Algebraic topology — homology groups

Homology groups

H_{n} (X)

are abelian groups associated to a topological space

X

that algebraically detect

n

-dimensional "holes." Key result:

H_{n} (X)

is the quotient of

n

-cycles by

n

-boundaries,

H_{n} = ker \partial_{n} / im \partial_{n + 1}

, and it is a homotopy invariant. Applies to simplicial complexes, CW complexes, manifolds, and arbitrary spaces (singular theory). Originated with Poincaré (1895); axiomatized by Eilenberg–Steenrod (1945–52).

Homology assigns to each topological space $X$ a sequence of abelian groups $H_{0} (X), H_{1} (X), H_{2} (X), \dots$ measuring connected components, loops, voids, and higher-dimensional analogues. The construction begins with a chain complex — a sequence of abelian groups linked by boundary maps $\partial$ satisfying $\partial \circ \partial = 0$ — and extracts homology as "cycles modulo boundaries." Homology matters because it is computable (unlike homotopy groups in most cases), functorial (continuous maps induce homomorphisms), and a homotopy invariant, making it the workhorse for distinguishing spaces. A surprising fact: although homotopy and homology both vanish in low dimensions for spheres, homology of $S^{n}$ is finitely generated and trivial to compute, while $π_{k} (S^{n})$ remains largely unknown — yet the Hurewicz theorem ties them together in the first nonvanishing degree.

The Core Insight

A "hole" is something with no boundary that is itself not the boundary of anything. The map $\partial$ encodes "taking the boundary"; cycles ( $ker \partial$ ) are boundaryless objects, and boundaries ( $im \partial$ ) are objects that bound. Because $\partial^{2} = 0$ , every boundary is a cycle, so the quotient $ker \partial / im \partial$ is well-defined and precisely measures cycles that fail to bound — the holes.

Formal Statement

Chain Complex and Homology

A chain complex $(C_{∙}, \partial_{∙})$ is a sequence of abelian groups $C_{n}$ (for $n \in Z$ ) with homomorphisms $\partial_{n} : C_{n} \to C_{n - 1}$ , called boundary operators, satisfying

\partial_{n} \circ \partial_{n + 1} = 0 for all n .

⟳This is the defining axiom; geometrically, the boundary of a boundary is empty.

The group of $n$ -cycles is $Z_{n} = ker \partial_{n} \subseteq C_{n}$ , and the group of $n$ -boundaries is $B_{n} = im \partial_{n + 1} \subseteq C_{n}$ . Equation [1] gives $B_{n} \subseteq Z_{n}$ , so the $n$ -th homology group is the quotient

H_{n} (C_{∙}) = Z_{n} / B_{n} = ker \partial_{n} / im \partial_{n + 1} .

⟳Well-defined because

B_{n} \leq Z_{n}

by [1]; elements are homology classes

[z] = z + B_{n}

Singular Homology

Let $Δ^{n} = {(t_{0}, \dots, t_{n}) \in R^{n + 1} : t_{i} \geq 0, \sum t_{i} = 1}$ be the standard $n$ -simplex. A singular $n$ -simplex in $X$ is a continuous map $σ : Δ^{n} \to X$ . The singular chain group $C_{n} (X)$ is the free abelian group on all singular $n$ -simplices. The boundary operator is

\partial_{n} σ = i = 0 \sum n (- 1)^{i} σ \circ d_{i},

⟳

d_{i} : Δ^{n - 1} \to Δ^{n}

is the

i

-th face inclusion omitting vertex

v_{i}

; the alternating sign enforces

\partial^{2} = 0

where $d_{i}$ inserts the $i$ -th face. The singular homology of $X$ is $H_{n} (X) = H_{n} (C_{∙} (X))$ . With coefficients in an abelian group $G$ , set $C_{n} (X; G) = C_{n} (X) \otimes G$ and $H_{n} (X; G) = H_{n} (C_{∙} (X; G))$ .

Reduced and Relative Homology

The reduced homology $\tilde{H}_{n} (X)$ uses the augmented complex

\dots \to C_{1} (X) \partial_{1} C_{0} (X) ε Z \to 0, ε (\sum n_{i} σ_{i}) = \sum n_{i} .

