Algebraic Topology: Leverage Map

A. EXISTENCE JUSTIFICATION

Geometry kept bumping into questions it couldn’t answer with its own tools:

• Why can’t you comb a hairy ball flat? (Every vector field on S² has a zero)
• Why can’t you continuously deform a coffee cup into a sphere? (But you can deform it into a donut)
• Why does the fundamental theorem of algebra work? (Every polynomial has a root)
• Why do some differential equations have no global solutions?

These questions aren’t about distances or angles—they’re about shape in a deeper sense. Algebraic topology exists because we needed algebraic invariants that detect topological features invisible to geometry: holes, twists, obstructions, global structure.

The core move: Assign algebraic objects (groups, rings, modules) to topological spaces in a way that:

1. Equivalent spaces get isomorphic algebra
2. Continuous maps induce algebraic homomorphisms
3. The algebra is computable

Then use algebra to prove topological theorems. If two spaces have different algebraic invariants, they can’t be homeomorphic (or homotopy equivalent). If an algebraic obstruction is nonzero, certain constructions are impossible.

The payoff: Hard geometric problems become routine algebra. “Can X be continuously deformed to Y?” becomes “Are their homology groups isomorphic?”

B. CORE OBJECTS & MORPHISMS

Morphisms:

• Continuous maps induce homomorphisms on all these invariants
• The assignment “space ↦ algebraic invariant” is a functor
• Homotopic maps induce the same homomorphism (homotopy invariance)

C. CENTRAL INVARIANTS

Hierarchy of “sameness”:

What the invariants detect:

What counts as “the same”:

For most purposes, homotopy equivalence. A space is “the same” as any space it can be continuously deformed to. The invariants are homotopy invariants—they see only the homotopy type.

This is why a coffee cup = donut (both homotopy equivalent to a circle), but ≠ sphere.

D. SIGNATURE THEOREMS

1. Brouwer Fixed Point Theorem

Every continuous map f: Dⁿ → Dⁿ (closed disk to itself) has a fixed point.

Why it matters: This is proved using homology. The argument: if no fixed point existed, you could construct a retraction of Dⁿ onto its boundary Sⁿ⁻¹. But such a retraction would induce a map Hₙ₋₁(Sⁿ⁻¹) → Hₙ₋₁(Dⁿ), which is ℤ → 0. The identity on the boundary would factor through 0—impossible.

The pattern: An algebraic impossibility proves a topological impossibility.

2. Fundamental Theorem of Algebra (topological proof)

Every non-constant polynomial has a complex root.

Why it matters: Purely algebraic statement, proved topologically. If p(z) has no root, then p/|p|: ℂ → S¹ is defined everywhere. Restricting to large circles, the map has winding number = degree(p) ≠ 0. But the map extends to the disk (ℂ ∪ {∞}), so the winding number must be 0. Contradiction.

The pattern: Homotopy groups obstruct extensions.

3. Hairy Ball Theorem

There is no continuous nonvanishing tangent vector field on S².

Why it matters: You can’t comb a hairy ball flat—somewhere the hair must have a cowlick (or bald spot). Proved via Euler characteristic: the sum of indices of zeros of any vector field on M equals χ(M). For S², χ = 2 ≠ 0, so zeros exist.

Generalizes: Odd-dimensional spheres do have nonvanishing vector fields. S¹ is obvious (tangent to the circle). S³ has three independent ones (the quaternion structure). This connects to deep questions about parallelizability.

4. Hurewicz Theorem

For a simply connected space, H₁ = 0 and H₂ ≅ π₂ (the first nontrivial homology equals the first nontrivial homotopy). More generally: if πᵢ = 0 for i < n, then Hₙ ≅ πₙ.

Why it matters: Homotopy groups are hard; homology is computable. Hurewicz lets you compute the first interesting πₙ from Hₙ. The connection between the two theories runs deep.

5. Long Exact Sequences

Every “good” short exact sequence of spaces or pairs gives a long exact sequence of homology/homotopy groups.

