{\psi})\\}$ be two minimal
$\mathcal{C}^{k}$-atlases on $\mathbb{L}$ corresponding to
$\mathcal{C}^{k}$-structures $\mathfrak{A}$ and $\mathfrak{B}$ respectively,
and $g_{\mathsf{A}}$ and $g_{\mathsf{B}}$ be $0$-extensions of their
respective transition maps $\psi\circ\varphi^{-1}$ and
$\bar{\psi}\circ\bar{\varphi}^{-1}$.

  1. 1)

Then the following correspondences
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ and
$\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}$ are well-defined
bijections:

$\displaystyle\begin{aligned}
&\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1},&&\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}(h)=\bar{\psi}\circ
h\circ\psi^{-1},\\\
&\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}\colon\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}},&\qquad\qquad&\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}(h)=\bar{\varphi}\circ
h\circ\psi^{-1},\\\ \end{aligned}$ $\displaystyle\begin{aligned}
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Thus, $\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}(h)$ is a local
presentation of
$h\in\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ with
respect to the charts $(V,\psi)$ and $(V,\bar{\psi})$, while
$\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}(h)$ is a local
presentation of
$h\in\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ with
respect to the charts $(V,\psi)$ and $(U,\bar{\varphi})$;

  2. 2)

If $\bar{\psi}$ and $\psi$ mutually preserve or reverse orientations, then

$\displaystyle\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}\bigl{(}\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\bigr{)}\subset\mathcal{D}^{k,+}(\mathbb{R},0),$
$\displaystyle\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}\bigl{(}\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\bigr{)}\subset\mathcal{D}^{k,-}(\mathbb{R},0).$

Similarly, if $\bar{\varphi}$ and $\psi$ mutually preserve or reverse
orientations, then

$\displaystyle\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}\bigl{(}\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\bigr{)}\subset\mathcal{D}^{k,+}(\mathbb{R},0),$
$\displaystyle\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}\bigl{(}\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\bigr{)}\subset\mathcal{D}^{k,-}(\mathbb{R},0);$

  3. 3)

If $\mathsf{A}=\mathsf{B}$, so $g_{\mathsf{A}}=g_{\mathsf{B}}$ and
$\mathfrak{A}=\mathfrak{B}$, then

$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{A}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$

is an isomorphism of groups. Moreover, the group
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ contains a subgroup
isomorphic with $\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$, and therefore
it is uncountable;

  4. 4)

If $\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ is non-
empty for some $*\in\\{\pm\\}$ and $\bullet\in\\{\mathrm{fix},\mathrm{ex}\\}$,
then it is uncountable.

###### Proof.

1) First we show that $\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is
a well-defined bijection. Let
$h\in\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$. Then
$h(U)=U$ and $h(V)=V$, whence we have the following commutative diagrams:

in which the lower arrows $a:=\bar{\varphi}\circ h\circ\varphi^{-1}$ and
$b:=\bar{\psi}\circ h\circ\psi^{-1}$ are local presentations of $h$ in the
corresponding charts. We need to show that
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}(h):=b\in\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$
and that correspondence $h\mapsto b$ is a bijection.

The assumption that $h$ is a $\mathcal{C}^{k}$-diffeomorphism of $\mathbb{L}$
means that $a,b\in\mathcal{D}^{k}(\mathbb{R},0)$. Notice that on
$\mathbb{R}\setminus 0$ we have the following identity:

$b=\bar{\psi}\circ
h\circ\psi^{-1}=(\bar{\psi}\circ\bar{\varphi}^{-1})\circ(\bar{\varphi}\circ
h\circ\varphi^{-1})\circ(\psi\circ\varphi^{-1})^{-1}.$

Taking $0$-extensions of these diffeomorphisms we obtain that

$b=g_{\mathsf{B}}\circ a\circ g_{\mathsf{A}}^{-1}\in
g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}.$

But $b\in\mathcal{D}^{k}(\mathbb{R},0)$ as well, whence
$b\in\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$,
and thus the map $\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is
well-defined.

Moreover, since $h(0)=0$, $h$ is uniquely determined by its restriction to
$\mathbb{L}\setminus\\{0\\}=V$, and therefore by the composition
$b=\bar{\psi}\circ h\circ\psi^{-1}$. Hence
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is injective.

Let us prove that $\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is
surjective. Let
$b\in\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$,
i.e.

$b=g_{\mathsf{B}}\circ a\circ
g_{\mathsf{A}}^{-1}=\bar{\psi}\circ\bar{\varphi}^{-1}\circ
a\circ\varphi\circ\psi^{-1}$

for some $a\in\mathcal{D}^{k}(\mathbb{R},0)$. The latter is equivalent to the
following identity:

$\bar{\varphi}^{-1}\circ a\circ\varphi=\bar{\psi}^{-1}\circ b\circ\psi.$ (5.1)

Define the map

$h\colon\mathbb{L}\to\mathbb{L},\qquad
h(x)=\begin{cases}\bar{\varphi}^{-1}\circ a\circ\varphi(x),&x\in U,\\\
\bar{\psi}^{-1}\circ b\circ\psi(x),&x\in V.\end{cases}$ (5.2)

Due to (5.1), the formulas for $h$ agree on $U\cap V$, and therefore $h$ is a
continuous bijection. Moreover, we have the following commutative diagrams:

which show that the local presentation of $h$ in the charts $U$ and $V$ are
$\mathcal{C}^{k}$-diffeomorphisms $a$ and $b$ respectively. Therefore $h$ is a
$\mathcal{C}^{k}$-diffeomorphism between $(\mathbb{L},\mathfrak{A})$ and
$(\mathbb{L},\mathfrak{B})$. This proves that
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is a bijection.