⟳

ε

counts points with sign;

\tilde{H}_{0} (X) \oplus Z ≅ H_{0} (X)

, killing the redundant component count for nonempty

X

For a subspace $A \subseteq X$ , the relative chain group is $C_{n} (X, A) = C_{n} (X) / C_{n} (A)$ , with induced boundary, giving relative homology $H_{n} (X, A)$ .

Eilenberg–Steenrod Axioms

A homology theory on the category of pairs $(X, A)$ is a sequence of functors $H_{n}$ with natural boundary maps $\partial : H_{n} (X, A) \to H_{n - 1} (A)$ satisfying:

Homotopy: $f ≃ g ⟹ f_{*} = g_{*}$ .
Exactness: the sequence $\dots \to H_{n} (A) \to H_{n} (X) \to H_{n} (X, A) \partial H_{n - 1} (A) \to \dots$ is exact.
Excision: for $\overline{Z} \subseteq int (A)$ , the inclusion induces $H_{n} (X ∖ Z, A ∖ Z) ≅ H_{n} (X, A)$ .
Dimension: $H_{n} (pt) = 0$ for $n \neq = 0$ .
Additivity: $H_{n} (⨆_{α} X_{α}) ≅ ⨁_{α} H_{n} (X_{α})$ .

On CW complexes these axioms determine $H_{*}$ up to natural isomorphism given the coefficient group $H_{0} (pt)$ .

Why It Holds / Derivation

Homotopy Invariance via Chain Homotopy

Claim: If

f, g : X \to Y

are homotopic, then

f_{*} = g_{*} : H_{n} (X) \to H_{n} (Y)

. Step 1. A homotopy

F : X \times I \to Y

with

F_{0} = f

F_{1} = g

exists. Build a prism operator

P_{n} : C_{n} (X) \to C_{n + 1} (Y)

by subdividing

Δ^{n} \times I

into

(n + 1)

-simplices:

P_{n} (σ) = i = 0 \sum n (- 1)^{i} F \circ (σ \times id)_{[v_{0}, \dots, v_{i}, w_{i}, \dots, w_{n}]} .

⟳

v_{j} = (x_{j}, 0)

w_{j} = (x_{j}, 1)

label the two copies of the simplex's vertices; the signs make the prism's boundary telescope.

Step 2. A direct (if tedious) computation of

\partial P + P \partial

on the prism's boundary yields the chain homotopy identity

\partial_{n + 1} P_{n} + P_{n - 1} \partial_{n} = g_{#} - f_{#},

⟳The boundary of the prism is top

-

bottom

-

sides; top and bottom give

g_{#} - f_{#}

, the sides give the

P \partial

term.

where $f_{#}, g_{#}$ are the induced chain maps.

Step 3. For a cycle

z

(

\partial z = 0

g_{#} (z) - f_{#} (z) = \partial P (z) + P (\partial z) = \partial P (z)

, a boundary. Hence

[g_{#} (z)] = [f_{#} (z)]

H_{n} (Y)

, so

f_{*} = g_{*}

■

The Long Exact Sequence of a Pair

Step 1. The short exact sequence of chain complexes

0 \to C_{∙} (A) i C_{∙} (X) j C_{∙} (X, A) \to 0

⟳Definitionally

C_{n} (X, A) = C_{n} (X) / C_{n} (A)

, giving exactness in each degree.

is exact in every degree.

Step 2. Apply the snake lemma / zig-zag construction: given a relative cycle

[c] \in H_{n} (X, A)

, lift

c

C_{n} (X)

; then

\partial c \in C_{n - 1} (X)

maps to

0

C_{n - 1} (X, A)

, so

\partial c \in C_{n - 1} (A)

defines a class. Set

\partial_{*} [c] = [\partial c]

. Step 3. A diagram chase shows

\partial_{*}

is well-defined and that the resulting long sequence

\dots \to H_{n} (A) i_{*} H_{n} (X) j_{*} H_{n} (X, A) \partial_{*} H_{n - 1} (A) \to \dots

⟳Exactness at each spot is six containment checks (e.g.

im j_{*} = ker \partial_{*}

), each a routine chase in [7].

is exact. $■$

Henri Poincaré | 1854–1912

Poincaré founded combinatorial topology ("analysis situs") in his 1895 memoir and five subsequent complements. He introduced Betti numbers and torsion coefficients, the boundary relation, and what is essentially the simplicial chain complex — though he worked without the formal language of groups, which Emmy Noether later supplied. His struggle to make these invariants rigorous (correcting errors across the complements) produced the Poincaré duality theorem and the fundamental group, and his famous conjecture about $S^{3}$ would drive a century of topology. He intuited homology as counting independent cycles long before the algebra crystallized.

