1. Fundamental groups

We first show that the loops with some fixed basepoint form a group under certain defined multiplication.

In other words, if two paths can continuously deformed to each other, they are homotopic.

One can prove that homotopy is an equivalence relation; in other words, it satisfies the following three properties:

• $\alpha\sim\alpha$
• $\alpha\sim\beta\implies\beta\sim\alpha$
• $\alpha\sim\beta,\beta\sim\gamma\implies\alpha\sim\gamma$

  Let $X$ be a topological space. The set of homotopy classes of loops at $x_{0}\in X$ is denoted by $\pi_{1}(X,x_{0})$, and is called the fundamental group (or 1st homotopy group) of $X$ at $x_{0}$. The product of homotopy classes $[\alpha],[\beta]$ is defined by

  $$\begin{equation} [\alpha]*[\beta]=[\alpha*\beta] \end{equation}$$

2.General properties of fundamental groups

Arcwise connected: there exists a path from any point to any point
Simply connected: any loop in X can be continuously shrunk to a point

2.1. Arcwise connectedness

According to the theorem, for arcwise connected $X$, we do not need t o specify the basepoint, and simply rewrite as $\pi_{1}(X)$.

2.2. Homotopic invariance of fundamental groups

The homotopic equivalence of paths and loops can be generalized to arbitary maps. If $f,g:X\to Y$ are continuous maps, if there exists continuous map $F:X\times I\to Y$ such that $F(x,0)=f(x),F(x,1)=g(x)$, then $f$ is said to be homotopic to $g$, denoted $f\sim g$. The map $F$ is called a homotopy between $f$ and $g$.

A fundamental group is invariant under homeomorphisms and hence is a topological invariant.

Fundamental groups are a weak classification of topological spaces compared with homeomorphisms. If X and Y have different fundamental groups, they are not homeomorphic.

$R$ is a retract but not deformation retract of $X$, since the hole in $X$ is an obstruction to continuous deformation of $id_{x}$ to retraction.

3 Fundamental group of torus

4. Fundamental groups of polyhedra

Free groups: a subset of a group $G$ is a free set of generators if any $g\in G$, $g\neq e$ can be uniquely written as $g=x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$, $n$ finite and $i_{k}\in\mathbb{Z}$. Adjacent x is not equal and $x^{0}$ term should be dropped. If so $G$ is called free group.
Given a set of generators $X$, we can construct a free group.The set of all reduced words form a well-defined free group denoted as $F[X]$. For example, $F[\{a\}]=\mathbb{Z}$.
For a group $G$ generated by $X$, any element $g\in G$ can be written as $g=x_{1}^{i_{1}}\dots x_{n}^{i_{n}}$. $G$ is not necessarily free. Let $F[X]$ be the free group generated by $X$, define a natural homomorphism $\varphi:F[X]\to G$,

$$\begin{equation} x_{1}^{i_{1}}\dots x_{n}^{i_{n}}\to x_{1}^{i_{1}}\dots x_{n}^{i_{n}}\in G \end{equation}$$

LHS is not unique. so $F[X]/ker \varphi \cong G$.

As noted earliear, the polyhedron $|K|$ is a nice approximation of a given topological space $X$ within a homeomorphism. Since fundamental groups are topological invariants, $\pi_{1}(X)=\pi_{1}(|K|)$. We assume that $X$ is arcwise connected space, drop base point. The problem becomes calculation of $\pi_{1}(|K|)$.

Define the edge path in simplicial complex $K$ as a sequence of vertices $v_{0}\dots v_{i}v_{i+1}\dots v_{k}$ of $|K|$. Each adjacent point pairs are a 1-simplex. We allow that $v_{i}=v_{i+1}$, in this case $v_{i}v_{i}$ is a 0-simplex. If $v_{0}=v_{k}=v$, then the path is a loop.

We classify loops into equivalence classes according to some equivalence relation.