The proof for $\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}$ is similar.
Let us just mention that if
$h\in\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$, then
$h(U)=V$ and $h(V)=U$, whence we have the following commutative diagrams:

in which the lower arrows $a:=\bar{\psi}\circ h\circ\varphi^{-1}$ and
$b:=\bar{\varphi}\circ h\circ\psi^{-1}$ are local presentations of $h$ in the
corresponding charts. Then
$\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}$ is defined by
$\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}(h)=b$. The verification
that $\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}$ is a well-defined
bijection is similar to the case of
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ and we leave it for the
reader.

2) This statement directly follows from the definition of bijections
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ and
$\xi^{\mathrm{\mathrm{ex}}}_{\mathsf{A},\mathsf{B}}$.

3) Suppose $\mathsf{A}=\mathsf{B}$, so $\psi=\bar{\psi}$ and
$g_{\mathsf{B}}=g_{\mathsf{A}}$. Then
$g_{\mathsf{A}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$ is a
conjugated to $\mathcal{D}^{k}(\mathbb{R},0)$ subgroup of
$\mathcal{W}^{k}(\mathbb{R},0)$, whence
$\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{A}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}$
is also a subgroup of $\mathcal{W}^{k}(\mathbb{R},0)$. Moreover,
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}(h)=g_{\mathsf{B}}\circ
h\circ g_{\mathsf{B}}^{-1}$ for all $h\in\mathcal{W}^{k}(\mathbb{R},0)$, and
therefore $\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is a
homomorphism of groups. Since
$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}$ is also a bijection, it
is an isomorphism.

Also, since by Lemma 5.2.3, $\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$ is
normal in $\mathcal{W}^{k}(\mathbb{R},0)$, we see that

$\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)\ =\
g_{\mathsf{A}}\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)g_{\mathsf{A}}^{-1}\
\subset\ \mathcal{D}^{k}(\mathbb{R},0)\ \cap\
g_{\mathsf{A}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}.$

Hence
$(\xi^{\mathrm{\mathrm{fix}}}_{\mathfrak{A},\mathfrak{A}})^{-1}(\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0))$
is the desired subgroup of
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ isomorphic to
$\mathcal{D}^{k}_{\mathrm{nb}}(\mathbb{R},0)$.

4) It follows from Lemma 3.5.2 that for each
$h\in\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ we have
three bijections:

$\textstyle{\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
k\,\mapsto\,k\circ
h^{-1}}$$\textstyle{\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\,\mapsto\,h\circ
g\ }$$\scriptstyle{g\,\mapsto\,h\circ g\circ
h^{-1}}$$\textstyle{\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{B}}(\mathbb{L})}$

Hence, if
$\mathcal{D}^{*,\bullet}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$,
then it is uncountable since
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A}}(\mathbb{L})$ is so. ∎

######  Corollary 5.3.3.

Let $k\in\mathbb{N}\cup\\{\infty\\}$ and $\mathfrak{A}$ and $\mathfrak{B}$ be
two $\mathcal{C}^{k}$-structures on $\mathbb{L}$.

  1. 1)

Then the following conditions are equivalent:

    1. (ifix)

$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$;

    2. (iifix)

there exist minimal atlases
$\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}\in\mathfrak{A}$ and
$\mathsf{B}=\\{(U,\bar{\varphi}),(V,\bar{\psi})\\}\in\mathfrak{B}$ such that
$g_{\mathsf{B}}=b\circ g_{\mathsf{A}}\circ a^{-1}$ for some
$a,b\in\mathcal{D}^{k}(\mathbb{R},0)$, i.e. the $0$-extensions
$g_{\mathsf{A}}$ and $g_{\mathsf{B}}$ of their respective transition maps
$\psi\circ\varphi^{-1}$ and $\bar{\psi}\circ\bar{\varphi}^{-1}$ belong to the
same $\mathcal{D}^{k}(\mathbb{R},0)$-double coset in
$\mathcal{W}^{k}(\mathbb{R},0)$;

    3. (iiifix)

property 1)(iifix) holds for any pair of minimal atlases
$\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$.

  2. 2)

Similarly, the following conditions are also equivalent:

    1. (iex)

$\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$;

    2. (iiex)

there exist minimal atlases $\mathsf{A}\in\mathfrak{A}$ and
$\mathsf{B}\in\mathfrak{B}$ such that $g_{\mathsf{B}}^{-1}=b\circ
g_{\mathsf{A}}\circ a^{-1}$ for some $a,b\in\mathcal{D}^{k}(\mathbb{R},0)$;

    3. (iiiex)

property 2)(iiex) holds for any pair of minimal atlases
$\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$.

  3. 3)

Hence, the following conditions are equivalent as well:

    1. (i)

$\mathcal{D}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\equiv\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\sqcup\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$;

    2. (ii)

there exist minimal atlases $\mathsf{A}\in\mathfrak{A}$ and
$\mathsf{B}\in\mathfrak{B}$ such that either $g_{\mathsf{B}}=b\circ
g_{\mathsf{A}}\circ a^{-1}$ or $g_{\mathsf{B}}^{-1}=b\circ g_{\mathsf{A}}\circ
a^{-1}$ for some $a,b\in\mathcal{D}^{k}(\mathbb{R},0)$, i.e. $g_{\mathsf{A}}$
and $g_{\mathsf{B}}$ belong to the same
$(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-double coset in
$\mathcal{W}^{k}(\mathbb{R},0)$;

    3. (iii)

property 3)(ii) holds for any pair of minimal atlases
$\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$.

###### Proof.

1) The implication 1)(iiifix)$\Rightarrow$1)(iifix) is trivial.