Building Intuition

Picture homology as a "hole detector" tuned to each dimension.

$H_{0}$ counts connected components: each piece of $X$ contributes a $Z$ , since points in different components are not joined by a $1$ -chain (path).
$H_{1}$ detects loops that don't bound a disk: on a torus, the two independent circles give $H_{1} ≅ Z^{2}$ .
$H_{2}$ detects enclosed voids: a hollow sphere $S^{2}$ encloses a $3$ -dimensional cavity, giving $H_{2} (S^{2}) ≅ Z$ .

The torsion subgroups encode subtler twisting: in $RP^{2}$ , going around the noncontractible loop twice bounds, so $H_{1} (RP^{2}) ≅ Z /2$ . The slogan: a cycle is a closed thing; a boundary is a filled thing; homology = closed but not filled.

The boundary-of-boundary principle: $\partial^2=0$ on a filled triangle

Worked Examples

Homology of the Circle $S^1$

Given: Model

S^{1}

as a CW complex with one

0

-cell

v

and one

1

-cell

e

. Steps:

The cellular chain complex is $\dots \to 0 \to C_{1} = Z ⟨ e ⟩ \partial_{1} C_{0} = Z ⟨ v ⟩ \to 0$ .
The $1$ -cell $e$ is a loop: both endpoints are $v$ , so $\partial_{1} e = v - v = 0$ .
Therefore $\partial_{1} = 0$ , giving $Z_{1} = Z$ , $B_{1} = 0$ , $Z_{0} = Z$ , $B_{0} = 0$ .
Compute:

H_{0} (S^{1}) = Z /0 = Z, H_{1} (S^{1}) = Z /0 = Z, H_{n} (S^{1}) = 0 (n \geq 2) .

⟳Both boundary maps vanish, so homology equals the chain groups themselves.

Answer:

H_{n} (S^{1}) ≅ {Z 0 n = 0, 1 else

This confirms one connected component and one independent loop.

Homology of the Real Projective Plane $\mathbb{RP}^2$

Given: CW structure: one

0

-cell, one

1

-cell, one

2

-cell, with the

2

-cell attached by the degree-

2

map

S^{1} \to S^{1}

(the attaching word

a^{2}

). Steps:

Cellular complex: $0 \to Z \partial_{2} Z \partial_{1} Z \to 0$ .
The $1$ -cell is a loop, so $\partial_{1} = 0$ .
The $2$ -cell wraps the $1$ -cell twice, so $\partial_{2} = \times 2$ (degree $2$ ):

\partial_{2} (1) = 2.

⟳The cellular boundary is computed by the degree of the attaching map onto each lower cell; here that degree is

2

Compute homologies:

H_{2} = ker (\times 2) = 0, H_{1} = Z /2 Z, H_{0} = Z .

⟳

ker (\times 2) = 0

Z

;

coker (\times 2) = Z /2

gives

H_{1}

Answer:

H_{0} = Z, H_{1} = Z /2 Z, H_{2} = 0

The torsion $Z /2$ reflects the orientation-reversing loop. Note $RP^{2}$ is non-orientable, signaled by $H_{2} = 0$ (no fundamental class over $Z$ ).

Common Misconceptions

Homology does not determine homotopy type

Equal homology groups do not imply homotopy equivalence. The Hurewicz theorem only links $π_{n}$ and $H_{n}$ in the first nonvanishing degree (and requires simple-connectivity for $n \geq 2$ ). Spaces can share all homology groups yet be homotopically distinct — homology forgets the group action of $π_{1}$ and higher Postnikov data. Conversely, homology is a strictly weaker (but far more computable) invariant than homotopy type.

Same homology, different homotopy type

Consider $X = S^{2} \lor S^{4}$ (wedge of spheres) and $Y = CP^{2}$ . Both have homology

Z, 0, Z, 0, Z

in degrees

0, 1, 2, 3, 4

. Yet they are not homotopy equivalent: the cup product

H^{2} \otimes H^{2} \to H^{4}

is nontrivial in

CP^{2}

(the generator squares to the top class) but zero in the wedge (cup products across a wedge vanish). Homology alone cannot see this; the ring structure of cohomology distinguishes them.