Define the product operation of edge paths

$$\begin{equation} \left\{v u_{1}\dots u_{k-1}v\right\}*\left\{v v_{1}\dots v_{i-1}v\right\}=\left\{v u_{1}\dots u_{k-1} v v_{1}\dots v_{i-1}v\right\} \end{equation}$$

The unit element is $\{v\}$ and inverse is defined as

$$\begin{equation} \left\{v u_{1}\dots u_{k}v\right\}^{-1} = \left\{v u_{k}\dots u_{1} v\right\} \end{equation}$$

The group is claled edge group of $K$ at $v$, denoted as $E(K;v)$.

Given an arcwise-connected simplicial complex $K$, there always exists subcomplex $L$ satisying:

• $L$ contains all vertices in $K$;
• The polyhedron $|L|$ is arcwise connected and simply connected.
  A 1d simplicial complex that is arcwise connected and simply connected is called a tree, A tree $T_{M}$ is a maximal tree of $K$ if it is not a proper subset of other trees.
  \fcolorbox{pink}{}{Lemma 4.3} A maximal tree contains all vertices of $K$ and satisfies the two conditions above.
  \fcolorbox{pink}{}{End Lemma}
  Suppose we obtained the subcomplex $L$. Since $|L|$ is simply connected, edge loops in $|L|$ do not contribute to $E(K;v)$. We can ignore the simplexes in $L$ in our calculation. Let $v_{0}=v,v_{1},\dots,v_{n}$ be vertices of $K$. Assign an object $g_{ij}$ for each ordered pair of vertices $v_{i},v_{j}$ if $\left<v_{i}v_{j}\right>$ is a 1-simplex of $K$. Let $G(K;L)$ be a group generated by all $g_{ij}$.
• Assign $g_{ij}=1$ if $\left<v_{i}v_{j}\right>\in L$
• If $\left<v_{i}v_{j}v_{k}\right>$ is a 2-simplex in $K$, there are no non-trivial loops around $v_{i}v_{j}v_{k}$, so we have $g_{ij}g_{jk}g_{ki}=1$.
  The generators $\{g_{ij}\}$ and the relations determined group $G(K;L)$.

Some more specific relations:

• $g_{ij}g_{ji}=g_{ii}=1$, so $g_{ji}=g_{ij}^{-1}$, we only assign $g_{ij}$ to $v_{i},v_{j}$ such that $\left<v_{i},v_{j}\right> \in K-L$, $i<j$

The rule is summarized as follows:

• 找一个triangulation $f:|K|\to X$
• 找到subcomplex $L$使得其道路连通，单连通且包含$K$的所有顶点
• 为每个$\left<v_{i},v_{j}\right> \in K-L$, $i<j$的1-simplex分配generator $g_{ij}$
• 为每个$\left<v_{i}v_{j}v_{k}\right>$的2-simplex施加关系$g_{ij}g_{jk}=g_{ik}$。如果$v_{i},v_{j},v_{k}$中的两个形成$L$的1-simplex，则对应生成元设置为1.
• 之后$\pi_{1}(X)$同构于$G(K;L)$，$G(K;L)$由$\{g_{ij}\}$生成。

  $G(K;L)$ is isomorphic to $E(K;v)\simeq \pi_{1}(|K|;v)$

定义$K^{(2)}$为$K$的2-skeleton，如果其只包含$K$的全部0,1,2-simplex. 我们有

$$\begin{equation} \pi_{1}(|K|) \cong \pi_{1}(|K^{(2)}|) \end{equation}$$

例如，$\left| K \right|$是三维球$D^{3}$的多面体，其边界$|L|$是$S^{2}$的多面体。由于$D^{3}$可收缩(contractible)，$\pi_{1}(|K|)\cong \{e\}$. 因此$\pi_{1}(S^{2})\cong\pi_{1}(K^{(2)})\cong \pi_{1}(|K|)\cong\{e\}$. In general, $n\geq 2$, $(n+1)$-simplex $\sigma_{n+1}$ and $\partial \sigma_{n+1}$ has the same 2-skeleton. If we find that $\sigma_{n+1}$ is contractible, and $\partial\sigma_{n+1}$ is a polyhedron of $S^{n}$, we have $\pi_{1}(S^{n})\cong\{e\},n\geq 2$

Relation between $H_{1}(K)$ and $\pi_{1}(|K|)$

For example, if $\pi_{1}(\text{2-bouquet})=(x,y;\emptyset)$, then $H_{1}(K)\cong \pi_{1}(\text{2-bouquet}) / F \cong (x,y;xyx^{-1}y^{-1})=\mathbb{Z}\oplus \mathbb{Z}$

5. Higher homotopy groups

The fundamental group classifies the homotopy classes of loops in a topological space $X$. We can also classify homotopy classes of the spheres in $X$ or homotopy classes of the tori in $X$. It turns out that the homotopy classes of $S^{n}(n\geq 2)$ form a group similar to the fundamental group.