Why it matters: Exact sequences are the main computational tool. If you know two out of three terms, you can deduce the third. Fiber bundles, pairs, covers—all give long exact sequences.

For a fiber bundle F → E → B: $$\cdots \to \pi_{n}(F) \to \pi_{n}(E) \to \pi_{n}(B) \to \pi_{n-1}(F) \to \cdots$$

E. BRIDGES TO OTHER DOMAINS

Pattern-linking gold:

The chain complex is the universal pattern: $$\cdots \xrightarrow{\partial} C_{n+1} \xrightarrow{\partial} C_n \xrightarrow{\partial} C_{n-1} \xrightarrow{\partial} \cdots$$

with ∂² = 0. Whenever you see this structure, homological algebra applies:

• Simplicial/singular chains (topology)
• de Rham complex (differential geometry)
• Koszul complex (algebra)
• Resolutions in homological algebra
• Cohomology of sheaves (algebraic geometry)

The condition ∂² = 0 is everywhere. It’s “the boundary of a boundary is empty” in topology, “d² = 0” for exterior derivative, “the image is in the kernel” abstractly. Homology = kernel/image = cycles/boundaries = closed but not exact.

F. COMMON MISCONCEPTIONS

1. “Homology counts holes” — Sort of, but “hole” is imprecise. H₁ detects 1-dimensional holes (like the hole in a torus’s tube), H₂ detects 2-dimensional voids (like the inside of a sphere). H₀ detects connected components—not “holes” in the usual sense.

2. “Homotopy equivalence means homeomorphism” — Homotopy equivalence is weaker. The letter “A” and a point are homotopy equivalent (A retracts to a point), but not homeomorphic. A disk is homotopy equivalent to a point.

3. “π₁ and H₁ are the same” — H₁ is the abelianization of π₁. They agree for nice spaces like surfaces, but π₁ can be nonabelian (figure-8 has π₁ = free group on 2 generators), while H₁ is always abelian.

4. “Higher homotopy groups are like fundamental groups but higher” — Higher πₙ are always abelian (for n ≥ 2) and much harder to compute. They don’t just “count higher holes”—they have subtle structure. Even πₙ(Sᵐ) is immensely complicated when n > m.

5. “Homology is a complete invariant” — Different spaces can have identical homology. The cohomology ring structure gives more information. Still not complete—homotopy theory sees even more.

6. “Algebraic topology is abstract nonsense” — The abstract machinery exists to prove concrete theorems. Fixed point theorems have applications from economics (Nash equilibrium) to differential equations. TDA is used in drug discovery, materials science, neuroscience.

7. “CW complexes are a technical convenience” — They’re the “standard form” for spaces in algebraic topology. Any reasonable space is homotopy equivalent to a CW complex. Working with CW complexes makes everything computable.

G. NOTATION SURVIVAL KIT

Key exact sequences:

For a pair (X, A): $$\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \to H_{n-1}(A) \to \cdots$$

For a fibration F → E → B: $$\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots$$

Mayer-Vietoris (X = A ∪ B): $$\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to \cdots$$

H. ONE WORKED MICRO-EXAMPLE

Computing H_*(S¹) via cellular homology:

Setup: Build S¹ as a CW complex:

• One 0-cell e⁰ (a point)
• One 1-cell e¹ (an interval, with both ends attached to e⁰)

Chain complex: $$0 \to C_1 \to C_0 \to 0$$

where C₁ = ℤ (generated by e¹) and C₀ = ℤ (generated by e⁰).

Boundary map: ∂(e¹) = endpoint - startpoint = e⁰ - e⁰ = 0.

So ∂: ℤ → ℤ is the zero map.

Homology:

• H₀ = C₀/im(∂) = ℤ/0 = ℤ (one connected component)
• H₁ = ker(∂)/im(from C₂ = 0) = ℤ/0 = ℤ (the circle has one 1-dimensional hole)

Interpretation: H₁(S¹) = ℤ detects the loop. The generator is the circle itself. Every loop wraps some integer number of times—the winding number.