1)(ifix)$\Rightarrow$1)(iiifix). Suppose
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$
and let $\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$ be any
minimal atlases. Then by Theorem 5.3.2, we have a bijection

$\xi^{\mathrm{\mathrm{fix}}}_{\mathsf{A},\mathsf{B}}\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\equiv\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}.$

Hence, the latter intersection is non-empty, i.e. $b=g_{\mathsf{B}}\circ
a\circ g_{\mathsf{A}}^{-1}$ for some $a,b\in\mathcal{D}^{k}(\mathbb{R},0)$,
which is the same as $g_{\mathsf{B}}=b\circ g_{\mathsf{A}}\circ a^{-1}$.

1)(iifix)$\Rightarrow$1)(ifix) Suppose there exist some minimal atlases
$\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$ such that
$g_{\mathsf{B}}=b\circ g_{\mathsf{A}}\circ a^{-1}$ for some
$a,b\in\mathcal{D}^{k}(\mathbb{R},0)$. Then

$b=g_{\mathsf{B}}\circ a\circ
g_{\mathsf{A}}^{-1}\in\mathcal{D}^{k}(\mathbb{R},0)\,\cap\,g_{\mathsf{B}}\,\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{-1}\neq\varnothing.$

Hence, by Theorem 5.3.2,
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})\neq\varnothing$
as well.

The proof of 2) is similar, and 3) is just a combination of 1) and 2). ∎

###  5.4. Classification of $\mathcal{C}^{k}$-structures on $\mathbb{L}$

Let $\mathcal{C}^{k}(\mathbb{L})$ be the set of all
$\mathcal{C}^{k}$-structures on $\mathbb{L}$. Then the group
$\mathcal{H}(\mathbb{L})$ of homeomorphisms of $\mathbb{L}$ acts on
$\mathcal{C}^{k}(\mathbb{L})$, as described in Lemma 3.4.4(3)(3b). For every
$\mathfrak{A}\in\mathcal{C}^{k}(\mathbb{L})$ let

  * •

$[\mathfrak{A}]$ be the orbit of $\mathfrak{A}$ with respect to that action of
$\mathcal{H}(\mathbb{L})$;

  * •

$[\mathfrak{A}]^{\mathrm{fix}}$ be the orbit of $\mathfrak{A}$ with respect to
the induced action of the subgroup $\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$
consisting of homeomorphisms fixing $0$ and $\bar{0}$.

Thus $[\mathfrak{A}]$ consist of all $\mathcal{C}^{k}$-structures on
$\mathbb{L}$ being $\mathcal{C}^{k}$-diffeomorphic to $\mathfrak{A}$, while
$[\mathfrak{A}]^{\mathrm{fix}}$ is a subset of $[\mathfrak{A}]$ consisting of
$\mathcal{C}^{k}$-structures being $\mathcal{C}^{k}$-diffeomorphic to
$\mathfrak{A}$ via a diffeomorphism fixing $0$ and $\bar{0}$.

Denote by $\mathcal{C}^{k}(\mathbb{L})/\mathcal{H}(\mathbb{L})$ and
$\mathcal{C}^{k}(\mathbb{L})/\mathcal{H}^{\mathrm{fix}}(\mathbb{L})$ the
corresponding orbits spaces. As a consequence we get the following theorem
clarifying the bijections mentioned in Theorem 1.2.1.

######  Theorem 5.4.1.

There are well-defined bijections

  * •

$\sigma\colon\mathcal{C}^{k}(\mathbb{L})/\mathcal{H}(\mathbb{L})\
\longrightarrow\
\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)^{\pm
1}\,/\,\mathcal{D}^{k}(\mathbb{R},0)$,

$\sigma([\mathfrak{A}])=\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}^{\pm
1}\,\mathcal{D}^{k}(\mathbb{R},0),\ \text{where $\mathsf{A}\in\mathfrak{A}$ is
any minimal atlas},$

  * •

$\sigma^{\mathrm{fix}}\colon\mathcal{C}^{k}(\mathbb{L})/\mathcal{H}^{\mathrm{fix}}(\mathbb{L})\
\longrightarrow\
\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)\,/\,\mathcal{D}^{k}(\mathbb{R},0)$,

$\sigma^{\mathrm{fix}}([\mathfrak{A}]^{\mathrm{fix}})=\mathcal{D}^{k}(\mathbb{R},0)\,g_{\mathsf{A}}\,\mathcal{D}^{k}(\mathbb{R},0),\
\text{where $\mathsf{A}\in\mathfrak{A}$ is any minimal atlas},$

associating to the isomorphism classes $[\mathfrak{A}]$ and
$[\mathfrak{A}]^{\mathrm{fix}}$ of a $\mathcal{C}^{k}$-structure
$\mathfrak{A}$ on $\mathbb{L}$ respectively $\mathcal{D}^{k}(\mathbb{R},0)$\-
and $(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-double cosets in
$\mathcal{W}^{k}(\mathbb{R},0)$ of the $0$-extension $g_{\mathsf{A}}$ of the
transition map of any minimal atlas $\mathsf{A}\in\mathfrak{A}$.

###### Proof.

It is just a rephrasing of Corollary 5.3.3. Namely, the statement that
$\sigma$ is a bijection means that two $\mathcal{C}^{k}$-structures
$\mathfrak{A}$ and $\mathfrak{B}$ on $\mathbb{L}$ are
$\mathcal{C}^{k}$-diffeomorphic iff for any pair of minimal atlases
$\mathsf{A}\in\mathfrak{A}$ and $\mathsf{B}\in\mathfrak{B}$ the $0$-extensions
$g_{\mathsf{A}}$ and $g_{\mathsf{B}}$ of their respective transition maps
belong to the same $(\mathcal{D}^{k}(\mathbb{R},0),\pm)$-double coset in
$\mathcal{W}^{k}(\mathbb{R},0)$. This is exactly the equivalence of 3)(i) and
3)(iii) of Corollary 5.3.3.

