03.17.01 · modern-geometry / algebraic-topology-depth

Homology and cohomology — the Eilenberg-Steenrod axioms and de Rham

shipped3 tiersLean: none

Anchor (Master): Hatcher Ch. 2–3; Spanier — Algebraic Topology (McGraw-Hill 1966) Ch. 4–6; Bredon — Topology and Geometry (Springer GTM 139, 1993) Ch. III–V; full proofs of Poincaré duality and de Rham's theorem

Intuition Beginner

Homology counts the holes of a space, one count for each dimension. Where the fundamental group records the loops you cannot shrink, homology records, in each dimension $n$ , how many independent " $n$ -dimensional holes" the space has. The counts are packaged as abelian groups $H_{0}$ , $H_{1}$ , $H_{2}$ , and so on.

The zeroth group $H_{0}$ counts connected pieces: a space with three disjoint chunks has $H_{0}$ equal to three copies of $Z$ . The first group $H_{1}$ counts loops that are not filled in by a surface. The second group $H_{2}$ counts enclosed voids, like the hollow inside of a sphere $S^{2}$ .

The difference from the fundamental group is that homology is abelian and computable. Where $π_{1}$ can be a wildly non-abelian group, $H_{1}$ is always abelian — it is the abelian shadow of $π_{1}$ . And higher homology $H_{2}, H_{3}, \dots$ captures higher-dimensional holes that the fundamental group never sees.

Visual Beginner

Picture three spaces side by side: the sphere $S^{2}$ , the torus $T^{2}$ , and the double torus (a figure-eight shaped doughnut with two holes). Each has one connected piece, so $H_{0}$ is $Z$ for all three. But the higher groups differ, and the differences measure the holes.

Space	$H_{0}$	$H_{1}$	$H_{2}$	What the counts mean
Sphere $S^{2}$	$Z$	$0$	$Z$	one void, no un-filled loops
Torus $T^{2}$	$Z$	$Z^{2}$	$Z$	two loops (longitude, meridian), one void
Double torus	$Z$	$Z^{4}$	$Z$	four loops, one void

The number of copies of $Z$ in $H_{1}$ is the number of independent one-dimensional holes; the number in $H_{2}$ is the number of independent enclosed voids.

Worked example Beginner

Compute the homology of the torus $T^{2}$ using its cell structure. Build the torus as a square whose edges are glued in pairs: the two horizontal edges glue to one loop $a$ , the two vertical edges glue to a second loop $b$ , and the corners all meet at a single point.

This cell structure has one $0$ -cell (the corner point), two $1$ -cells (the loops $a$ and $b$ ), and one $2$ -cell (the square itself). So the chain groups are $C_{0} = Z$ , $C_{1} = Z^{2}$ , and $C_{2} = Z$ .

Now read off the boundary maps. Each $1$ -cell starts and ends at the single $0$ -cell, and the two endpoint signs cancel. So the boundary map from $C_{1}$ down to $C_{0}$ sends every $1$ -cell to zero. The boundary of the $2$ -cell, read around the square, is $a + b - a - b$ , which equals zero. So both boundary maps vanish.

Reading off the homology: $H_{0} = Z$ (one connected piece), $H_{1} = Z^{2}$ (the two free loops $a$ and $b$ ), and $H_{2} = Z$ (the whole torus surface, with zero boundary, is a genuine two-dimensional hole). The same recipe on the sphere $S^{2}$ — built from one $0$ -cell and one $2$ -cell with nothing in between — gives $H_{0} = Z$ , $H_{1} = 0$ , $H_{2} = Z$ .

Check your understanding Beginner

Formal definition Intermediate+

A chain complex $(C_{*}, \partial)$ over a commutative ring $R$ is a sequence of $R$ -modules and $R$ -linear maps

\dots \partial_{n + 2} C_{n + 1} \partial_{n + 1} C_{n} \partial_{n} C_{n - 1} \partial_{n - 1} \dots

with $\partial_{n} \circ \partial_{n + 1} = 0$ for every $n$ . Elements of $C_{n}$ are $n$ -chains; elements of $ker \partial_{n}$ are $n$ -cycles; elements of $im \partial_{n + 1}$ are $n$ -boundaries. The condition $\partial^{2} = 0$ says every boundary is a cycle. The $n$ -th homology is

H_{n} (C_{*}) := ker \partial_{n} / im \partial_{n + 1} .

