A bit of everything

# All about $1$-manifolds

While $1$-manifolds are the almost most trivial manifolds, second only to $0$-manifolds (of which there are only one equivalence class of connected manifolds and almost no structure), they still contain quite a lot of interesting facts which are easily enough solved thanks to their simplicity. They can also be made quite complex by keeping things as general as possible, as well as generalizing beyond manifolds themselves to more complex structures. With this in mind, it is possible to find $1$-manifolds that are not orientable, not metrizable or with more than one differential structure.

## 1. $1$-manifolds

As $1$-manifolds are $1$-dimensional manifolds, their definition follows from this, ie :

• For every open set $U$, there is a homeomorphism $\phi_U$ from $U$ to an open subset $\mathcal{O} \subset \mathbb{R}$.
• For every overlapping open sets $U,V$, $U \cap V \neq \varnothing$, the transition function $\phi_U^{-1} \circ \phi_V$ is a homeomorphism

Similar definitions apply to manifolds with boundaries, where the charts are mapped to subsets of $\mathbb{R}_{\geq 0}$. Manifolds with corners are equivalent to manifolds with boundaries in $1$ dimension, so further generalization in that direction won't be necessary.

While it is possible with any open subset of $\mathbb{R}$, it is also possible to do it using connected open sets of $\mathbb{R}$, as for any manifold.

Theorem : A $1$-manifold admits an atlas comprised entirely of maps to either $\mathbb{R}$ or open balls $(a,b)$, $a,b \in \mathbb{R}$.

Proof : As open balls form a basis for the topology of $\mathbb{R}$, any open set $\mathcal{O} \subset \mathbb{R}$ can be written as the union of open balls.

$$\mathcal{O} = \bigcup_{i\in I} B_{x_i, r_i}$$

Then taking the original atlas $\{ (U_\alpha, \phi_\alpha) \}$, we can construct an atlas composed of charts built from the decomposition of each $U_\alpha$ into open balls $B_{\alpha, x_i, r_i}$ with charts made from restriction of the original charts to this domain,

$$\{ (\phi_{\alpha}(B_{\alpha, x_i, r_i}), \phi_{\alpha}|_{B_{\alpha, x_i, r_i}}) \}$$

As open balls $(a,b)$ are homeomorphic to $\mathbb{R}$, for instance via the homeomorphism

$$f(x) = \tan(\pi(\frac{x - a}{b-a}) - \frac{\pi}{2})$$

It is possible to make them all map to $\mathbb{R}$ via the chart

$$f \circ \phi_{\alpha}|_{B_{\alpha, x_i, r_i}}$$

Due to this, it will be useful to define some types of intervals more precisely.

Definition : The unit interval is the interval $(0,1)$. Sets homeomorphic to the unit interval will be called $O$-sets, while sets homeomorphic to the half-open unit interval $[0,1)$ will be called $H$-sets. Intervals of either types will be called $I$-sets. Charts to such intervals are correspondingly called $O$-charts, $H$-charts and $I$-charts.

Due to the above theorem, any $1$-manifold can be covered in $O$-charts, by taking the open balls $(a -R, a + R)$ and rescaling them via

$$g(x) = \frac{1}{2} (\frac{x - a}{R} + 1)$$

Via similar proofs, a $1$-manifold with boundaries can be covered with $H$-charts.

Definition : An open subinterval is upper if it is of the form $(a,1)$, and lower if it is of the form $(0,b)$. It is outer if it is either upper or lower.

Theorem : An outer interval $I$ contains a sequence $(x_n)_{n \in \mathbb{N}}$ without a limit point in $(0,1)$.

Proof : Simple sequences of $(0,1)$ which do not converge to a point in $(0,1)$ are, for instance,

\begin{eqnarray} y_n &=& \frac{1}{n}\\ z_n &=& 1 - \frac{1}{n+1} \end{eqnarray}

which converge respectively to $0$ and $1$. Taking some sequences $y_n$, $z_n$ that converge to either value, we can consider the subsequences such that, if the interval has $0$ as an infimum, we consider

$$y'_n = \frac{1}{[n\text{th number such that 1/n \in I }]}$$

As this is a converging sequence, there exists an $N$ such that for every $n > N$, $x_n < b$, so this is equivalent to the subsequence $(y_n)_{n \in [N...]}$., which also converges to $0$. A similar method is used for $z'_n$ and upper subintervals.

## 2. A few basic $1$-manifolds

The empty set $\varnothing$ is, as with every dimension, a vacuous example of a $1$-manifold, with the topology $\varnothing$ and the atlas $\varnothing$. While a valid $1$-manifold, it is generally disregarded.

### 2.1. The real line

The simplest $1$-manifold is the real line, which can be simply constructed by using the identity map over the reals.

$$\mathbb{R}_M = \{ (\mathbb{R}, \mathrm{Id}) \}$$

It can be given the more complete atlas

$$\mathbb{R}_M = \{ (\mathbb{R}, \mathrm{Id}) \} \cup \{ ((a,b), \mathrm{Id}) | \forall a, b \in \mathbb{R} \}$$

As it is second-countable, we can also use this chart which forms a countable basis of the topology :

$$\mathbb{R}_M = \{ ( B_{x,r}, \mathrm{Id}) | \forall x, r \in \mathbb{Q} \}$$

The real line inherits from $\mathbb{R}$ all of its important topological properties.

