Homology & Cohomology

Homology and cohomology provide computable algebraic invariants of topological spaces. Simplicial and singular homology count "holes" of each dimension, while de Rham cohomology uses differential forms to detect topological features. The de Rham theorem establishes their equivalence for smooth manifolds, bridging topology and analysis.

Historical Context

Homology was introduced by Poincaré (1895) using simplicial decompositions. Emmy Noether (1925) revolutionized the subject by recognizing that homology groups are the correct algebraic structures, replacing Betti numbers and torsion coefficients. Samuel Eilenberg and Norman Steenrod axiomatized homology theories in their 1952 book, showing that many different constructions (simplicial, singular, cellular) all give the same invariants.

Georges de Rham proved his famous theorem (1931) that the cohomology defined by differential forms equals singular cohomology with real coefficients. This result bridges analysis and topology, allowing topological invariants to be computed by integrating differential forms.

In physics, de Rham cohomology is the natural framework for electromagnetism (closed but not exact forms correspond to magnetic charges) and for classifying conserved quantities via Noether's theorem. The BRST cohomology of gauge theories is a direct generalization.

Derivation 1: Simplicial Homology

Given a simplicial complex $K$, the chain complex is:

$\cdots \xrightarrow{\partial_{n+1}} C_n(K) \xrightarrow{\partial_n} C_{n-1}(K) \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\partial_1} C_0(K) \to 0$

where $C_n(K)$ is the free abelian group on oriented $n$-simplices and the boundary operator is:

$\partial_n[v_0, \ldots, v_n] = \sum_{i=0}^n (-1)^i [v_0, \ldots, \hat{v}_i, \ldots, v_n]$

The key property $\partial_{n-1} \circ \partial_n = 0$ (boundary of a boundary is zero) ensures that $\text{im}(\partial_{n+1}) \subseteq \ker(\partial_n)$. The homology groups are:

$\boxed{H_n(K) = \ker(\partial_n) / \text{im}(\partial_{n+1}) = Z_n / B_n}$

Elements of $Z_n = \ker(\partial_n)$ are cycles; elements of$B_n = \text{im}(\partial_{n+1})$ are boundaries. Homology measures cycles that are not boundaries.

Betti numbers: $b_n = \text{rank}(H_n)$ counts the number of independent $n$-dimensional holes. The Euler characteristic is the alternating sum:$\chi = \sum_n (-1)^n b_n$.

Derivation 2: de Rham Cohomology

On a smooth manifold $M$, the de Rham complex is:

$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \cdots$

The property $d^2 = 0$ gives the de Rham cohomology:

$H^k_{dR}(M) = \frac{\ker(d: \Omega^k \to \Omega^{k+1})}{\text{im}(d: \Omega^{k-1} \to \Omega^k)} = \frac{\text{closed } k\text{-forms}}{\text{exact } k\text{-forms}}$

Examples

$H^0_{dR}(M) = \mathbb{R}^{\pi_0(M)}$: locally constant functions (one copy of $\mathbb{R}$ per component)

$H^1_{dR}(S^1) = \mathbb{R}$: generated by $d\theta$ (closed but not exact globally)

$H^2_{dR}(S^2) = \mathbb{R}$: generated by the area form $\sin\theta\,d\theta \wedge d\phi$

Electromagnetism: Maxwell's equations in vacuum state that$dF = 0$ and $d*F = 0$. The first equation means $F$ is a closed 2-form. If $H^2_{dR}(M) = 0$, then $F = dA$ globally. But if $H^2 \neq 0$ (e.g., around a monopole), $F$ can represent a nontrivial cohomology class—this is the magnetic charge.

Derivation 3: The de Rham Theorem

The de Rham theorem establishes a canonical isomorphism:

$\boxed{H^k_{dR}(M) \cong H^k_{\text{sing}}(M, \mathbb{R})}$

The isomorphism is given by integration: a closed $k$-form$\omega$ defines a linear functional on $k$-cycles via:

$\sigma \mapsto \int_\sigma \omega$

Stokes's theorem ensures this is well-defined on cohomology/homology: if $\omega = d\alpha$(exact), then $\int_\sigma d\alpha = \int_{\partial\sigma} \alpha = 0$ for any cycle $\sigma$.

Hodge Theory Refinement

On a compact Riemannian manifold, the Hodge theorem provides a canonical representative for each cohomology class: the unique harmonic form:

$H^k_{dR}(M) \cong \mathcal{H}^k(M) = \{\omega : d\omega = 0, \delta\omega = 0\}$

where $\delta = (-1)^{nk+n+1}*d*$ is the codifferential. Harmonic forms are both closed and coclosed, and minimize the $L^2$ norm in their cohomology class.

