Algebraic topology — homology groups

⚡ Key Takeaways

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, $H_n = \ker\partial_n / \operatorname{im}\partial_{n+1}$ , 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).

💡 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 ( $\operatorname{im}\partial$ ) are objects that bound. Because $\partial^2 = 0$ , every boundary is a cycle, so the quotient $\ker\partial/\operatorname{im}\partial$ is well-defined and precisely measures cycles that fail to bound — the holes.

DefinitionChain Complex and Homology

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

1

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

Derivation hint: 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 $B_n = \operatorname{im}\partial_{n+1} \subseteq C_n$ . Equation [1] gives $B_n \subseteq Z_n$ , so the $n$ -th homology group is the quotient

2

\quad H_n(C_\bullet) = Z_n / B_n = \ker\partial_n / \operatorname{im}\partial_{n+1}.

Derivation hint: Well-defined because $B_n \le Z_n$ by [1]; elements are homology classes $[z] = z + B_n$ .

DefinitionSingular Homology

Let $\Delta^n = \{(t_0,\dots,t_n) \in \mathbb{R}^{n+1} : t_i \ge 0,\ \sum t_i = 1\}$ be the standard $n$ -simplex. A singular $n$ -simplex in $X$ is a continuous map $\sigma: \Delta^n \to X$ . The singular chain group $C_n(X)$ is the free abelian group on all singular $n$ -simplices. The boundary operator is

3

\quad \partial_n \sigma = \sum_{i=0}^{n} (-1)^i\, \sigma \circ d_i,

Derivation hint: $d_i : \Delta^{n-1}\to\Delta^n$ 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 $H_n(X) = H_n(C_\bullet(X))$ . With coefficients in an abelian group $G$ , set $C_n(X;G) = C_n(X) \otimes G$ and $H_n(X;G) = H_n(C_\bullet(X;G))$ .

DefinitionReduced and Relative Homology

The reduced homology $\tilde H_n(X)$ uses the augmented complex

4

\quad \cdots \to C_1(X) \xrightarrow{\partial_1} C_0(X) \xrightarrow{\ \varepsilon\ } \mathbb{Z} \to 0,\qquad \varepsilon\!\left(\textstyle\sum n_i \sigma_i\right) = \textstyle\sum n_i.

Derivation hint: $\varepsilon$ counts points with sign; $\tilde H_0(X) \oplus \mathbb{Z} \cong H_0(X)$ , 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 $H_n(X,A)$ .

TheoremEilenberg–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: 1. Homotopy: $f \simeq g \implies f_* = g_*$ . 2. Exactness: the sequence $\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \xrightarrow{\partial} H_{n-1}(A) \to \cdots$ is exact. 3. Excision: for $\overline{Z} \subseteq \operatorname{int}(A)$ , the inclusion induces $H_n(X\setminus Z, A\setminus Z) \xrightarrow{\cong} H_n(X,A)$ . 4. Dimension: $H_n(\mathrm{pt}) = 0$ for $n \ne 0$ . 5. Additivity: $H_n(\bigsqcup_\alpha X_\alpha) \cong \bigoplus_\alpha H_n(X_\alpha)$ . On CW complexes these axioms determine $H_*$ up to natural isomorphism given the coefficient group $H_0(\mathrm{pt})$ .

ProofHomotopy 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 $\Delta^n \times I$ into $(n+1)$ -simplices:

5

\quad P_n(\sigma) = \sum_{i=0}^{n} (-1)^i\, F \circ (\sigma \times \mathrm{id})\big|_{[v_0,\dots,v_i,\,w_i,\dots,w_n]}.

Derivation hint: $v_j = (x_j,0)$ , $w_j = (x_j,1)$ 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

6

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

Derivation hint: 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 $[g_\#(z)] = [f_\#(z)]$ in $H_n(Y)$ , so $f_* = g_*$ . $\blacksquare$

∎

ProofThe Long Exact Sequence of a Pair

Step 1. The short exact sequence of chain complexes

7

\quad 0 \to C_\bullet(A) \xrightarrow{\ i\ } C_\bullet(X) \xrightarrow{\ j\ } C_\bullet(X,A) \to 0

Derivation hint: 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 $c$ to $C_n(X)$ ; then $\partial c \in C_{n-1}(X)$ maps to $0$ in $C_{n-1}(X,A)$ , 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

8

\quad \cdots \to H_n(A) \xrightarrow{i_*} H_n(X) \xrightarrow{j_*} H_n(X,A) \xrightarrow{\partial_*} H_{n-1}(A) \to \cdots

Derivation hint: Exactness at each spot is six containment checks (e.g. $\operatorname{im} j_* = \ker\partial_*$ ), each a routine chase in [7]. is exact. $\blacksquare$

∎

mathematician · 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.