Similarly, existence of the bijection $\sigma^{\mathrm{fix}}$ is the same as
the equivalence 1)(ifix)$\Leftrightarrow$1)(iiifix). ∎

###  5.5. Special minimal atlases

Our next aim is to provide examples of $\mathcal{C}^{k}$-structures
$\mathfrak{A}$ and $\mathfrak{B}$ on $\mathbb{L}$ for which either of sets
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ and
$\mathcal{D}^{\mathrm{ex}}_{\mathfrak{A},\mathfrak{B}}(\mathbb{L})$ can be
empty or non-empty. For that reason it will be convenient to work only with
$\mathcal{C}^{k}$-structures admitting special minimal atlases defined as
follows.

Recall that $U=\mathbb{R}$ and
$V=(\mathbb{R}\setminus\\{0\\})\cup\\{\bar{0}\\}$ by the construction of
$\mathbb{L}$. Hence, we have the following homeomorphisms: the identity map
$\mathrm{id}_{u}\colon U\to\mathbb{R}$, and

$\mathrm{id}_{v}\colon
V\to\mathbb{R},\qquad\mathrm{id}_{v}(x)=\begin{cases}0,&x=\bar{0},\\\
x=\mathrm{id}_{u}(x),&x\in V\setminus\bar{0}.\end{cases}$

In particular, $\mathrm{id}_{u}=\mathrm{id}_{v}$ on $U\setminus
0=V\setminus\bar{0}$.

Then for each $h\in\mathcal{W}^{k}(\mathbb{R},0)$ one can define the following
minimal atlas

$\mathsf{S}(h):=\\{(U,\mathrm{id}_{u}),(V,h\circ\mathrm{id}_{v})\\}$

whose transition map is

$h\circ\mathrm{id}_{v}\circ\mathrm{id}_{u}^{-1}|_{\mathbb{R}\setminus
0}=h|_{\mathbb{R}\setminus 0}.$

Hence, its $0$-extension is $h$, i.e. $g_{\mathsf{S}(h)}=h$.

The atlas $\mathsf{S}(h)$ will be called the special minimal atlas of
$h\in\mathcal{W}^{k}(\mathbb{R},0)$. Denote by $\mathfrak{S}(h)$ the
corresponding $\mathcal{C}^{k}$-structure on $\mathbb{L}$ defined by the atlas
$\mathsf{S}(h)$, so it is determined by $h$. The following lemma collects
simple facts about special atlases.

######  Lemma 5.5.1.

  1. (1)

Let $\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}$ be a minimal atlas on $\mathbb{L}$
such that $\varphi=\mathrm{id}_{u}$. Then $\mathsf{A}$ is a special minimal
atlas corresponding to the homeomorphism
$\psi\circ\mathrm{id}_{v}^{-1}\in\mathcal{W}^{k}(\mathbb{R},0)$, i.e.
$\mathsf{A}=\mathsf{S}(\psi\circ\mathrm{id}_{v}^{-1})$;

  2. (2)

For a $\mathcal{C}^{k}$-structure $\mathfrak{A}$ on $\mathbb{L}$ the following
conditions are equivalent:

    1. (a)

$\mathfrak{A}$ has a special minimal $\mathcal{C}^{k}$-atlas;

    2. (b)

there exists a minimal atlas
$\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}\in\mathfrak{A}$ such that
$\varphi\colon U=\mathbb{R}\to\mathbb{R}$ is a
$\mathcal{C}^{k}$-diffeomorphism;

    3. (c)

for every minimal atlas $\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}\in\mathfrak{A}$
we have that $\varphi\colon U=\mathbb{R}\to\mathbb{R}$ is a
$\mathcal{C}^{k}$-diffeomorphism;

  3. (3)

Let $\mathfrak{A}$ be a $\mathcal{C}^{k}$-structure on $\mathbb{L}$,
$\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}\in\mathfrak{A}$ some minimal atlas, and
$g_{\mathsf{A}}$ be the $0$-extension of its transition map
$\psi\circ\varphi^{-1}$; Then the $\mathcal{C}^{k}$-structures $\mathfrak{A}$
and $\mathfrak{S}(g_{\mathsf{A}})$ are diffeomorphic via a preserving
orientation diffeomorphism $h$ fixing $0$ and $\bar{0}$, i.e.
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A},\mathfrak{S}(g_{\mathsf{A}})}(\mathbb{L})\neq\varnothing$;
For instance, if $\varphi$ preserves orientation, then the following map
$h\colon\mathbb{L}\to\mathbb{L}$ defined by

$h|_{U}=\mathrm{id}_{u}^{-1}\circ\varphi\colon U\to U\qquad\text{and}\qquad
h(\bar{0})=\bar{0}$

belongs to
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{A},\mathfrak{S}(g_{\mathsf{A}})}(\mathbb{L})$;

  4. (4)

Let $g,h\in\mathcal{D}^{k}(\mathbb{R},0)$. Then
$\mathfrak{S}(h)=\mathfrak{S}(g)$ if and only if $g\circ
h^{-1}\in\mathcal{D}^{k}(\mathbb{R},0)$. In particular, there is a bijection
between $\mathcal{C}^{k}$-structures on $\mathbb{L}$ admitting special minimal
atlas and right cosets
$\mathcal{D}^{k}(\mathbb{R},0)\setminus\mathcal{W}^{k}(\mathbb{R},0)$;

