Homotopy Groups

Homotopy groups measure the topological complexity of a space by classifying maps from spheres into it, up to continuous deformation. The fundamental group $\pi_1$ detects loops that cannot be contracted, while higher homotopy groups $\pi_n$ detect higher-dimensional "holes." In physics, homotopy groups classify topological defects, instantons, and the global structure of gauge theories.

Historical Context

Henri Poincaré introduced the fundamental group in his 1895 "Analysis Situs," launching algebraic topology. He formulated the famous Poincaré conjecture: every simply connected closed 3-manifold is homeomorphic to $S^3$ (proved by Perelman in 2003).

Higher homotopy groups $\pi_n$ were defined by Witold Hurewicz (1935). Unlike $\pi_1$, they are always abelian for $n \geq 2$. Heinz Hopf's discovery (1931) that$\pi_3(S^2) = \mathbb{Z}$ (via the Hopf fibration) was completely unexpected—it showed that higher spheres can have nontrivial mappings between them.

In physics, the systematic use of homotopy for classifying topological defects was pioneered by Toulouse and Kléman (1976) and Mermin (1979), with applications to superfluids, liquid crystals, and cosmological defects.

Derivation 1: The Fundamental Group $\pi_1$

The fundamental group $\pi_1(X, x_0)$ consists of homotopy classes of loops based at $x_0$:

$\pi_1(X, x_0) = \{[\gamma] : \gamma: [0,1] \to X, \gamma(0) = \gamma(1) = x_0\}$

with group operation given by concatenation of loops and inverse by traversing in reverse.

Winding Number

For $X = S^1$, every loop has a well-defined winding number:

$\text{wind}(\gamma) = \frac{1}{2\pi}\oint_\gamma d\theta = \frac{1}{2\pi i}\oint \frac{dz}{z} \in \mathbb{Z}$

This establishes $\pi_1(S^1) \cong \mathbb{Z}$. The winding number is a homomorphism from loops to integers, preserved under homotopy.

SO(3) and spinors: $\pi_1(SO(3)) = \mathbb{Z}_2$ means there is a loop in rotation space that cannot be contracted (the "Dirac belt trick"). This necessitates the double cover $SU(2) \to SO(3)$ and explains why fermions pick up a minus sign under $2\pi$ rotation.

Derivation 2: Higher Homotopy Groups

The $n$-th homotopy group classifies maps from $S^n$ into $X$:

$\pi_n(X, x_0) = [(S^n, *), (X, x_0)] = \{[f: S^n \to X] : f(*) = x_0\}$

Key Properties

$\pi_n(X) \text{ is abelian for } n \geq 2$

Unlike $\pi_1$, which can be non-abelian (e.g., free groups)

$\pi_n(S^n) = \mathbb{Z}$ for all $n \geq 1$

Generated by the identity map (degree 1)

$\pi_k(S^n) = 0$ for $k < n$

Low-dimensional spheres cannot wrap around higher-dimensional ones

$\pi_3(S^2) = \mathbb{Z}$ (Hopf invariant)

Generated by the Hopf fibration—the first surprise in homotopy theory

Degree of a map: For $f: S^n \to S^n$, the degree$\deg(f) = \int_{S^n} f^*\omega / \int_{S^n} \omega$ (ratio of volumes) gives the homotopy class. Equivalently, it counts (with signs) the preimages of a regular value.

Derivation 3: The Long Exact Sequence of a Fibration

Given a fiber bundle $F \hookrightarrow E \xrightarrow{\pi} B$, there is a long exact sequence in homotopy:

$\cdots \to \pi_n(F) \xrightarrow{i_*} \pi_n(E) \xrightarrow{\pi_*} \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots$

Application: Hopf Fibration

For $S^1 \hookrightarrow S^3 \to S^2$:

$\cdots \to \pi_3(S^1) \to \pi_3(S^3) \to \pi_3(S^2) \to \pi_2(S^1) \to \pi_2(S^3) \to \pi_2(S^2) \to \pi_1(S^1) \to \pi_1(S^3) \to \pi_1(S^2) \to \cdots$

Substituting known values:

$\cdots \to 0 \to \mathbb{Z} \to \pi_3(S^2) \to 0 \to 0 \to \mathbb{Z} \to \mathbb{Z} \to 0 \to 0 \to \cdots$

Exactness at $\pi_3(S^2)$ gives $\pi_3(S^2) \cong \mathbb{Z}$: the Hopf invariant classifies maps $S^3 \to S^2$. The connecting map $\mathbb{Z} \to \mathbb{Z}$ at$\pi_2(S^2) \to \pi_1(S^1)$ is an isomorphism.

