For the rest of the note we will analyse ${\pi }_1({\mathbb{S}}^1,1)$ on a close examination of the expoontial quotient map $\epsilon :\mathbb{R}\to {\mathbb{S}}^1\subseteq \mathbb{C}$ defined $\epsilon (r):=\exp (2\pi ir)$.

Lifting Properties of the Circle

Def: If $B$ is a topological space and $\phi :B\to {\mathbb{S}}^1$ is a continuous map a lift of $\phi$ is a continuous map $\tilde{\phi }:B\to \mathbb{R}$ such that the diagram commutes

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\begin{tikzcd}[row sep=2cm, column sep=2cm]
     & \mathbb{R}\arrow[d,"\varepsilon"] \\
     B\arrow[r,"\varphi"']\arrow[ur,dashed,"\tilde\varphi"] & \mathbb{S}^1
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Prop: Each point $z\in {\mathbb{S}}^1$ has a neighbourhood $U$ such that ${\epsilon }^{-1}[U]$ is a countable union of disjoint intervals ${\tilde{U}}_n$ with the property that $\epsilon {|}_{{\tilde{U}}_n}:{\tilde{U}}_n\to U$ is a homeomorphism.

Local Section Property of the Circle: Let $U\subseteq {\mathbb{S}}^1$ be any evenly covered open subset. For any $z\in U$ and any $r$ in the fiber of $\epsilon$ over $z$, there is a local section $\sigma$ of $\epsilon$ over U$ sich that $\sigma (z)=r$.

Unique Lifting Property of the Circle: Let $B$ be a connected topological space. Suppose $\phi :B\to {\mathbb{S}}^1$ is continuos and ${\tilde{\phi }}_1,{\phi }_2:B\to \mathbb{R}$ are lifts of $\phi$ that agree at some point of $B$. Then ${\tilde{\phi }}_1$ is equal to ${\tilde{\phi }}_2$.

Path Lifting Property of the Circle Suppose $f:I\to {\mathbb{S}}^1$ is any path, and $r_0\in \mathbb{R}$ is any point in the fiber of $\epsilon$ over $f(0)$. Then there exists a unique lift $\tilde{f}:I\to \mathbb{R}$ of $f$ such that $\tilde{f}(0)=r_0$, and any other lifft differs from $\tilde{f}$ by an addition of an integer constant.

Homotopy Lifting Property of the Circle: Let $B$ be a locally connected topological space. Suppose that ${\phi }_0,{\phi }_1:B\to {\mathbb{S}}^1$ are continuous maps, $H:B\times I\to {\mathbb{S}}^1$ is a homotopy from ${\phi }_0$ to ${\phi }_1$, and ${\phi }_0:B\to \mathbb{R}$ is any lift of ${\phi }_0$. Then there exists a unique lift $H$ to a homotopy $\tilde{H}$ satisfying ${\tilde{H}}_0={\tilde{\phi }}_0$. If $H$ is stationary on some subset $A\subseteq B$, then so is $\tilde{H}$.

Path Homotopy Criterion for the Circle: Suppose $f_0$ and $f_1$ are paths in ${\mathbb{S}}^1$ with the same initial point and the same terminal point, and ${\tilde{f}}_0,{\tilde{f}}_1:I\to \mathbb{R}$ are lifts of $f_0$ and $f_1$ with the same initial point. The $f_0\sim f_1$ iff ${\tilde{f}}_0$ and \tilde $f_1$ have the same terminal point.

Fundamental Group

Def: Suppose $f:I\to {\mathbb{S}}^1$ is a loop based at a point $z_0\in {\mathbb{S}}^1$. If $\tilde{f}:I\to \mathbb{R}$ is any lift of $f$ then we define the winding number of $f$ to be $N(f):=\tilde{f}(1)-\tilde{f}(0)$. Note that it is well defined, by the Path Lifting property of the circle.

Prop: A rotation of ${\mathbb{S}}^1$ is a map $\rho :{\mathbb{S}}^1\to {\mathbb{S}}^1$ of the form $\rho (z)e^{i\theta }z$ for some fixed $e^{i\theta }\in {\mathbb{S}}^1$. Then $N(\rho \circ f)=N(f)$ for every loop $f\in {\mathbb{S}}^1$ and every rotation $\rho$.