  5. (5)

For each $g,h\in\mathcal{W}^{k}(\mathbb{R},0)$ there are bijections:

$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(g),\mathfrak{S}(h)}(\mathbb{L})\xrightarrow{\leavevmode\nobreak\
\xi^{\mathrm{}}_{\mathsf{S}(g),\mathsf{S}(h)}\leavevmode\nobreak\
}\mathcal{D}^{k}(\mathbb{R},0)\ \cap\
h\,\mathcal{D}^{k}(\mathbb{R},0)\,g^{-1}\xleftarrow{\leavevmode\nobreak\
\xi^{\mathrm{}}_{\mathsf{S}(g^{-1}),\mathsf{S}(h)}\leavevmode\nobreak\
}\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(g^{-1}),\mathfrak{S}(h)}(\mathbb{L});$

  6. (6)

If $b\in\mathcal{W}^{k}(\mathbb{R},0)$ but
$b\not\in\mathcal{D}^{k}(\mathbb{R},0)$, then the $\mathcal{C}^{k}$-structures
$\mathfrak{S}(\mathrm{id}_{\mathbb{R}})$ and $\mathfrak{S}(b)$ are not
$\mathcal{C}^{k}$-diffeomorphic, i.e.
$\mathcal{D}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L})=\varnothing$.

###### Proof.

Statements (1) is trivial, while (5) follows from Theorem 5.3.2.

(2) The implication (2)(c)$\Rightarrow$(2)(b) is also trivial.

(2)(b)$\Rightarrow$(2)(a) Let
$\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}\in\mathfrak{A}$ be a
$\mathcal{C}^{k}$-atlas such that $\varphi\colon U=\mathbb{R}\to\mathbb{R}$ is
a $\mathcal{C}^{k}$-diffeomorphism. Then
$\\{(U,\varphi^{-1}\circ\varphi=\mathrm{id}_{u}),(V,\psi)\\}$ is a special
minimal $\mathcal{C}^{k}$-atlas $\mathcal{C}^{k}$-compatible with
$\mathsf{A}$, and therefore belonging to $\mathfrak{B}$.

(2)(a)$\Rightarrow$(2)(c) Let
$\mathsf{S}(a)=\\{(U,\mathrm{id}_{u}),(V,a\circ\mathrm{id}_{v})\\}\in\mathfrak{A}$
be a special minimal atlas, and $\mathsf{A}=\\{(U,\varphi),(V,\psi)\\}$ be any
other minimal atlas in $\mathfrak{A}$. Then they must be
$\mathcal{C}^{k}$-compatible, i.e. $\varphi\circ\mathrm{id}_{u}^{-1}=\varphi$
is a $\mathcal{C}^{k}$-diffeomorphism.

(3) By Theorem 5.3.2 we have a bijection
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{A},\mathfrak{S}(g_{\mathsf{A}})}(\mathbb{L})\equiv\mathcal{D}^{k}(\mathbb{R},0)\cap
g_{\mathsf{A}}\mathcal{D}^{k}(\mathbb{R},0)g_{\mathsf{A}}^{-1}$. The latter
intersection contains $\mathrm{id}_{\mathbb{R}}$, and therefore it is non-
empty, whence $\mathfrak{A}$ and $\mathfrak{S}(g_{\mathsf{A}})$ are
$\mathcal{C}^{k}$-diffeomorphic via a diffeomorphism fixing $0$ and $\bar{0}$.

(4) Consider the corresponding special minimal atlases

$\displaystyle\mathsf{S}(g)$
$\displaystyle:=\\{(U,\mathrm{id}_{u}),(V,g\circ\mathrm{id}_{v})\\},$
$\displaystyle\mathsf{S}(h)$
$\displaystyle:=\\{(U,\mathrm{id}_{u}),(V,h\circ\mathrm{id}_{v})\\}.$

Since they have the same first chart $(U,\mathrm{id}_{u})$, the
$\mathcal{C}^{k}$-structures $\mathsf{S}(g)$ and $\mathsf{S}(h)$ coincide if
and only if their second charts are $\mathcal{C}^{k}$-compatible. The latter
means that the corresponding transition map
$(h\circ\mathrm{id}_{v})\circ(g\circ\mathrm{id}_{v})^{-1}=h\circ
g^{-1}\in\mathcal{D}^{k}(\mathbb{R},0)$.

(6) By Theorem 5.3.2 we have the following bijections:

$\displaystyle\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L})$
$\displaystyle\stackrel{{\scriptstyle\xi^{\mathrm{}}_{\mathsf{S}(\mathrm{id}_{\mathbb{R}}),\mathsf{S}(b)}}}{{\equiv}}\mathcal{D}^{k}(\mathbb{R},0)\cap
b\mathcal{D}^{k}(\mathbb{R},0)\mathrm{id}_{\mathbb{R}}^{-1}=\mathcal{D}^{k}(\mathbb{R},0)\cap
b\mathcal{D}^{k}(\mathbb{R},0)=$ (5.3)
$\displaystyle=\mathcal{D}^{k}(\mathbb{R},0)\cap
b\mathcal{D}^{k}(\mathbb{R},0)\mathrm{id}_{\mathbb{R}}\stackrel{{\scriptstyle(\xi^{\mathrm{}}_{\mathsf{S}(\mathrm{id}_{\mathbb{R}}^{-1}),\mathsf{S}(b)})^{-1}}}{{\equiv}}\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L}).$

Since $b\not\in\mathcal{D}^{k}(\mathbb{R},0)$, the adjacent classes
$\mathcal{D}^{k}(\mathbb{R},0)$ and $b\mathcal{D}^{k}(\mathbb{R},0)$ do not
intersect each other, so all the sets in (5.3) are empty. Hence,