Singular homology. For a topological space $X$ 02.01.01, the standard $n$ -simplex is $Δ^{n} = {(t_{0}, \dots, t_{n}) \in R^{n + 1} : t_{i} \geq 0, \sum t_{i} = 1}$ . A singular $n$ -simplex is any continuous map $σ : Δ^{n} \to X$ . The singular chain group $C_{n} (X; R)$ is the free $R$ -module on the set of singular $n$ -simplices. The boundary operator is the alternating restriction to faces:

\partial_{n} (σ) = i = 0 \sum n (- 1)^{i} σ ∣_{[v_{0}, \dots, \overset{v}{^}_{i}, \dots, v_{n}]},

where the hat denotes omission of the $i$ -th vertex. A signed-pair cancellation argument gives $\partial_{n - 1} \circ \partial_{n} = 0$ , so $(C_{*} (X; R), \partial)$ is a chain complex. Its homology is the singular homology $H_{n} (X; R)$ .

Simplicial and cellular homology. If $X$ is a $Δ$ -complex or a simplicial complex, the chain complex is the free module on the combinatorial simplices, with boundary given by the same signed face formula; this is simplicial homology 03.12.12. If $X$ is a CW complex 03.12.10 with $n$ -skeleton $X^{n}$ , the cellular chain complex has $C_{n}^{CW} (X) = H_{n} (X^{n}, X^{n - 1}; R)$ , a free $R$ -module on the $n$ -cells, with boundary the connecting map of the triple $(X^{n}, X^{n - 1}, X^{n - 2})$ . The cellular boundary formula computes this degree by degree: the coefficient of a cell $e_{β}^{n}$ in $\partial (e_{α}^{n})$ is the degree of the composite $S_{α}^{n - 1} \to X^{n - 1} \to S_{β}^{n - 1}$ .

Relative homology and pairs. For a pair $(X, A)$ with $A \subset X$ , the relative chain complex is the quotient $C_{*} (X, A) = C_{*} (X) / C_{*} (A)$ , and $H_{n} (X, A; R)$ is its homology. The short exact sequence of chain complexes $0 \to C_{*} (A) \to C_{*} (X) \to C_{*} (X, A) \to 0$ induces a long exact sequence of a pair.

Cohomology. Applying the contravariant functor $Hom_{R} (-, M)$ to a chain complex reverses the arrows and produces a cochain complex $(C^{*}, d)$ with $d^{n} : C^{n} \to C^{n + 1}$ and $d \circ d = 0$ . The $n$ -th cohomology is $H^{n} (C^{*}) = ker d^{n} / im d^{n - 1}$ . For singular cochains $C^{n} (X; M) = Hom (C_{n} (X), M)$ this gives singular cohomology $H^{n} (X; M)$ .

The cup product. The diagonal map $Δ : X \to X \times X$ and the cochain cross product combine to give a product on cohomology. For $[φ] \in H^{p} (X; R)$ and $[ψ] \in H^{q} (X; R)$ , the cup product is

[φ] ⌣ [ψ] := [Δ^{*} (φ \times ψ)] \in H^{p + q} (X; R) .

Under $⌣$ , the direct sum $H^{*} (X; R) = ⨁_{n} H^{n} (X; R)$ becomes a graded-commutative ring, the cohomology ring. Cohomology, not homology, carries this multiplicative structure: this is the decisive reason cohomology is the more powerful invariant for applications.

de Rham cohomology. On a smooth manifold $M$ [02.01.01, 48.01.01], the exterior derivative $d : Ω^{p} (M) \to Ω^{p + 1} (M)$ on differential $p$ -forms satisfies $d \circ d = 0$ , giving the de Rham complex $(Ω^{*} (M), d)$ . Its cohomology is the de Rham cohomology $H_{dR}^{*} (M)$ , a real vector space. de Rham's theorem identifies it with singular cohomology with real coefficients: $H_{dR}^{*} (M) ≅ H^{*} (M; R)$ .