Next we define group operations. $\alpha*\beta$ is defined by

$$\begin{equation} \alpha*\beta(s_{1},\dots,s_{n})=\begin{cases} \alpha(2s_{1},\dots,s_{n}), & 0\leq s_{1}\leq \frac{1}{2} \\ \beta(2s_{1}-1,\dots,s_{n}), & \frac{1}{2}\leq s_{1} \leq 1 \end{cases} \end{equation}$$

and

$$\begin{equation} \alpha ^{-1}(s_{1},\dots,s_{n})=\alpha(1-s_{1},\dots,s_{n}) \end{equation}$$

We have $\alpha ^{-1}*\alpha\sim c_{x_{0}}$, where $c_{x_{0}}$ is a constant $n$-loop, $c_{x_{0}}(s_{1},\dots,s_{n})=x_{0}$.

Let us consider $\pi_{0}(X,x_{0})$. In this case $I^{0}=\{0\}$ and $\partial I^{0}=\emptyset$, Let $\alpha,\beta:\{0\}\to X$ be such that $\alpha(0)=x,\beta(0)=y$. We define $\alpha\sim\beta$ if there exists continuous map $F:\{0\}\times I\to X$ such that $F(0,0)=x,F(0,1)=y$. This shows that $\alpha\sim\beta$ iff $x,y$ are connected by a curve in $X$. This equivalence relation is independe-nt of $x_{0}$ and denotes the number of arcwise connected components of $X$.

6. General properties of higher homotopy groups

1. Abelian: $\alpha * \beta \sim \beta * \alpha$, or $[\alpha]*[\beta]=[\beta]*[\alpha]$
2. If a topological space $X$ is arcwise connected, $\pi_{n}(X,x_{0})\cong \pi_{n}(X,x_{1})$ for any $x_{0},x_{1}\in X$
3. If $f:X\to Y$ is a homotopy equivalence, then $\pi_{n}(X,x_{0})\cong\pi_{n}(Y,f(x_{0}))$. If $X$ is contractible, then $\pi_{n}(X,x_{0})=\{e\},n>1$.
4. Let $X,Y$ be arcwise connected topological spaces, then $\pi_{n}(X\times Y)\cong \pi_{n}(X)\oplus \pi_{n}(Y)$
5. Let $X,Y$ be arcwise connected topological spaces, then$\pi_{n}(X\times Y) \cong \pi_{n}(X)\oplus \pi_{n}(Y)$

8. Orders in condensed matter system

1. Order parameter

Let $H$ be a Hamiltonian describing a condensed matter system. Assume $H$ is invariant under a certain symmetry operation. The ground state need not preserve the symmetry of $H$. If so the system undergoes spontaneous symmetry breakdown.
Consider heisenberg hamiltonian

$$\begin{equation} H=-J \sum_{(i,j)} \mathbf{S}_{i}\cdot \mathbf{S}_{j} + \mathbf{h}\cdot \sum_{i} \mathbf{S}_{i} \end{equation}$$

which describes $N$ ferromagnetic Heisenberg spins $\{\mathbf{S}_{i}\}$, where $(i,j)$ sums over nearest neighbours, $h$ is the external magnetic field. The free energy $F$ is defined by $\exp(-\beta F)=Z$, and average magnetization ($\beta?$)

$$\begin{equation} \mathbf{m} = \frac{1}{N} \sum_{i}\left<\mathbf{S}_{i}\right> = \frac{1}{N\beta} \frac{ \partial F }{ \partial \mathbf{h} } \end{equation}$$

Consider $\mathbf{h}\to0$ limit. Although $H$ is invariant under $SO(3)$ rotations for all $S_{i}$ in this limit, under large $\beta$ limit, the system does not observe $SO(3)$ symmetry. It is called spontaneous magnetization. The maximum temperature such that $\mathbf{m}\neq 0$ is called critical temperature.
$\mathbf{m}$ is the order parameter describing the phase transition between ordered state ($\mathbf{m}\neq 0$) and disordered state ($\mathbf{m}\neq 0$).