Reduced vs. unreduced $H_0$

For a path-connected nonempty space, $H_{0} (X) = Z$ but $\tilde{H}_{0} (X) = 0$ . Mixing these conventions causes off-by- $Z$ errors in long exact sequences and Mayer–Vietoris. Always state which convention is in force; many theorems (e.g. the suspension isomorphism $\tilde{H}_{n} (X) ≅ \tilde{H}_{n + 1} (Σ X)$ ) hold only for reduced homology.

Connections

Homology sits at the center of algebraic topology's web:

Cohomology groups are the dual theory; the Universal Coefficient Theorem relates $H^{n} (X; G)$ to $Hom (H_{n}, G)$ and $Ext (H_{n - 1}, G)$ .
Homotopy groups connect via the Hurewicz theorem: the first nonzero $π_{n}$ abelianizes to $H_{n}$ .
Poincaré duality for closed oriented $n$ -manifolds gives $H_{k} (M) ≅ H^{n - k} (M)$ , capping with the fundamental class.
CW complexes provide cellular homology, agreeing with singular homology but vastly more computable.
Spectral sequences (Serre, Leray) compute homology of fibrations and total spaces.
Homological algebra abstracts the chain-complex machinery (derived functors, Tor, Ext) used throughout.

Real-World Applications

Topological Data Analysis (Persistent Homology): point clouds are filtered by scale; persistence barcodes track $H_{n}$ across scales to find robust features in genomics, neuroscience, and materials science (Carlsson, Edelsbrunner).
Sensor Networks: $H_{1}$ of the Rips/Čech complex detects coverage holes in distributed sensor fields without coordinate data (de Silva–Ghrist).
Robotics & Motion Planning: configuration-space homology identifies obstructions and distinct path classes; $H_{1}$ counts topologically inequivalent routes.
Condensed Matter Physics: topological insulators are classified by $K$ -theory and (co)homology of the Brillouin zone; Chern numbers are cohomology classes detecting protected edge states.
Digital Image Analysis: cubical homology computes Betti numbers of 2D/3D images for shape recognition and porous-media characterization.
Cosmology: the homology (Betti numbers) of the cosmic web characterizes large-scale structure — voids ( $H_{2}$ ), filaments/loops ( $H_{1}$ ).
Chemistry & Biology: homology of molecular surfaces and protein structures quantifies cavities and channels relevant to drug binding.

Historical Development

1857–1871: Riemann and Betti study connectivity numbers of surfaces and higher manifolds, prefiguring Betti numbers.
1895: Poincaré's Analysis Situs introduces combinatorial topology, the boundary operator, Betti numbers, and torsion coefficients.
1899–1904: Poincaré's five Complements repair errors, develop simplicial methods, and state Poincaré duality and the Poincaré conjecture.
1925–1928: Emmy Noether reframes Betti numbers and torsion as homology groups, importing the language of abelian groups and homomorphisms (via Hopf, Alexandroff).
1933: Čech and Vietoris develop homology for general (non-triangulated) spaces.
1944: Eilenberg introduces singular homology, freeing the theory from triangulations.
1945: Eilenberg and Steenrod formulate the axiomatic characterization of homology theories.
1952: Eilenberg–Steenrod's Foundations of Algebraic Topology publishes the definitive axiomatic treatment.
1950s–1960s: Generalized homology theories (K-theory, bordism, stable homotopy) emerge from the axioms via Brown representability.

Subtleties & Limits

Computability limits: While homology is algorithmically computable for finite complexes, the homology of spaces given by other presentations (e.g., word problems) can be undecidable; finite presentation does not guarantee a tractable chain complex.
Extraordinary theories & coefficients: The Eilenberg–Steenrod axioms minus the dimension axiom admit generalized theories (K-theory, cobordism); classifying all such theories ties to deep stable homotopy questions, including the Telescope Conjecture (resolved negatively in 2023 by Burklund–Hahn–Levy–Schlank).
Integral vs. field coefficients: Torsion phenomena make integral homology strictly richer; the precise interplay (e.g., the failure of the Künneth formula to split naturally, anomalous Bocksteins) remains a source of computational subtlety.
Homology vs. homotopy gap: Reconstructing homotopy type from homological data (the realization problem for Postnikov towers, obstruction theory with $k$ -invariants) is in general intractable; the homotopy groups of spheres remain unknown in infinitely many degrees despite trivial homology.