Homotopy Classification of Loops in ${\mathbb{S}}^1$ Two loops in ${\mathbb{S}}^1$ based at the same point are path-homotopy iff they have the same winding number.

Let $\omega :I\to {\mathbb{S}}^1$ be the loop $\omega (s)=e^{2\pi is}$ based at $1$, which traverses the circle once counterclockwise at constant speed.

Fundamental Group of the Circle: The group ${\pi }_1({\mathbb{S}}^1,1)$ is an infinite cyclic group generated by $[\omega ]$, and thus ${\pi }_1({\mathbb{S}}^1,1)\cong \mathbb{Z}$.

Fundamental Group of the Punctured Plane: The fundamental group ${\pi }_1(\mathbb{C}\setminus \{0\},1)$ is an infinite cyclic group generated by the path class of the loop $\omega$.

Def: If $f:I\to \mathbb{C}\setminus \{0\}$ is any loop, we define the winding number of $f$ to be the winding number of the loop $r\circ f$ where $r:\mathbb{C}\setminus \{0\}\to {\mathbb{S}}^1$ is the retraction $r(z)=z/|z|$.

Classification of Loops in the Punctured Plane: Two loops in $\mathbb{C}\setminus \{0\}$ based at the same point are path-homotopic iff they have the same winding number.

Fundamental Groups of Tori: Let ${\mathbb{T}}^n:={\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1$ be the $n$-dimensional torus with $p=(1,\dots ,1)$ as a base point, and for each $j=1,\dots ,n$ let ${\omega }_j$ denoted the standard loop in the $j$th copy of ${\mathbb{S}}^1$: $$\omega_j (s) = (1,\dots, 1,e^{2\pi i s},1\dots, 1).$$The map $\phi :{\mathbb{Z}}^n\to {\pi }_1({\mathbb{T}}^n,p)$ given by $\phi (k_1,\dots ,k_n)=[\omega {]}^{k_1}\cdots [{\omega }_n{]}^{k_n}$$ is an isomorphism, and ${\pi }_1({\mathbb{T}}^n,p)\cong {\mathbb{Z}}^n$.

Cor: For $n\ge 2$, the $n$-sphere is not homeomorphic to the $n$-torus.

Degree Theory for the Circle

Def: If $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ is a continuous map, we define the degree of $\phi$ to be the winding number of the loop $\phi \circ \omega$. This integer is denoted by $\mathrm{deg}\phi$.

For any continuous map $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$, let ${\rho }_{\phi }:{\mathbb{S}}^1\to {\mathbb{S}}^1$ be the rotation that takes $\phi (1)\mapsto 1$, namely ${\rho }_{\phi }(z)=z/\phi (1)$.

Prop: Suppose $f:I\to \mathbb{C}\setminus \{0\}$ is continuously differentiable loop. Then the winding number is given by $$N(f)= \frac1{2\pi i} \int_0^1 \frac{f'(s)}{f(s)}, ds.$$
Def: A vector field on ${\mathbb{R}}^n$ is a continuous map $V:{\mathbb{R}}^n\to {\mathbb{R}}^n$. If $V$ is a vector field, a point $p\in {\mathbb{R}}^n$ is called a singular point of $V$ if $V(p)=0$, and a regular point if $V(p)\ne 0$. A singular point is isolated if it has neighbourhood no other singular points. Suppose $V$ is a vector field on ${\mathbb{R}}^n$, and let ${\mathcal{R}}_V\subseteq {\mathbb{R}}^2$ denote the set of regular points of $V$. For any loop $f:I\to {\mathcal{R}}_V$, define the winding number of $V$ around $f$, denoted by $N(V,f)$, to be the winding number of the loop $V\circ f:I\to \mathbb{R}\setminus \{0\}$.

• $N(V,f)$ depends only on the path class of $f$.
• Suppose $p$ is an isolated singular point of $V$. Then $N(V,f)$ is independent of $\epsilon$ for $\epsilon$ sufficiently small, where $f_{\epsilon }(s)=p+\epsilon \omega (s)$, and $\omega$ is the standard counterclockwise loop around the unit circle. This integer is called the index of $V$ at $p$, and is denoted by $\text{Ind}(V,p)$.
• If $V$ has finitely many singular points in the closed unit disk, all in the interior, then $N(V,\omega )$ is equal to the sum of indices of $V$ at the interior singular points.