Physical application: For the gauge group breaking$G \to H$, the exact sequence of $H \hookrightarrow G \to G/H$ determines which topological defects exist: monopoles from $\pi_2(G/H)$, strings from $\pi_1(G/H)$, and domain walls from $\pi_0(G/H)$.

Derivation 4: The Hurewicz Theorem

The Hurewicz theorem connects homotopy to homology. If$X$ is $(n-1)$-connected (i.e., $\pi_k(X) = 0$ for $k < n$), then:

$\boxed{h: \pi_n(X) \xrightarrow{\cong} H_n(X, \mathbb{Z})}$

The Hurewicz map sends $[f: S^n \to X]$ to $f_*[S^n]$, the image of the fundamental class of $S^n$ under $f$.

Applications

$\pi_1(X)^{\text{ab}} \cong H_1(X, \mathbb{Z})$ always (abelianization of $\pi_1$)

$\pi_n(S^n) \cong H_n(S^n) = \mathbb{Z}$ (recovers the degree)

$\pi_2(\mathbb{CP}^n) \cong H_2(\mathbb{CP}^n) = \mathbb{Z}$ (since $\pi_1 = 0$)

Computational tool: Homology is generally much easier to compute than homotopy (it has excision, Mayer-Vietoris, etc.). The Hurewicz theorem lets us compute the first nontrivial homotopy group from homology. Beyond the Hurewicz dimension, the relationship becomes more subtle and requires spectral sequences.

Derivation 5: Topological Defects and Symmetry Breaking

When a symmetry group $G$ is broken to a subgroup $H$, the order parameter takes values in the vacuum manifold $\mathcal{M} = G/H$. Topological defects correspond to non-contractible maps from spheres surrounding the defect:

$\pi_0(G/H) \neq 0: \text{ domain walls (codimension 1)}$

$\pi_1(G/H) \neq 0: \text{ cosmic strings / vortices (codimension 2)}$

$\pi_2(G/H) \neq 0: \text{ monopoles (codimension 3)}$

$\pi_3(G/H) \neq 0: \text{ textures / skyrmions}$

Example: GUT Monopoles

In Grand Unified Theories with $SU(5) \to SU(3) \times SU(2) \times U(1)$:

$\pi_2(SU(5)/[SU(3) \times SU(2) \times U(1)]) = \mathbb{Z}$

This predicts stable magnetic monopoles with mass $\sim M_{GUT}/\alpha \sim 10^{16}$ GeV ('t Hooft-Polyakov monopoles). Their non-observation is one motivation for cosmic inflation.

Kibble mechanism: During a cosmological phase transition with symmetry breaking, topological defects form at a density determined by the correlation length at the time of the transition. This was proposed by Tom Kibble (1976) and has been tested in condensed matter analogs (superfluid helium, liquid crystals).

Interactive Simulation

This simulation computes winding numbers for maps $S^1 \to S^1$, visualizes Hopf fibration fibers, plots the topological degree of maps between spheres, and displays vortex configurations corresponding to different homotopy classes.

Homotopy Groups: Winding Numbers, Hopf Fibration & Topological Defects

Python

script.py217 lines

#!/usr/bin/env python3
"""
homotopy_groups.py - Fundamental Group, Higher Homotopy, and Exact Sequences
Computes winding numbers, visualizes homotopy classes, and applies to physics
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# ============================================================
# 1. Fundamental group pi_1
# ============================================================
print("=" * 72)
print("  HOMOTOPY GROUPS: TOPOLOGICAL CLASSIFICATION OF MAPPINGS")
print("=" * 72)
print()

print("  FUNDAMENTAL GROUP pi_1")
print("  " + "-" * 50)
print()

spaces = [
    ("R^n", "0 (trivial)", "contractible"),
    ("S^1", "Z", "circle: winding numbers"),
    ("S^2", "0", "simply connected"),
    ("S^n (n>1)", "0", "simply connected"),
    ("T^2", "Z x Z", "two independent loops"),
    ("RP^2", "Z_2", "loop squares to identity"),
    ("SO(3)", "Z_2", "Dirac belt trick"),
    ("U(1)", "Z", "same as S^1"),
    ("SU(2)", "0", "same as S^3"),
]

print("  pi_1 of common spaces:")
for name, pi1, note in spaces:
    print("    {}: pi_1 = {} ({})".format(name, pi1, note))
print()