Key theorem with proof Intermediate+

Theorem (long exact sequence of a pair). Let $(X, A)$ be a pair of topological spaces with $A \subset X$ , and fix a coefficient ring $R$ . There is a long exact sequence

\dots ⟶ H_{n} (A; R) i_{*} H_{n} (X; R) j_{*} H_{n} (X, A; R) \partial H_{n - 1} (A; R) ⟶ \dots

where $i_{*}$ is induced by the inclusion $i : A ↪ X$ , $j_{*}$ by the quotient $j : (X, \emptyset) \to (X, A)$ , and $\partial$ is the connecting homomorphism. The sequence is natural in the pair.

This is the content of the exactness axiom of Eilenberg-Steenrod: the proof derives it from a purely algebraic fact about short exact sequences of chain complexes.

Proof. The inclusion $i : A ↪ X$ induces an injective chain map $i_{#} : C_{*} (A) \to C_{*} (X)$ (a singular simplex of $A$ is, by inclusion, a singular simplex of $X$ ). The quotient map $j : C_{*} (X) \to C_{*} (X) / C_{*} (A) = C_{*} (X, A)$ is surjective. These fit into a short exact sequence of chain complexes

0 ⟶ C_{*} (A) i_{#} C_{*} (X) j_{#} C_{*} (X, A) ⟶ 0

that commutes with the boundary operators (a cycle relation is preserved by both maps).

The long exact sequence is then the standard zig-zag lemma for a short exact sequence of chain complexes. Define the connecting homomorphism $\partial : H_{n} (X, A) \to H_{n - 1} (A)$ by the recipe: lift a relative cycle $c \in C_{n} (X)$ with $j_{#} (c)$ a relative cycle (so $\partial c \in C_{n - 1} (A)$ ); since $c$ is a relative cycle, $\partial c$ lies in $ker j_{#} = im i_{#}$ , i.e. $\partial c = i_{#} (a)$ for some $a \in C_{n - 1} (A)$ . Because $\partial (\partial c) = 0$ and $i_{#}$ is injective, $a$ is a cycle in $C_{n - 1} (A)$ . Set $\partial ([j_{#} c]) := [a] \in H_{n - 1} (A)$ .

Well-definedness. If $c$ is replaced by $c + \partial b$ for $b \in C_{n} (X)$ , then $\partial (c + \partial b) = \partial c$ , so $a$ is unchanged. If $j_{#} (c)$ is replaced by $j_{#} (c) + j_{#} (\partial b^{'}) = j_{#} (c + \partial b^{'})$ with $b^{'} \in C_{n} (X)$ , again $\partial$ is unaffected. If a representative differs by a relative boundary $j_{#} (\partial \tilde{c})$ with $\tilde{c} \in C_{n + 1} (X)$ , then $a$ changes by the boundary $\partial (i_{#}^{- 1} \partial \tilde{c})$ , which is zero in homology. So $\partial$ is well-defined.

Exactness at $H_{n} (X)$ . $im i_{*} = ker j_{*}$ . A class $[c] \in H_{n} (X)$ maps to zero in $H_{n} (X, A)$ iff $j_{#} (c) = \partial d$ for some relative chain $d \in C_{n + 1} (X, A)$ ; lifting $d$ to $\tilde{d} \in C_{n + 1} (X)$ gives $c - \partial \tilde{d} \in ker j_{#} = im i_{#}$ , so $[c] = i_{*} [a]$ for some $[a] \in H_{n} (A)$ . The reverse inclusion $im i_{*} \subset ker j_{*}$ holds because $j_{#} \circ i_{#} = 0$ .

Exactness at $H_{n} (X, A)$ . $im j_{*} = ker \partial$ . A class $[j_{#} c]$ has $\partial ([j_{#} c]) = 0$ iff the corresponding cycle $a \in C_{n - 1} (A)$ is a boundary, $a = \partial a^{'}$ ; adjusting $c$ by the chain that $a^{'}$ bounds gives a cycle of $X$ , hence lies in $im j_{*}$ . Again $im j_{*} \subset ker \partial$ is direct from $i_{#} \circ \partial = 0$ on chains.

Exactness at $H_{n - 1} (A)$ . $im \partial = ker i_{*}$ . A class $[a] \in H_{n - 1} (A)$ lies in $im \partial$ iff some $c \in C_{n} (X)$ has $\partial c = i_{#} (a)$ ; this is exactly the condition that $i_{*} [a] = [i_{#} a] = [\partial c] = 0$ in $H_{n} (X)$ . Hence $im \partial = ker i_{*}$ .