• For small $T$, minimizing $F$ means minimizing $\left<H\right>$, the spins tend to be in the same direction
• For large $T$, minimizing $F$ means maximizing $S$, the spin tend to be disordered.
  Consider the direction of $\mathbf{m}$. In ground state, $\mathbf{m}$ is expected to be independent of position $\mathbf{x}$ at ground state. Use $(\theta,\phi)$ to describe direction of $\mathbf{m}$. In excited states the function $\mathbf{m}(\mathbf{x})$ depends on $\mathbf{x}$ and cannot be obtained by small perturbations from ground state. For $2d$, the Heisenberg ferromagnet may admit an excitation called Belavin-Polyakov monopole. The $m$ approaches constant vector so the energy does not diverge. This configuration cannot be deformed to uniform one with $\mathbf{m}$ far from origin kept fixed. These excitations are called topological excitations. The $\mathbf{m}(\mathbf{x})$ defines map $S^{2}\to S^{2}$ and are classified by $\pi_{2}(S^{2})=\mathbb{Z}$.

10 Textures in Superfluid He3-A

From NMR and other observations, the superfluid is in the spin-triplet, p-wave state. Instead of field operators $\Psi_{\alpha\beta}(\vec{x})=\Delta ^{\mu}(\vec{x}) i(\sigma_{\mu}\cdot\sigma_{2})_{\alpha\beta}$

The most general form of triplet superfluid order parameter is

$$\begin{equation} \left<c_{\alpha,\vec{k}}c_{\beta,-\vec{k}}\right> \propto \sum_{\mu=1}^{3} (i\sigma_{2}\sigma_{\mu}) _{\alpha\beta} d_{\mu}(\vec{k}) \end{equation}$$

where $\alpha,\beta$ are spin indices. The Cooper pair forms in the p-wave state, so $d_{\mu}(\vec{k})\propto Y_{1m}\sim k_{i}$,

$$\begin{equation} d_{\mu}(\vec{k})= \sum_{i=1}^{3} \Delta_{0}A_{\mu i}k_{i} \end{equation}$$

The A-phase order parameter takes the form $A_{\mu i}= d_{\mu}(\Delta_{1}+i\Delta_{2})_{i}$, $\vec{d}$ is a unit vector along which the spin projection of Cooper pair vanishes. $(\Delta_{1},\Delta_{2})$ is a pair of orthonormal unit vectors. $\vec{d}$ takes value from $S^{2}$. Let $\vec{l}\equiv\Delta_{1}\times\Delta_{2}$, the $(\Delta_{1},\Delta_{2},\vec{l})$ forms an orthonormal frame at each point of medium. Since any orthonormal frame is obtained from fixed frame by a rotation, the order parameter of He-3A is $S^{2}\times SO(3)$. The vector $\vec{l}$ is the axis of angular momentum of Cooper pair.

Neglect variation of $\hat{d}$ for simplicity (in fact, $\hat{d}$ is locked along $\hat{l}$ due to dipole force). The order parameter assumes the form $A_{i}=\Delta_{0}(\hat{\Delta}_{1}+\hat{\Delta}_{2})_{i}$, where $\hat{\Delta}_{1},\hat{\Delta}_{2},\hat{l}\equiv\hat{\Delta}_{1}\times\hat{\Delta}_{2}$ forms an orthonormal frame at each point of the medium. Take a standard frame $e_{1},e_{2},e_{3}$, the frame is obtained by applying $g\in SO(3)$ to $(e_{1},e_{2},e_{3})$. So $(\hat{\Delta}_{1}(\vec{x}),\hat{\Delta}_{2}(\vec{x}),\hat{l}(\vec{x}))$ defines a map $\psi:X\to SO(3)$, called texture of superfluid He-3. The relevant homotopy groups for classifying defects are $\pi_{n}(SO(3))$.