Micro-example 2: The torus T² = S¹ × S¹

CW structure:

• One 0-cell (vertex)
• Two 1-cells a, b (meridian and longitude)
• One 2-cell (the surface, attached along aba⁻¹b⁻¹)

Chain complex: $$0 \to C_2 = \mathbb{Z} \xrightarrow{\partial_2} C_1 = \mathbb{Z}^2 \xrightarrow{\partial_1} C_0 = \mathbb{Z} \to 0$$

Boundary maps:

• ∂₁(a) = ∂₁(b) = 0 (both loops start and end at same vertex)
• ∂₂(e²) = a + b - a - b = 0 (the attaching map goes around aba⁻¹b⁻¹, which sums to zero in abelian group)

Homology:

• H₀ = ℤ (connected)
• H₁ = ker(∂₁)/im(∂₂) = ℤ²/0 = ℤ² (two independent loops: a and b)
• H₂ = ker(∂₂) = ℤ (the torus encloses a void—but unlike the sphere, you can reach it via the hole!)

Euler characteristic: χ = 1 - 2 + 1 = 0. Matches the formula χ = 2 - 2g for genus g = 1.

Micro-example 3: Why π₁(S¹) = ℤ matters

The covering space perspective:

ℝ → S¹ via x ↦ e^(2πix) is a covering map. The fiber over any point is ℤ (the integers lifted above that point).

Lifting paths: A loop in S¹ lifts to a path in ℝ. The path starts at 0 and ends at… some integer n. This integer is the winding number, and it’s the element of π₁ the loop represents.

Why this connects to everything:

The fundamental group classifies covering spaces:

• Subgroups of π₁(X) ↔ covering spaces of X
• The universal cover has trivial π₁

For S¹:

• Subgroups of ℤ are nℤ for n = 0, 1, 2, …
• The n-fold cover is S¹ → S¹ by z ↦ zⁿ
• The universal cover is ℝ (π₁(ℝ) = 0)

HoTT connection: In HoTT, the circle S¹ is defined with one point base and one path loop : base = base. The type base = base (self-identities of base) turns out to be equivalent to ℤ—that’s the fundamental group, emerging from the type structure.

HoTT connection (deep):

HoTT is essentially algebraic topology internalized:

• Types = spaces (up to homotopy)
• Identity types = path spaces
• π_n(A) = ||Ωⁿ A|| (truncated iterated loop space)
• Higher inductive types = CW complex constructors

The univalence axiom is a topological statement: the path space of the universe (equality of types) is equivalent to the space of equivalences. This is wild from a classical logic perspective but natural topologically.

Our HoTT-protein-folding paper link: the space of folds has topological structure. Different folding pathways that reach the same fold are homotopic. The fundamental group of the fold space could detect topologically distinct folding mechanisms.

Convergence Thesis:

If optimal cognitive architectures are constrained to a manifold, that manifold has homology. The Betti numbers (ranks of homology groups) are invariants of “the shape of possible minds.” Different homology → different computational topology → different capabilities.

Topological obstructions might explain impossibility results: “No architecture can do X” because it would require a section of a bundle that doesn’t exist (like a nonvanishing vector field on S²).

Persistent homology / TDA:

For your pattern-linking across domains: persistent homology tracks features across scales. If you embed different domains into a common representation space, TDA can detect shared topological features—holes, clusters, voids that appear at similar scales.

This is a formal way to find “structural isomorphism” between different data types.

Spectral sequences:

The power tool of algebraic topology. A spectral sequence is like “iterated homology”—you compute approximations that converge to the true answer. The Serre spectral sequence computes homology of fiber bundles; the Adams spectral sequence computes homotopy groups of spheres.

You don’t need to compute these, but knowing they exist tells you: homological algebra has multi-scale structure, and hard computations can sometimes be broken into tractable layers.

Next up: Lie Theory—where symmetry meets geometry, and representation theory finds its natural home. This will tie together the representation theory and differential geometry pieces into a unified picture.