Naturality in the pair follows because every step — the lift, the boundary, the quotient — is functorial in chain maps. $□$

The other four Eilenberg-Steenrod axioms are: homotopy (homotopic maps induce equal maps on homology), excision (if $Z \subset A$ with $\overline{Z} \subset int A$ , the inclusion $(X - Z, A - Z) \to (X, A)$ induces an isomorphism on homology), dimension ( $H_{0} (pt) = R$ , $H_{n} (pt) = 0$ for $n > 0$ ), and additivity (homology of a disjoint union is the direct sum). The five axioms together characterise ordinary homology on CW-pairs up to natural isomorphism ^{[EilenbergSteenrod1952]}.

Bridge. The long exact sequence of a pair builds toward the Mayer–Vietoris sequence and the cellular boundary formula, and it appears again in the construction of the Serre and Leray–Serre spectral sequences [03.13.01, 03.13.02], where it is iterated through the exact-couple machinery. The foundational reason this unit's machinery organises all of algebraic topology is that exactness, excision, and homotopy invariance are precisely the three properties any "homology-like" theory must satisfy; this is exactly the content of the Eilenberg-Steenrod axioms, and the bridge is that every generalised cohomology theory — K-theory, cobordism, sheaf cohomology — is obtained by varying these axioms.

Exercises Intermediate+

Exercise 3 (medium, proof). $H_{0}$ is path components.

Let $X$ be a topological space. Prove that $H_{0} (X; R)$ is the free $R$ -module on the set of path components of $X$ .

Hint

A $0$ -cycle is a formal $R$ -linear combination of points. Two points lie in the same $0$ -homology class iff they are joined by a path. Show a path $γ : [0, 1] \to X$ is a singular $1$ -simplex with $\partial γ = γ (1) - γ (0)$ .

Answer

$C_{0} (X; R)$ is the free $R$ -module on the points of $X$ . Since $C_{- 1} = 0$ , every $0$ -chain is a cycle, so $H_{0} = C_{0} / im \partial_{1}$ . A singular $1$ -simplex is a path $γ : Δ^{1} \to X$ , identified with a path $[0, 1] \to X$ after reparametrisation; its boundary is $\partial_{1} (γ) = γ (1) - γ (0)$ . So in the quotient, the endpoints of any path are identified.

Pick a representative point $x_{α}$ in each path component $α$ . For any point $x$ in component $α$ , a path from $x_{α}$ to $x$ gives $[x] = [x_{α}]$ in $H_{0}$ . Different components cannot be identified (there is no path between them). So $H_{0} (X; R) ≅ ⨁_{α} R$ , one summand per path component. $□$

Exercise 5 (medium, proof). Mayer–Vietoris by excision.

Let $X = A \cup B$ with $A, B$ open. Derive the Mayer–Vietoris long exact sequence:

\dots \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 \dots

from the long exact sequence of the pair $(A \cup B, A)$ together with excision.

Hint

Excision of $B$ from $(X, A)$ — equivalently the hypothesis $\overline{B} \subset int A$ replaced by the open-cover form — gives $H_{n} (X, A) ≅ H_{n} (B, A \cap B)$ . Then splice the long exact sequence of $(X, A)$ with that of $(B, A \cap B)$ .

Answer

The open-cover form of excision gives $H_{n} (X, A) ≅ H_{n} (B, A \cap B)$ because $X - B \subset A$ and the cover hypothesis is satisfied. Write the long exact sequence of the pair $(B, A \cap B)$ :

\dots \to H_{n} (A \cap B) \to H_{n} (B) \to H_{n} (B, A \cap B) \to H_{n - 1} (A \cap B) \to \dots

Substituting $H_{n} (B, A \cap B) ≅ H_{n} (X, A)$ and splicing with the long exact sequence of $(X, A)$ , which contains $H_{n} (A) \to H_{n} (X) \to H_{n} (X, A)$ , replaces the relative term by a map out of $H_{n} (X)$ . The diagonal map $H_{n} (A \cap B) \to H_{n} (A) \oplus H_{n} (B)$ (inclusion of each piece) and the subtraction map $H_{n} (A) \oplus H_{n} (B) \to H_{n} (X)$ emerge from the splicing. The result is the displayed Mayer–Vietoris sequence. $□$

Exercise 6 (hard, proof). de Rham via Mayer–Vietoris.