$\mathcal{D}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L})\
=\
\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L})\
\sqcup\
\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(\mathrm{id}_{\mathbb{R}}),\mathfrak{S}(b)}(\mathbb{L})\
=\ \varnothing.$

Therefore, there are no diffeomorphisms between
$\mathfrak{S}(\mathrm{id}_{\mathbb{R}})$ and $\mathfrak{S}(b)$ (neiter fixing
$0$ and $\bar{0}$ nor exchanging them). ∎

Statement (3) of Lemma 5.5.1 shows that in order to classify
$\mathcal{C}^{k}$-structures on $\mathbb{L}$ up to a
$\mathcal{C}^{k}$-diffeomorphism, it suffices to consider only
$\mathcal{C}^{k}$-structures having special minimal atlases. Also, due to
statement (6), $\mathbb{L}$ admits at least two non-diffeomorphic
$\mathcal{C}^{k}$-structures.

Our final step is to show that in fact, there are uncountably many pair-wise
non-diffeomorphic $\mathcal{C}^{k}$-structures on $\mathbb{L}$.

Recall that for each $a>0$ we have defined a homeomorphism $w_{a}$ by formula
(1.2), so $w_{a}(x)=x$ for $x\leq 0$, and $w_{a}(x)=ax$ for $x\geq 0$. The
following simple statement allows to classify $\mathcal{C}^{k}$-structures
$\mathfrak{S}(w_{a})$ on $\mathbb{L}$ corresponding to special minimal atlases
$\mathsf{S}(w_{a})$ of $w_{a}$ for all $a>0$.

######  Lemma 5.5.2.

Let $a,b>0$, $f\in\mathcal{D}^{k}(\mathbb{R},0)$, $\tau_{1}=f^{\prime}(0)$,
and

$q:=w_{b}\circ f\circ w_{a}\in\mathcal{W}^{k}(\mathbb{R},0).$

  1. (a)

Suppose $\tau_{1}>0$; Then $q$ is $\mathcal{C}^{1}$ if and only if $a=1/b$.

  2. (b)

Let $\tau_{1}<0$. Then $q$ is $\mathcal{C}^{1}$ if and only if $a=b$;

  3. (c)

Suppose $q$ is $\mathcal{C}^{1}$ and let $n\in\mathbb{N}$ be such that $2\leq
n\leq k$. Assume also that $a,b\neq 1$. Then $q$ is $\mathcal{C}^{n}$ if and
only if $f^{\prime\prime}(0)=\cdots=f^{(n)}(0)=0$.

###### Proof.

Since $f$ is of class $\mathcal{C}^{k}$, we have that
$f(x)=\tau_{1}x+\tau_{2}x^{2}+\cdots+\tau_{n}x^{n}+g(x)$, where
$\tau_{i}=\tfrac{1}{i!}f^{(i)}(0)$, $i=1,\ldots,n$, and
$g\colon\mathbb{R}\to\mathbb{R}$ is a $\mathcal{C}^{k}$-function such that
$\lim\limits_{x\to 0}\bigl{(}g(x)/x^{n}\bigr{)}=0$. It follows that if
$\tau_{1}>0$, then

$q(x)=\begin{cases}f(x)=\tau_{1}x+\tau_{2}x^{2}+\cdots\tau_{n}x^{n}+g(x),&x\leq
0,\\\\[2.84526pt] bf(ax)=ba\tau_{1}x+ba^{2}\tau_{2}x^{2}+\cdots
ba^{n}\tau_{n}x^{n}+bg(ax),&x\geq 0.\end{cases}$ (5.4)

On the other hand, if $\tau_{1}<0$, then

$q(x)=\begin{cases}bf(x)=b\tau_{1}x+b\tau_{2}x^{2}+\cdots
b\tau_{n}x^{n}+bg(x),&x\leq 0,\\\\[2.84526pt]
f(ax)=a\tau_{1}x+a^{2}\tau_{2}x^{2}+\cdots a^{n}\tau_{n}x^{n}+g(ax),&x\geq
0.\end{cases}$ (5.5)

Thus if $\tau_{1}>0$, then due to (5.4), $q$ is $\mathcal{C}^{1}$ iff $ba=1$.
Moreover, in that case $q$ is $\mathcal{C}^{n}$ iff $a^{i-1}\tau_{i}=\tau_{i}$
for $i=2,\ldots,n$. Since $a\neq 0$, the latter is possible iff $\tau_{i}=0$
for $i=2\ldots,n$. This proves (a) and a part of (c) for $\tau_{1}>0$.

Similarly, if $\tau_{1}<0$, then due to (5.5), $q$ is $\mathcal{C}^{1}$ iff
$b=a$. Moreover, again that case $q$ is $\mathcal{C}^{n}$ iff
$a^{i-1}\tau_{i}=\tau_{i}$ for $i=2,\ldots,n$, which is again equivalent to
the assumption that $\tau_{i}=0$ for $i=2\ldots,n$. This proves (b) and
completes (c) for $\tau_{1}<0$. ∎

######  Corollary 5.5.3.

For all $a,b\neq 1$ we have that

$\displaystyle\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}$
$\displaystyle=\begin{cases}\varnothing,&a\neq b,1/b,\\\\[2.84526pt]
\mathcal{E}^{k,+}(\mathbb{R},0),&ab=1,\\\\[2.84526pt]
\mathcal{E}^{k,-}(\mathbb{R},0),&a=b.\end{cases}$

###### Proof.

1) If $a\neq b,1/b$, then by Lemma 5.5.2 for any
$f\in\mathcal{D}^{k}(\mathbb{R},0)$ the homeomorphism $q=w_{b}\circ f\circ
w_{a}$ is not even $\mathcal{C}^{1}$. In particular,
$\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}=\varnothing$.