# ============================================================
# 2. Winding number computation
# ============================================================
print("  WINDING NUMBER COMPUTATION")
print("  " + "-" * 50)
print()

# Compute winding numbers for various maps S^1 -> S^1
t = np.linspace(0, 2*np.pi, 1000)

print("  Map f: S^1 -> S^1, f(t) = (cos(nt), sin(nt))")
print("  Winding number = (1/2pi) integral d(arg(f))")
print()

for n in range(-3, 4):
    x = np.cos(n * t)
    y = np.sin(n * t)
    # Compute winding number by integrating d(theta)
    dtheta = np.diff(np.unwrap(np.arctan2(y, x)))
    winding = np.sum(dtheta) / (2 * np.pi)
    print("    n = {}: winding number = {:.6f}".format(n, winding))
print()

# ============================================================
# 3. Higher homotopy groups
# ============================================================
print("  HIGHER HOMOTOPY GROUPS")
print("  " + "-" * 50)
print()

homotopy_data = [
    ("pi_2(S^2)", "Z", "Hopf degree"),
    ("pi_3(S^2)", "Z", "Hopf fibration"),
    ("pi_3(S^3)", "Z", "degree of map"),
    ("pi_4(S^3)", "Z_2", "exotic"),
    ("pi_n(S^n)", "Z", "degree (stable)"),
    ("pi_3(SU(2))", "Z", "instanton number"),
    ("pi_2(S^2/Z_2)", "Z", "monopole number"),
    ("pi_1(U(1))", "Z", "vortex number"),
]

print("  Key homotopy groups in physics:")
for group, value, application in homotopy_data:
    print("    {} = {} ({})".format(group, value, application))
print()

# ============================================================
# 4. Exact sequence for Hopf fibration
# ============================================================
print("  LONG EXACT SEQUENCE: HOPF FIBRATION")
print("  " + "-" * 50)
print()
print("  Fibration: S^1 -> S^3 -> S^2")
print("  Long exact sequence in homotopy:")
print("    ... -> pi_n(S^1) -> pi_n(S^3) -> pi_n(S^2) -> pi_{n-1}(S^1) -> ...")
print()
print("  n=3: pi_3(S^1) -> pi_3(S^3) -> pi_3(S^2) -> pi_2(S^1)")
print("       0 -> Z -> pi_3(S^2) -> 0")
print("       => pi_3(S^2) = Z  (the Hopf invariant)")
print()
print("  n=2: pi_2(S^1) -> pi_2(S^3) -> pi_2(S^2) -> pi_1(S^1)")
print("       0 -> 0 -> Z -> Z")
print("       The connecting map Z -> Z is multiplication by 0")
print("       (confirmed: pi_2(S^2) = Z)")
print()

# ============================================================
# 5. Defects classified by homotopy
# ============================================================
print("  TOPOLOGICAL DEFECTS IN PHYSICS")
print("  " + "-" * 50)
print()

defects = [
    ("Vortex (superfluid)", "pi_1(S^1) = Z", "quantized circulation"),
    ("Monopole (GUT)", "pi_2(G/H)", "topological magnetic charge"),
    ("Skyrmion (nuclear)", "pi_3(SU(2)) = Z", "baryon number"),
    ("Domain wall", "pi_0(Z_2) = Z_2", "discrete symmetry breaking"),
    ("Cosmic string", "pi_1(U(1)) = Z", "symmetry breaking in early universe"),
    ("Instanton", "pi_3(SU(2)) = Z", "vacuum tunneling"),
]

print("  Topological defects classified by homotopy groups:")
for name, group, meaning in defects:
    print("    {}: {} ({})".format(name, group, meaning))
print()

# ============================================================
# Visualization
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#111111')
    for spine in ax.spines.values():
        spine.set_color('#2dd4bf')
    ax.tick_params(colors='#99f6e4')
    ax.grid(True, color='#134e4a', alpha=0.4)

# Plot 1: Winding number maps
ax1 = axes[0, 0]
t_plot = np.linspace(0, 2*np.pi, 500)
colors = ['#f472b6', '#6b7280', '#2dd4bf', '#f59e0b', '#a78bfa']
for i, n in enumerate([-2, -1, 0, 1, 2]):
    x = np.cos(n * t_plot)
    y = np.sin(n * t_plot)
    ax1.plot(x, y, color=colors[i], linewidth=2, alpha=0.7,
             label='n = {}'.format(n))
ax1.set_aspect('equal')
ax1.set_title('Winding Number Maps S^1 -> S^1', color='#99f6e4', fontsize=12, fontweight='bold')
ax1.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4', fontsize=8)
ax1.set_xlabel('x', color='#5eead4')
ax1.set_ylabel('y', color='#5eead4')