Using the Mayer–Vietoris sequence and induction on the number of sets in a good cover, prove de Rham's theorem for a smooth manifold $M$ admitting a finite good cover: $H_{dR}^{*} (M) ≅ H^{*} (M; R)$ .

Hint

Both theories satisfy the homotopy axiom (homotopic maps agree) and both give the same answer on a contractible open set (namely $R$ in degree $0$ and $0$ elsewhere). Both have a Mayer–Vietoris sequence for an open cover. Induct on the number of good-cover sets, using the five-lemma.

Answer

Let $f : H_{dR}^{*} (-) \to H^{*} (-; R)$ be the de Rham map (integration of forms over cycles descends to a map on cohomology). Both sides are homotopy functors: a homotopy equivalence induces an isomorphism on each, and both send a point to $R$ in degree $0$ and zero elsewhere. Both satisfy the Mayer–Vietoris property for a two-set open cover: de Rham by the partition-of-unity construction on forms, singular by Exercise 5. The de Rham map is a chain map between the two Mayer–Vietoris sequences.

Induct on the number $k$ of sets in a finite good cover. For $k = 1$ , $M$ is diffeomorphic to $R^{n}$ (contractible), so both sides are $R$ in degree $0$ and zero elsewhere, and $f$ is an isomorphism. For the inductive step, write $M = U \cup V$ with $U$ a union of $k - 1$ good-cover sets and $V$ a single good-cover set, so $U$ , $V$ , and $U \cap V$ each admit good covers with fewer than $k$ sets. The de Rham map gives a morphism of Mayer–Vietoris sequences relating the de Rham and singular theories of $(U, V, U \cap V, M)$ . By the inductive hypothesis it is an isomorphism on $U$ , $V$ , and $U \cap V$ ; by the five-lemma it is an isomorphism on $M$ . $□$

This is the standard Bott–Tu proof.

Exercise 7 (hard, symbolic). Homology of real projective space.

Using the CW structure on $RP^{n}$ with one cell in each dimension $0, 1, \dots, n$ , compute the cellular chain complex with $Z$ and $Z /2$ coefficients, and read off $H_{k} (RP^{n}; Z)$ and $H_{k} (RP^{n}; Z /2)$ .

Hint

The cellular boundary $C_{k} \to C_{k - 1}$ is multiplication by $1 + (- 1)^{k}$ . So it is zero for $k$ odd and multiplication by $2$ for $k$ even.

Answer

With $Z$ coefficients, $C_{k} = Z$ for $0 \leq k \leq n$ , and $\partial_{k}$ is multiplication by $1 + (- 1)^{k}$ : zero for $k$ odd, $2$ for $k$ even. Reading off:

$H_{0} (RP^{n}; Z) = Z$ ;
$H_{k} (RP^{n}; Z) = 0$ for $k$ odd, $0 < k < n$ ;
$H_{k} (RP^{n}; Z) = Z /2$ for $k$ even, $0 < k < n$ ;
$H_{n} (RP^{n}; Z) = Z$ if $n$ is odd, and $0$ if $n$ is even.

With $Z /2$ coefficients, every boundary map is zero (since $2 = 0$ ), so $H_{k} (RP^{n}; Z /2) = Z /2$ for $0 \leq k \leq n$ . This uniform answer is the shadow of Steenrod-squaring structure on the mod- $2$ cohomology ring.

Lean formalization Intermediate+

lean_status: none — see the unit metadata Mathlib gap analysis note. Mathlib's homological-algebra layer (Mathlib.Algebra.Homology.HomologicalComplex, Mathlib.CategoryTheory.Abelian.Basic) provides chain complexes, the snake lemma, and the long exact sequence of a short exact sequence of complexes at the level of abstract homological algebra. Simplicial sets and their realisation are present in Mathlib.AlgebraicTopology.SimplicialObject.

What is not yet formalised at the level this unit requires: singular homology $H_{n} (X; G)$ as the homology of the singular chain complex of a topological space, stated with the Eilenberg-Steenrod axioms as a theorem package; the cellular boundary formula and the isomorphism between cellular and singular homology of a CW complex; the cup product on singular cohomology; de Rham's theorem as an explicit isomorphism; and Poincaré duality for closed orientable manifolds. Until these are in place, downstream units invoke the statements by hand.