2) Suppose $a=1/b$. Again, let $f\in\mathcal{D}^{k}(\mathbb{R},0)$ and
$q=w_{b}\circ f\circ w_{a}$.

If $f\in\mathcal{D}^{k,-}(\mathbb{R},0)$, then by (b), $q$ is not
$\mathcal{C}^{1}$ since $a\neq b$. On the other hand, if
$f\in\mathcal{D}^{k,+}(\mathbb{R},0)$, then by (a) and (c), $q$ is
$\mathcal{C}^{k}$ iff $f\in\mathcal{E}^{k,+}(\mathbb{R},0)$. In that case, it
also follows from (5.4) that $q\in\mathcal{E}^{k,+}(\mathbb{R},0)$. In other
words,

$\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}\,=\,\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{E}^{k,+}(\mathbb{R},0)w_{a}\,\subset\,\mathcal{E}^{k,+}(\mathbb{R},0).$

To prove the inverse inclusion take any
$\bar{q}\in\mathcal{E}^{k,+}(\mathbb{R},0)$, and let
$\bar{f}=w_{1/b}\circ\bar{q}\circ w_{1/a}$. Since $1/a=1/(1/b)$, the already
proved part of case 2) applied to $\bar{q}$ implies that
$\bar{f}\in\mathcal{E}^{k,+}(\mathbb{R},0)$. Hence,

$\bar{q}=w_{b}\circ\bar{f}\circ w_{a}\in\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{E}^{k,+}(\mathbb{R},0)w_{a}.$

3) The proof that $\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}=\mathcal{E}^{k,-}(\mathbb{R},0)$ for
$a=b\neq 1$ is similar. ∎

##  6\. Proof of Theorem 1.2.3

Denote by $\mathfrak{S}(a)$ the $\mathcal{C}^{k}$-structure on $\mathbb{L}$
corresponding to the special minimal atlas $\mathsf{S}(w_{a})$ of the
homeomorphism $w_{a}$. We need to describe the sets of diffeomorphisms between
$\mathcal{C}^{k}$-structures $\mathfrak{S}(a)$ and $\mathfrak{S}(b)$ for all
$a,b>0$. Let

$\displaystyle\mathsf{S}(w_{a})=\\{(U,\mathrm{id}_{u}),(V,w_{a}\circ\mathrm{id}_{v})\\},$
$\displaystyle\mathsf{S}(w_{b})=\\{(U,\mathrm{id}_{u}),(V,w_{b}\circ\mathrm{id}_{v})\\},$

be special minimal atlases of $w_{a}$ and $w_{b}$ respectively. Then by
Theorem 5.3.2 we have the following bijections:

$\displaystyle\xi^{\mathrm{\mathrm{fix}}}\colon$
$\displaystyle\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(b)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}^{-1},$
$\displaystyle\xi^{\mathrm{\mathrm{fix}}}(h)$
$\displaystyle=w_{b}\circ\mathrm{id}_{v}\circ h\circ\mathrm{id}_{v}^{-1}\circ
w_{a}^{-1},$ $\displaystyle\xi^{\mathrm{\mathrm{ex}}}\colon$
$\displaystyle\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(b)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a},$
$\displaystyle\xi^{\mathrm{\mathrm{ex}}}(h)$
$\displaystyle=\mathrm{id}_{u}\circ h\circ\mathrm{id}_{v}^{-1}\circ
w_{a}^{-1}.$

given by the lower arrows of the corresponding diagrams:

and thus associating to each
$h\in\mathcal{D}_{\mathfrak{S}(a),\mathfrak{S}(b)}(\mathbb{L})$ its local
presentation from chart $V$. Moreover, Corollary 5.5.3 gives an explicit
description of the images of $\xi^{\mathrm{\mathrm{fix}}}$ and
$\xi^{\mathrm{\mathrm{ex}}}$.

(4) If $a\not=b,1/b$, then by Corollary 5.5.3 the intersections
$\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{b}\mathcal{D}^{k}(\mathbb{R},0)w_{a}^{\pm 1}$ are empty, whence
$\mathcal{D}_{\mathfrak{S}(a),\mathfrak{S}(b)}(\mathbb{L})=\varnothing$ as
well.

(1) Let $a=1$, so $w_{a}=\mathrm{id}_{\mathbb{R}}$. Then

$w_{1}\mathcal{D}^{k}(\mathbb{R},0)w_{1}^{-1}=\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{1}\mathcal{D}^{k}(\mathbb{R},0)w_{1}=\mathcal{D}^{k}(\mathbb{R},0),$

whence we above bijections reduce to the following ones:

$\displaystyle\xi^{\mathrm{\mathrm{fix}}}\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(1)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0),$
$\displaystyle\xi^{\mathrm{\mathrm{ex}}}\colon\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(1)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0).$

Let $\zeta\colon\mathbb{L}\to\mathbb{L}$ be the involution exchanging $0$ and
$\bar{0}$ and fixed on $\mathbb{R}\setminus 0$. One easily checks $\zeta$ is a
self-diffeomorphism of
$\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(1)}(\mathbb{L})$. Namely, the local
presentations of $\zeta$ with respect to the special minimal atlas
$\mathsf{S}(w_{1})=\\{(U,\mathrm{id}_{u}),(V,w_{1}\circ\mathrm{id}_{v}=\mathrm{id}_{v})\\}$
are given by the following commutative diagrams in which both lower arrows are
$\mathrm{id}_{\mathbb{R}}$:

In particular, $\xi^{\mathrm{\mathrm{ex}}}(\zeta)=\mathrm{id}_{\mathbb{R}}$.