Derivation 4: Poincaré Duality

For a closed oriented $n$-manifold $M$, Poincaré dualitygives a canonical isomorphism:

$H^k(M) \cong H_{n-k}(M)$

In de Rham terms, the pairing is given by the wedge product and integration:

$H^k(M) \times H^{n-k}(M) \to \mathbb{R}, \quad ([\alpha], [\beta]) \mapsto \int_M \alpha \wedge \beta$

This pairing is non-degenerate, establishing the isomorphism. In particular, $b_k = b_{n-k}$(the Betti numbers are palindromic).

Intersection Form

For a 4-manifold, the intersection form on $H^2(M^4)$:

$Q(\alpha, \beta) = \int_{M^4} \alpha \wedge \beta$

is a symmetric bilinear form that completely classifies simply connected closed 4-manifolds up to homeomorphism (Freedman, 1982). For smooth 4-manifolds, additional constraints from Donaldson theory apply.

String theory: Poincaré duality on a Calabi-Yau 3-fold gives the mirror symmetry between type IIA and IIB string theories: the interchange$H^{1,1} \leftrightarrow H^{2,1}$ exchanges Kähler and complex structure moduli.

Derivation 5: The Mayer-Vietoris Sequence

If $M = U \cup V$ with $U, V$ open, the Mayer-Vietoris sequence is a long exact sequence:

$\cdots \to H^k(M) \to H^k(U) \oplus H^k(V) \to H^k(U \cap V) \xrightarrow{\delta} H^{k+1}(M) \to \cdots$

Example: $H^*(S^n)$

Cover $S^n$ by two open hemispheres $U, V$ (each contractible, so $H^k = 0$ for $k > 0$), with $U \cap V \simeq S^{n-1}$:

$0 \to H^k(S^n) \to 0 \oplus 0 \to H^k(S^{n-1}) \xrightarrow{\cong} H^{k+1}(S^n) \to 0$

This gives the inductive computation: $H^k(S^n) \cong H^{k-1}(S^{n-1})$ for $0 < k \leq n$, confirming $H^k(S^n) = \mathbb{R}$ for $k = 0, n$ and zero otherwise.

Computational power: Mayer-Vietoris is the standard tool for computing cohomology by decomposing spaces. Together with homotopy invariance and the Künneth formula ($H^*(M \times N) \cong H^*(M) \otimes H^*(N)$), it covers most practical calculations.

Interactive Simulation

This simulation computes Betti numbers for common topological spaces, verifies the Gauss-Bonnet theorem via the Euler characteristic, demonstrates the simplicial chain complex for the tetrahedron, and plots how Betti numbers vary with genus.

Homology & Cohomology: Betti Numbers, de Rham & Poincare Duality

Python

script.py236 lines

#!/usr/bin/env python3
"""
homology_cohomology.py - Simplicial Homology, de Rham Cohomology, and Poincare Duality
Computes Betti numbers, de Rham groups, and verifies Poincare duality
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# ============================================================
# 1. Betti numbers of common spaces
# ============================================================
print("=" * 72)
print("  HOMOLOGY AND COHOMOLOGY: ALGEBRAIC TOPOLOGY TOOLKIT")
print("=" * 72)
print()

print("  BETTI NUMBERS OF COMMON SPACES")
print("  " + "-" * 50)
print()
print("  Betti number b_k = rank H_k(M, Z) = dim H^k_{dR}(M)")
print()

spaces_betti = [
    ("Point", [1]),
    ("S^1", [1, 1]),
    ("S^2", [1, 0, 1]),
    ("S^n", [1, 0, 0, "...", 0, 1]),
    ("T^2", [1, 2, 1]),
    ("T^3", [1, 3, 3, 1]),
    ("CP^1 = S^2", [1, 0, 1]),
    ("CP^2", [1, 0, 1, 0, 1]),
    ("Sigma_g", [1, "2g", 1]),
    ("K3", [1, 0, 22, 0, 1]),
]

for name, betti in spaces_betti:
    betti_str = ", ".join(str(b) for b in betti)
    chi = 0
    for i, b in enumerate(betti):
        if isinstance(b, int):
            chi += (-1)**i * b
    print("    {}: b = ({})  chi = {}".format(name, betti_str, chi))
print()