Advanced results Master

Cellular homology computes singular homology. For any CW complex $X$ , the cellular chain complex $C_{*}^{CW} (X)$ has homology naturally isomorphic to the singular homology of $X$ . The proof uses the long exact sequences of the triples $(X^{n}, X^{n - 1}, X^{n - 2})$ to identify $H_{n} (X^{n}, X^{n - 1}) ≅ H_{n} (X)$ after quotienting by the adjacent skeleton, and then reads off the cellular boundary formula. This is what makes CW complexes the workhorse setting for homology computation: a finite CW structure reduces $H_{*} (X)$ to linear algebra over a finitely generated free module.

The Hurewicz theorem. For a path-connected space $X$ , the natural map $h : π_{1} (X, x_{0}) \to H_{1} (X; Z)$ — send a loop to the homology class of the singular $1$ -cycle it traces — is the abelianisation: $H_{1} (X; Z) ≅ π_{1} (X, x_{0}) / [π_{1}, π_{1}]$ . More generally, if $X$ is $(n - 1)$ -connected for $n \geq 2$ (so $π_{k} (X) = 0$ for $k < n$ ), the Hurewicz homomorphism $π_{n} (X) \to H_{n} (X; Z)$ is an isomorphism, and $π_{n + 1} (X) \to H_{n + 1} (X; Z)$ is surjective. The Hurewicz theorem is the precise sense in which homology is the first homotopy-theoretic shadow of a space ^{[Hurewicz1935]}.

The universal coefficient theorems. Homology and cohomology with arbitrary coefficient groups are recovered from integral homology by algebraic operations. The universal coefficient theorem for homology gives a short exact sequence $0 \to H_{n} (X; Z) \otimes G \to H_{n} (X; G) \to Tor (H_{n - 1} (X; Z), G) \to 0$ that splits (unnaturally). The universal coefficient theorem for cohomology gives $0 \to Ext (H_{n - 1} (X; Z), G) \to H^{n} (X; G) \to Hom (H_{n} (X; Z), G) \to 0$ , also splitting. The $Ext$ term is the algebraic signature of cohomology being the linear dual of homology with a one-degree shift.

Poincaré duality. For a closed orientable $n$ -manifold $M$ , cap product with a fundamental class $[M] \in H_{n} (M; Z)$ gives an isomorphism $H^{k} (M; Z) \cap [M] H_{n - k} (M; Z)$ for every $k$ . In de Rham language, the wedge product paired with integration gives a perfect pairing $H_{dR}^{k} (M) \times H_{dR}^{n - k} (M) \to R$ . Poincaré duality is the statement that, on a closed orientable manifold, homology and cohomology are reflections of each other across the middle dimension. It underlies the intersection form on $H^{n /2}$ of even-dimensional manifolds, and hence the classification of simply connected $4$ -manifolds (Freedman, Donaldson) ^{[Poincare1895]}.

The Künneth formula. For spaces $X, Y$ over a field $k$ , $H^{n} (X \times Y; k) ≅ ⨁_{p + q = n} H^{p} (X; k) \otimes H^{q} (Y; k)$ , and the cohomology ring is the graded tensor product. Over $Z$ the formula acquires a $Tor$ correction. The Künneth formula is the homological shadow of the product of spaces, and the Leray–Hirsch / splitting-principle theorems are its fibrewise refinements.

de Rham's theorem and Hodge theory. de Rham's theorem identifies $H_{dR}^{*} (M) ≅ H^{*} (M; R)$ . When $M$ carries a Riemannian metric 48.01.01, the Hodge theorem identifies each de Rham class with a unique harmonic representative. Combined with Poincaré duality, this gives the Hodge decomposition on a Kähler manifold, the gateway to modern complex and algebraic geometry.