Moreover, as noted in Lemma 5.1.2, $\zeta$ commutes with all homeomorphisms of
$\mathbb{L}$, and in particular with
$\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(1)}(\mathbb{L})$. Hence the map
$\eta\colon\mathcal{D}^{k}(\mathbb{R},0)\times\mathbb{Z}_{2}\to\mathcal{D}(\mathbb{L})$,
$\eta(h,\delta)\mapsto\zeta^{\delta}\circ h$, is an isomorphism of groups. It
is aslo evident, that is the identities (1.3) hold.

(2) Suppose $a\neq 1$. Note also that $w_{a}^{-1}=w_{1/a}$. Then by Theorem
5.3.2 and Corollary 5.5.3 the above bijections reduce to the following ones

$\displaystyle\xi^{\mathrm{\mathrm{fix}}}$
$\displaystyle\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{a}\mathcal{D}^{k}(\mathbb{R},0)w_{1/a}=\mathcal{E}^{k,+}(\mathbb{R},0),$
$\displaystyle\xi^{\mathrm{\mathrm{ex}}}$
$\displaystyle\colon\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{a}\mathcal{D}^{k}(\mathbb{R},0)w_{a}=\mathcal{E}^{k,-}(\mathbb{R},0),$

which send preserving (resp. reversing) orientation diffeomorphisms of
$\mathbb{L}$ to diffeomorphisms of $\mathbb{R}$ of the same type. Hence
$\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L})=\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L})=\varnothing$,
and

$\mathcal{D}_{\mathfrak{S}(a)}(\mathbb{L})=\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L}){}\sqcup\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L}){}.$

Finally, consider the following homeomorphism
$p\colon\mathbb{L}\to\mathbb{L}$:

$p(x)=\begin{cases}-x/\sqrt{a},&x\in(-\infty;0),\\\ \bar{0},&x=0,\\\
0,&x=\bar{0},\\\ -x\sqrt{a},&x\in(0;\infty).\end{cases}$

It exchanges $0$ and $\bar{0}$ and one easily checks that $p$ has order $2$.
We claim that $p$ is the desired $\mathcal{C}^{k}$-diffeomorphism of
$\mathfrak{S}(a)$ belonging to
$\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a)}(\mathbb{L})$.

To check that, let us will compute local presentations of $p$ in each of the
charts of the special minimal atlas
$\mathsf{S}(w_{a})=\\{(U,\mathrm{id}_{u}),(V,w_{a}\circ\mathrm{id}_{v})\\}$ of
$w_{a}$. Note that $p$ exchanges the connected components of
$\mathbb{L}\setminus\\{0,\bar{0}\\}=\mathbb{R}\setminus 0$. Then the local
presentation of $p$ with respect to the charts $(U,\mathrm{id}_{u})$ and
$(V,w_{a}\circ\mathrm{id}_{v})$ on the components $(-\infty;0)$ and
$(0;\infty)$ are given by the following two diagrams:

The formulas for the lower arrows are the same, and thus the local
presentation of $p$ with respect to the charts $(U,\mathrm{id}_{u})$ and
$(V,w_{a}\circ\mathrm{id}_{v})$ is the map $x\mapsto-\sqrt{a}x$.

Similarly, the local presentation of $p$ with respect to the charts
$(V,w_{a}\circ\mathrm{id}_{v})$ and $(U,\mathrm{id}_{u})$ on the components
$(-\infty;0)$ and $(0;\infty)$ are given by:

Hence, the local presentation of $p$ with respect to the charts
$(V,w_{a}\circ\mathrm{id}_{v})$ and $(U,\mathrm{id}_{u})$ is the map
$x\mapsto-x/\sqrt{a}$.

Thus $p$ is $\mathcal{C}^{k}$ in both charts of the atlas $\mathsf{S}(w_{a})$,
and therefore it is a self-diffeomorphism of $\mathfrak{S}(a)$.

(3) Again if $a\not=1$, then by Theorem 5.3.2 and Corollary 5.5.3 the above
bijections reduce to the following ones

$\displaystyle\xi^{\mathrm{\mathrm{fix}}}$
$\displaystyle\colon\mathcal{D}^{\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{1/a}\mathcal{D}^{k}(\mathbb{R},0)w_{a}^{-1}=\mathcal{E}^{k,-}(\mathbb{R},0),$
$\displaystyle\xi^{\mathrm{\mathrm{ex}}}$
$\displaystyle\colon\mathcal{D}^{\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})\to\mathcal{D}^{k}(\mathbb{R},0)\cap
w_{1/a}\mathcal{D}^{k}(\mathbb{R},0)w_{a}=\mathcal{E}^{k,+}(\mathbb{R},0),$

which send preserving (resp. reversing) orientation diffeomorphisms of
$\mathbb{L}$ to diffeomorphisms of $\mathbb{R}$ of the same type.

Hence
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\mathcal{D}^{-,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\varnothing$,
and

$\mathcal{D}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L})=\mathcal{D}^{-,\mathrm{fix}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L}){}\sqcup\mathcal{D}^{+,\mathrm{ex}}_{\mathfrak{S}(a),\mathfrak{S}(1/a)}(\mathbb{L}){}.$

The latter statement on bijections of those sets with
$\mathcal{D}^{+,\mathrm{fix}}_{\mathfrak{S}(a)}(\mathbb{L})$ follows from
Lemma 3.5.2. This completes Theorem 1.2.3.

### Acknowledgement

This work was supported by grants from the Simons Foundation (1030291,
1290607, S.I.M. and L.M.V.).

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