# ============================================================
# 2. Simplicial homology computation (tetrahedron)
# ============================================================
print("  SIMPLICIAL HOMOLOGY OF THE TETRAHEDRON (= S^2)")
print("  " + "-" * 50)
print()
print("  Vertices: v0, v1, v2, v3")
print("  Edges: 6 (every pair connected)")
print("  Faces: 4 (every triple)")
print()

# Chain complex: C_0 = Z^4, C_1 = Z^6, C_2 = Z^4
# Boundary matrices
print("  Boundary operator d_1 (edges -> vertices):")
print("    6x4 matrix with entries +/-1")
print("  Boundary operator d_2 (faces -> edges):")
print("    4x6 matrix with entries +/-1")
print()

# Ranks: rank(d_2) = 3, rank(d_1) = 3
# H_0 = ker(d_0)/im(d_1) = Z^4 / Z^3 = Z
# H_1 = ker(d_1)/im(d_2) = Z^3 / Z^3 = 0
# H_2 = ker(d_2)/im(d_3) = Z / 0 = Z

print("  H_0 = Z  (connected)")
print("  H_1 = 0  (no 1-cycles)")
print("  H_2 = Z  (one 2-cycle = the whole sphere)")
print("  chi = 1 - 0 + 1 = 2  (= V - E + F = 4 - 6 + 4 = 2)")
print()

# ============================================================
# 3. de Rham cohomology
# ============================================================
print("  DE RHAM COHOMOLOGY")
print("  " + "-" * 50)
print()
print("  H^k_{dR}(M) = {closed k-forms} / {exact k-forms}")
print("                = ker(d: Omega^k -> Omega^{k+1}) / im(d: Omega^{k-1} -> Omega^k)")
print()

# Volume form on S^2
theta_vals = np.linspace(0.01, np.pi - 0.01, 200)
phi_vals = np.linspace(0, 2*np.pi, 200)

print("  Example: H^2_{dR}(S^2) = R")
print("  Generator: omega = sin(theta) d(theta) ^ d(phi)")
print("  integral_{S^2} omega = 4*pi")
print()

# Verify: omega is closed (d(omega) = 0 since it is a top form on S^2)
# Verify: omega is NOT exact (integral != 0)
vol = np.trapz(np.sin(theta_vals), theta_vals) * 2 * np.pi
print("  Numerical integral: {:.6f}".format(vol))
print("  Exact: 4*pi = {:.6f}".format(4 * np.pi))
print()

# ============================================================
# 4. Poincare duality
# ============================================================
print("  POINCARE DUALITY")
print("  " + "-" * 50)
print()
print("  For a closed oriented n-manifold M:")
print("    H^k(M) = H^{n-k}(M)")
print("    b_k = b_{n-k}")
print()

print("  Verification:")
manifolds = [
    ("S^2 (n=2)", [1, 0, 1]),
    ("T^2 (n=2)", [1, 2, 1]),
    ("CP^2 (n=4)", [1, 0, 1, 0, 1]),
    ("S^3 (n=3)", [1, 0, 0, 1]),
    ("S^2 x S^2 (n=4)", [1, 0, 2, 0, 1]),
]

for name, betti in manifolds:
    n = len(betti) - 1
    dual_ok = all(betti[k] == betti[n-k] for k in range(n+1))
    print("    {}: b = {}, Poincare dual: {}".format(
        name, betti, "verified" if dual_ok else "FAILED"))
print()

# ============================================================
# 5. de Rham theorem verification
# ============================================================
print("  DE RHAM THEOREM")
print("  " + "-" * 50)
print()
print("  Theorem: H^k_{dR}(M, R) = H^k_{sing}(M, R)")
print("  The de Rham cohomology equals singular cohomology with real coefficients.")
print()
print("  The isomorphism is given by integration:")
print("    [omega] maps to (sigma -> integral_sigma omega)")
print("  where sigma is a singular k-cycle.")
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: Betti numbers comparison
ax1 = axes[0, 0]
names = ['pt', 'S^1', 'S^2', 'T^2', 'S^3', 'CP^2']
chi_vals = [1, 0, 2, 0, 0, 3]
b_totals = [1, 2, 2, 4, 2, 3]
x_pos = np.arange(len(names))
ax1.bar(x_pos - 0.2, chi_vals, 0.35, color='#2dd4bf', alpha=0.7, label='chi')
ax1.bar(x_pos + 0.2, b_totals, 0.35, color='#f59e0b', alpha=0.7, label='Sum b_k')
ax1.set_xticks(x_pos)
ax1.set_xticklabels(names, color='#99f6e4')
ax1.set_ylabel('Value', color='#5eead4')
ax1.set_title('Euler Characteristic and Total Betti', color='#99f6e4', fontsize=12, fontweight='bold')
ax1.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