🧠 Intuition

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

$H_0$ counts connected components: each piece of $X$ contributes a $\mathbb{Z}$ , 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 $H_1 \cong \mathbb{Z}^2$ .
$H_2$ detects enclosed voids: a hollow sphere $S^2$ encloses a $3$ -dimensional cavity, giving $H_2(S^2) \cong \mathbb{Z}$ .

The torsion subgroups encode subtler twisting: in $\mathbb{RP}^2$ , going around the noncontractible loop twice bounds, so $H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2$ . The slogan: a cycle is a closed thing; a boundary is a filled thing; homology = closed but not filled.

Figure

The boundary-of-boundary principle:

\partial^2=0

on a filled triangle

ExampleHomology of the Circle

S^1

Given: Model $S^1$ as a CW complex with one $0$ -cell $v$ and one $1$ -cell $e$ .

Steps: 1. The cellular chain complex is $\cdots \to 0 \to C_1 = \mathbb{Z}\langle e\rangle \xrightarrow{\partial_1} C_0 = \mathbb{Z}\langle v\rangle \to 0$ . 2. The $1$ -cell $e$ is a loop: both endpoints are $v$ , so $\partial_1 e = v - v = 0$ . 3. Therefore $\partial_1 = 0$ , giving $Z_1 = \mathbb{Z}$ , $B_1 = 0$ , $Z_0 = \mathbb{Z}$ , $B_0 = 0$ . 4. Compute:

9

\quad H_0(S^1) = \mathbb{Z}/0 = \mathbb{Z}, \quad H_1(S^1) = \mathbb{Z}/0 = \mathbb{Z}, \quad H_n(S^1) = 0\ (n\ge 2).

Derivation hint: Both boundary maps vanish, so homology equals the chain groups themselves.

Answer:

\boxed{H_n(S^1) \cong \begin{cases}\mathbb{Z} & n = 0,1\\ 0 & \text{else}\end{cases}}

This confirms one connected component and one independent loop.

ExampleHomology 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: 1. Cellular complex: $0 \to \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z} \xrightarrow{\partial_1} \mathbb{Z} \to 0$ . 2. The $1$ -cell is a loop, so $\partial_1 = 0$ . 3. The $2$ -cell wraps the $1$ -cell twice, so $\partial_2 = \times 2$ (degree $2$ ):

10

\quad \partial_2(1) = 2.

Derivation hint: The cellular boundary is computed by the degree of the attaching map onto each lower cell; here that degree is $2$ . 4. Compute homologies:

11

\quad H_2 = \ker(\times 2) = 0,\quad H_1 = \mathbb{Z}/2\mathbb{Z},\quad H_0 = \mathbb{Z}.

Derivation hint: $\ker(\times 2)=0$ in $\mathbb{Z}$ ; $\operatorname{coker}(\times 2)=\mathbb{Z}/2$ gives $H_1$ .

Answer:

\boxed{H_0 = \mathbb{Z},\quad H_1 = \mathbb{Z}/2\mathbb{Z},\quad H_2 = 0}

The torsion $\mathbb{Z}/2$ reflects the orientation-reversing loop. Note $\mathbb{RP}^2$ is non-orientable, signaled by $H_2 = 0$ (no fundamental class over $\mathbb{Z}$ ).

⚠️ Homology does not determine homotopy type

Equal homology groups do not imply homotopy equivalence. The Hurewicz theorem only links $\pi_n$ and $H_n$ in the first nonvanishing degree (and requires simple-connectivity for $n\ge 2$ ). Spaces can share all homology groups yet be homotopically distinct — homology forgets the group action of $\pi_1$ and higher Postnikov data. Conversely, homology is a strictly weaker (but far more computable) invariant than homotopy type.

CounterexampleSame homology, different homotopy type

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

\mathbb{Z}, 0, \mathbb{Z}, 0, \mathbb{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 $\mathbb{CP}^2$ (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, $H_0(X) = \mathbb{Z}$ but $\tilde H_0(X) = 0$ . Mixing these conventions causes off-by- $\mathbb{Z}$ errors in long exact sequences and Mayer–Vietoris. Always state which convention is in force; many theorems (e.g. the suspension isomorphism $\tilde H_n(X) \cong \tilde H_{n+1}(\Sigma X)$ ) hold only for reduced homology.

connection

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 $\operatorname{Hom}(H_n,G)$ and $\operatorname{Ext}(H_{n-1},G)$ .
Homotopy groups connect via the Hurewicz theorem: the first nonzero $\pi_n$ abelianizes to $H_n$ .
Poincaré duality for closed oriented $n$ -manifolds gives $H_k(M) \cong H^{n-k}(M)$ , 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.

🔬 Where this appears

⏳ History

- 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.

open-problem

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.

Algebraic topology — homology groups

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also