Lemma: If $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ is continuous, the degree of $\phi$ is equal to the degree of the endomorphism $({\rho }_{\phi }\circ \phi {)}_{\ast }$. In particular, if $\phi (1)$, then $\mathrm{deg}\phi =\mathrm{deg}{\phi }_{\ast }$.

Properties of Degree:

• Homotopic continuous maps have the same degree.
• If $\phi ,\psi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ are continuous maps, then $\mathrm{deg}(\psi \circ \phi )=(\mathrm{deg}\psi )(\mathrm{deg}\phi )$.

Degrees of Maps:

• The identity map of the circle has degree $1$
• Every constant map has degree $0$.
• Every rotation has degree $1$.
• For each $n\in \mathbb{Z}$, let $p_n:{\mathbb{S}}^1\to {\mathbb{S}}^1$ be the $n$th power map, defined in complex notation $p_n(z):=z^n$, we see that $p_n$ has degree $n$.
• The conjugation map $c:{\mathbb{S}}^1\to {\mathbb{S}}^1$, given by $c(z)=\overline{z}$. We see that $c=p_{-1}$, and thus have degree $-1$
• The antipodal map is the map $\alpha :{\mathbb{S}}^1\to {\mathbb{S}}^1$ is given by $\alpha (z):=-z$, is a rotation, and thus have degree $1$.

Homotopy Classification of Maps of the Circle: Two continuous maps from ${\mathbb{S}}^1$ to itself are homotopic iff they have the same degree.

Th: Let $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$. If $\mathrm{deg}\phi \ne 0$, then $\phi$ is surjective.

Th: Let $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ be continuous. If $\mathrm{deg}\phi \ne 1$, then $\phi$ has a fixed point.

Th: Let $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ be continuous. If $\mathrm{deg}\phi \ne \pm 1$, then $\phi$ is not injective.

Prop: Let $\phi ,\psi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ be continuous. If $\mathrm{deg}\phi \ne \mathrm{deg}\psi$, then there's a $z\in {\mathbb{S}}^1$ such that $\phi (z)=-\psi (z)$.

Invariance of Dimension, $2$-dimensional Case: A non empty topological space cannot be a $2$-manifold and an $n$-manifold for some $n>2$.

Invariance of the Boundary, $2$-dimensional Case: Suppose $M$ is a $2$-dimensional manifold with boundary. A point of $M$ cannot be both a boundary point and interior point.

Prop: A continuous map $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ has an extension to a continuous map $\mathrm{\Phi }:{\overline{\mathbb{B}}}^2\to {\mathbb{S}}^1$ iff it has degree $0$.

The Brouwer Fixed Point Theorem, $2$-dimensional Case: Every continuous map $f:{\overline{\mathbb{B}}}^2\to {\overline{\mathbb{B}}}^2$ has a fixed point.

Homotopy Classification of Torus Maps: For each continuous map $\phi :{\mathbb{T}}^2\to {\mathbb{T}}^2$, there is a $2\times 2$ integer matrix $D(\phi )$, with the following properties:

• Two continuous maps $\phi$ and $\psi$ are homotopic off $D(\phi )=D(\psi )$.
• $D(\psi \circ \phi )$ is equal to the matrix product $D(\psi )D(\phi )$
• For every $2\times 2$ integer matrix $E$, there is a continuous map $\phi :{\mathbb{T}}^2\to {\mathbb{T}}^2$ such that $D(\phi )=E$.
• If $\phi$ is homotopic to a homeomorphism iff $D(\phi )$ is invertible over the integers.

Even and Odd Functions

Def: Let $f:{\mathbb{S}}^1\to {\mathbb{S}}^1$ be a continuous functions. We say $f$ is odd if $f(-z)=f(z)$ for all $z\in {\mathbb{S}}^1$, and even if $f(z)=f(-z)$.

Prop: Every odd map has an odd degree. This result relies on a more general covering maps theory.

Prop: Every even map has an even degree,