# Plot 2: Volume form on S^2 (sin(theta) as function of theta)
ax2 = axes[0, 1]
ax2.plot(theta_vals, np.sin(theta_vals), color='#2dd4bf', linewidth=2.5)
ax2.fill_between(theta_vals, np.sin(theta_vals), alpha=0.15, color='#2dd4bf')
ax2.set_xlabel('theta', color='#5eead4')
ax2.set_ylabel('sin(theta)', color='#5eead4')
ax2.set_title('Volume Form Density on S^2', color='#99f6e4', fontsize=12, fontweight='bold')
ax2.text(np.pi/2, 0.5, 'integral = 4*pi', color='#f59e0b', fontsize=12, ha='center')

# Plot 3: Betti numbers of Sigma_g (genus g surface)
ax3 = axes[1, 0]
genus = np.arange(0, 8)
b0 = np.ones_like(genus)
b1 = 2 * genus
b2 = np.ones_like(genus)
chi_genus = b0 - b1 + b2
ax3.plot(genus, b0, 'o-', color='#2dd4bf', linewidth=2, label='b_0')
ax3.plot(genus, b1, 's-', color='#f472b6', linewidth=2, label='b_1')
ax3.plot(genus, b2, '^-', color='#a78bfa', linewidth=2, label='b_2')
ax3.plot(genus, chi_genus, 'D-', color='#f59e0b', linewidth=2, label='chi')
ax3.set_xlabel('Genus g', color='#5eead4')
ax3.set_ylabel('Value', color='#5eead4')
ax3.set_title('Betti Numbers vs Genus', color='#99f6e4', fontsize=12, fontweight='bold')
ax3.legend(facecolor='#111111', edgecolor='#134e4a', labelcolor='#99f6e4')

# Plot 4: Chain complex diagram
ax4 = axes[1, 1]
ax4.set_xlim(0, 10)
ax4.set_ylim(0, 10)
ax4.set_aspect('equal')

# Draw chain complex boxes
chain_labels = ['C_0 = Z^4', 'C_1 = Z^6', 'C_2 = Z^4', '0']
homology_labels = ['H_0 = Z', 'H_1 = 0', 'H_2 = Z', '']
x_positions = [1, 3.5, 6, 8.5]
y_main = 7
y_hom = 3

for i, (cl, hl, xp) in enumerate(zip(chain_labels, homology_labels, x_positions)):
    ax4.text(xp, y_main, cl, color='#2dd4bf', fontsize=10, ha='center',
             bbox=dict(boxstyle='round', facecolor='#111111', edgecolor='#2dd4bf', alpha=0.8))
    if hl:
        ax4.text(xp, y_hom, hl, color='#f59e0b', fontsize=10, ha='center',
                 bbox=dict(boxstyle='round', facecolor='#111111', edgecolor='#f59e0b', alpha=0.8))
    if i < len(x_positions) - 1:
        ax4.annotate('', xy=(x_positions[i+1]-0.8, y_main), xytext=(xp+0.8, y_main),
                     arrowprops=dict(arrowstyle='->', color='#99f6e4', lw=1.5))
        if i < 3:
            ax4.text((xp + x_positions[i+1])/2, y_main + 0.5, 'd_{}'.format(i+1),
                     color='#99f6e4', fontsize=9, ha='center')

ax4.set_title('Chain Complex of Tetrahedron', color='#99f6e4', fontsize=12, fontweight='bold')
ax4.axis('off')

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="Homology and Cohomology 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

Simplicial Homology

Counts cycles modulo boundaries: $H_n = \ker\partial_n / \text{im}\,\partial_{n+1}$. Betti numbers$b_n$ count independent $n$-holes, and the Euler characteristic is their alternating sum.

de Rham Cohomology

Closed forms modulo exact forms: $H^k_{dR} = \ker d / \text{im}\,d$. Detects the same topological information as singular cohomology, by the de Rham theorem.

Poincaré Duality

On a closed oriented $n$-manifold: $H^k \cong H_{n-k}$. The wedge-integration pairing is non-degenerate, giving palindromic Betti numbers.

Mayer-Vietoris

The fundamental computational tool for (co)homology. Decomposes a space into overlapping pieces and relates their (co)homology via a long exact sequence.

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