### 2.2. The $1$-sphere

The $1$-sphere, or circle, can be constructed a number of different ways, either by performing a gluing of two manifolds with boundaries homeomorphic to $[a,b]$, or by considering the manifold with two open sets mapping to $O_1 = (a_1,b_1), O_2 = (a_2, b_2)$, $U_1 = \phi_1(O_1)$, $U_2 = \phi_2(O_2)$ with some overlap \begin{eqnarray} \phi_1^{-1} (U_1 \cap U_2) &=& (b_1 - \varepsilon_{12}, b_1) \cup (a_1, a_1 + \varepsilon_{11})\\ \phi_2^{-1} (U_1 \cap U_2) &=& (b_2 - \varepsilon_{22}, b_1) \cup (a_2, a_2 + \varepsilon_{21}) \end{eqnarray}

The precise values of all those parameters doesn't matter much, as they can always be changed by homeomorphism, as long as their order is preserved.

### 2.3. The long ray

The long ray is a $1$-manifold that is not paracompact. It is constructed by considering the product

$$\omega_1 \times [0,1)$$

with $\omega_1$ the first uncountable ordinal. It is equipped with the lexicographical order $\leq$, that is

$$\forall (a,b), (a',b') \in \omega_1 \times [0,1), (a,b) < (a',b') \Leftrightarrow (a < a' \vee (a = a' \wedge b \leq b'))$$

It is then equipped with the order topology of this order, and the $0$ element ($\{0 \} \times \{0\}$) removed.

Theorem : The long ray is a manifold.

Proof : There is a function mapping points of the long ray to $H$-intervals by considering

\begin{eqnarray} J : \mathbb{LR} &\to& [0,1)\\ (\omega, x) \to x \end{eqnarray}

### The long line

The long line is constructed similarly to the real line from two half-lines : simply take the adjunction of two closed long rays. It can be done simply by taking two copies of the long ray and adding a single chart to bridge the two.

### The line with two origins

The line with two origins is constructed from two (or arbitrarily many, for a line with arbitrarily many origins) of $\mathbb{R}$

$$A = \bigsqcup_{i \in I} \mathbb{R}$$

before identifying them for any non-zero value. That is, we take consider the equivalence relation

$$\forall a, b \in A, a \sim b \Leftrightarrow \exists x \neq 0 \in \mathbb{R}, a = (i, x) \wedge b = (j, x)$$

So that $a$ and $b$ have the same value in $\mathbb{R}$ outside of their index $i$ and $j$, except for $0$. The line with multiple origins is then the quotient by that equivalence relation

$$\mathbb{R}_{\div} = A / \sim$$

If $I$ is a set of cardinality $n$, we call it the line with $n$ origins. Due to this, it's already easy to see that there will be an uncountable number of $1$-manifolds.

## Structures on $1$-manifolds

As with all manifolds, a number of structures can be defined on $1$-manifolds, with the added benefit of being unique in quite a variety of cases.

### Differential structures

A differential structure

### Metric

If a Riemannian metric tensor can be defined on a $1$-manifold, it is always the flat metric (up to diffeomorphism). This can easily be seen by considering the expression of the metric in a coordinate patch with coordinate $x$,

\begin{equation} ds^2 = f(x) dx^2 \end{equation}

with $f(x)$ a nowhere vanishing positive function to keep the metric Riemannian, in which case one can define a new coordinate patch via

\begin{equation} x' = \int \frac{dx}{\sqrt{f(x)}} \end{equation}

### Clifford algebra and spinors

The Clifford algebra in $1$ dimension is easy enough to work out. The number of $p$-forms is simply the $0$-forms and $1$-forms, so that any polyvector can be written as

\begin{equation} A = a_0 + a_1 dx \end{equation}

and, from $v^2 = Q(v)$,

\begin{eqnarray} AB &=& (a_0 + a_1 dx)(b_0 + b_1 dx)\\ &=& (a_0 b_0 + a_1 b_1) + (a_0 b_1 + a_1 b_0) dx \end{eqnarray}

## Classification of $1$-manifolds

Depending on the definition of a manifold, the classification of $1$-manifolds can be either fairly trivial or almost impossible.

### Manifolds as Hausdorff, second-countable spaces

Theorem : A connected, Hausdorff, second-countable $1$-manifold that contains two $O$-charts $U,V$ such that $U \cap V$ has two disconnected components is homeomorphic to the circle.

Proof : If we call the two components $W$ and $Z$, let's pick two $O$-mappings $\phi$ and $\psi$ such that $\phi(W)$ and $\psi(W)$ are lower and $\phi(Z)$ and $\psi(Z)$ are upper.

There are only two Hausdorff, second-countable $1$-dimensional manifolds without boundaries (up to homeomorphism), which are the line and the circle.

## Bibliography

1. David Gale, The Classification of $1$-Manifolds : A Take-Home Exam, 1987
2. $1$-manifolds

Last updated : 2020-02-03 10:50:16