# Plot 2: Hopf fibration fibers (stereographic projection)
ax2 = axes[0, 1]
phi_vals = np.linspace(0, 2*np.pi, 200)
for z2_phase in np.linspace(0, np.pi, 8):
    # Points on a Hopf fiber in S^3
    z1 = np.cos(z2_phase) * np.exp(1j * phi_vals)
    z2 = np.sin(z2_phase) * np.ones_like(phi_vals)
    # Stereographic projection from S^3 to R^3 (take first two coords)
    w = 1 / (1 - z2.real + 1e-6)
    x_proj = z1.real * w
    y_proj = z1.imag * w
    ax2.plot(x_proj, y_proj, linewidth=1.5, alpha=0.6)
ax2.set_xlim(-5, 5)
ax2.set_ylim(-5, 5)
ax2.set_aspect('equal')
ax2.set_title('Hopf Fibers (Stereographic)', color='#99f6e4', fontsize=12, fontweight='bold')
ax2.set_xlabel('x', color='#5eead4')
ax2.set_ylabel('y', color='#5eead4')

# Plot 3: Degree of map from winding integral
ax3 = axes[1, 0]
n_range = np.arange(-5, 6)
degrees = n_range.copy()
colors_bar = ['#f472b6' if d < 0 else '#34d399' if d > 0 else '#6b7280' for d in degrees]
ax3.bar(n_range, np.abs(degrees), color=colors_bar, alpha=0.7, edgecolor='#2dd4bf')
for i, (n, d) in enumerate(zip(n_range, degrees)):
    ax3.text(n, abs(d) + 0.15, str(d), ha='center', color='#99f6e4', fontsize=9)
ax3.set_xlabel('Map parameter n', color='#5eead4')
ax3.set_ylabel('|Degree|', color='#5eead4')
ax3.set_title('Topological Degree of f_n: S^1 -> S^1', color='#99f6e4', fontsize=12, fontweight='bold')

# Plot 4: Vortex configurations
ax4 = axes[1, 1]
x_grid = np.linspace(-3, 3, 20)
y_grid = np.linspace(-3, 3, 20)
X, Y = np.meshgrid(x_grid, y_grid)
for n in [1, 2]:
    R_grid = np.sqrt(X**2 + Y**2)
    theta_grid = np.arctan2(Y, X)
    # Vortex: order parameter phase = n * theta
    vx = -n * np.sin(n * theta_grid) / (R_grid + 0.3)
    vy = n * np.cos(n * theta_grid) / (R_grid + 0.3)
    if n == 1:
        ax4.quiver(X, Y, vx, vy, color='#2dd4bf', alpha=0.6, scale=15)
    else:
        ax4.quiver(X + 0.08, Y + 0.08, vx, vy, color='#f472b6', alpha=0.4, scale=15)
ax4.set_aspect('equal')
ax4.set_title('Vortex Configurations (n=1,2)', color='#99f6e4', fontsize=12, fontweight='bold')
ax4.set_xlabel('x', color='#5eead4')
ax4.set_ylabel('y', color='#5eead4')

plt.tight_layout(pad=2.0)
buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print('<img src="data:image/png;base64,' + img_b64 + '" alt="Homotopy Groups Visualization" />')
buf.close()
print()
print("Visualization complete.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Fundamental Group

$\pi_1$ classifies loops up to homotopy. For $S^1$: winding numbers ($\mathbb{Z}$). For $SO(3)$: the $\mathbb{Z}_2$ explaining spinor sign flips under $2\pi$ rotation.

Higher Homotopy Groups

$\pi_n(X)$ classifies maps $S^n \to X$. Always abelian for $n \geq 2$. The surprise$\pi_3(S^2) = \mathbb{Z}$ is generated by the Hopf fibration.

Exact Sequences

Fibrations yield long exact sequences connecting the homotopy groups of fiber, total space, and base. A powerful computational tool and classifier of symmetry-breaking defects.

Topological Defects

Homotopy groups of $G/H$ classify defects in ordered media: domain walls ($\pi_0$), vortices ($\pi_1$), monopoles ($\pi_2$), and skyrmions ($\pi_3$).

← Topology in Physics Next: Homology & Cohomology →