Synthesis. The Eilenberg-Steenrod axioms are the foundational reason every "homology-like" theory shares the same formal behaviour, and the long exact sequence of a pair is the central insight that propagates local data into global invariants; this is exactly the structure that iterates into the spectral sequences of 03.13.01, generalises to generalised cohomology theories and sheaf cohomology 04.03.01, and is dual to the cup-product ring structure that makes cohomology the richer invariant. Putting these together, the bridge is that the five axioms — homotopy, exactness, excision, dimension, additivity — pin down ordinary homology up to natural isomorphism on CW-pairs, and every departure from "ordinary" (K-theory, cobordism, sheaf cohomology, de Rham) is a controlled variation of one axiom while preserving the others. The Hurewicz theorem ties homology back to homotopy [03.12.01, 03.12.19], and Poincaré duality ties cohomology to the geometry of manifolds 48.01.01; both are instances of the same axiom-driven machinery.

Full proof set Master

Proposition ( $\partial^{2} = 0$ for singular chains). For every $n \geq 1$ and every singular $n$ -simplex $σ : Δ^{n} \to X$ , $\partial_{n - 1} (\partial_{n} σ) = 0$ . Hence $\partial \circ \partial = 0$ on all of $C_{*} (X)$ .

Proof. Write $F_{i} : Δ^{n - 1} \to Δ^{n}$ for the $i$ -th face inclusion (omitting the $i$ -th vertex), so $\partial_{n} σ = \sum_{i = 0}^{n} (- 1)^{i} σ \circ F_{i}$ . Then

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

where $F_{j}^{(i)}$ is the $j$ -th face of $Δ^{n - 1}$ embedded into $Δ^{n}$ through $F_{i}$ . The standard identity on face inclusions is $F_{i} \circ F_{j}^{(i)} = F_{j + 1} \circ F_{j}^{(j + 1 ?)}$ for $i \leq j$ , equivalently the two composites $F_{i} \circ F_{j + 1}$ (omitting $i$ then $j + 1$ ) and $F_{j + 1} \circ F_{i}$ (omitting $j + 1$ then $i$ , relabelled) both omit the same pair of vertices. Concretely $F_{i} \circ F_{j - 1} = F_{j} \circ F_{i}$ for $i < j$ : both maps omit the vertices $v_{i}$ and $v_{j}$ , in either order. The two terms in the double sum corresponding to the pair $(i, j)$ with $i < j$ thus have signs $(- 1)^{i + (j - 1)}$ and $(- 1)^{j + i}$ , which are negatives of each other. They cancel. Every term is paired with a unique cancelling partner, so the sum is zero. $□$

Proposition (naturality of the connecting homomorphism). A map of pairs $f : (X, A) \to (Y, B)$ induces a commutative square

H_{n} (X, A) f_{*} ↓ H_{n} (Y, B) \partial \partial H_{n - 1} (A) ↓ (f ∣_{A})_{*} H_{n - 1} (B)

Proof. Chase a class $[c] \in H_{n} (X, A)$ represented by a relative cycle $c \in C_{n} (X)$ with $\partial c \in C_{n - 1} (A)$ . Its boundary under $\partial$ is $[a]$ where $a \in C_{n - 1} (A)$ is the unique chain with $i_{#} a = \partial c$ . Apply $f_{*}$ : $f_{#} (c) \in C_{n} (Y)$ is a relative cycle, and $\partial (f_{#} c) = f_{#} (\partial c) = f_{#} (i_{#} a) = i_{#} ((f ∣_{A})_{#} a)$ . So the boundary of $f_{#} (c)$ in $C_{n - 1} (B)$ is $(f ∣_{A})_{#} (a)$ , giving $\partial (f_{*} [c]) = (f ∣_{A})_{*} [a] = (f ∣_{A})_{*} (\partial [c])$ . $□$

Proposition (Hurewicz map is a homomorphism). For a path-connected space $X$ with basepoint $x_{0}$ , the map $h : π_{1} (X, x_{0}) \to H_{1} (X; Z)$ sending $[γ]$ to the homology class of the singular $1$ -cycle $γ$ is a group homomorphism, and its kernel is the commutator subgroup $[π_{1}, π_{1}]$ .

Sketch. That $h$ is a homomorphism: the concatenation $γ_{1} \cdot γ_{2}$ is, as a singular $1$ -chain, homologous to $γ_{1} + γ_{2}$ (a reparametrisation of $[0, 1]$ realises the difference as a boundary of a singular $2$ -simplex). So $h ([γ_{1}] [γ_{2}]) = h ([γ_{1}]) + h ([γ_{2}])$ .

The kernel: $H_{1}$ is abelian, so the commutator subgroup is contained in $ker h$ . For the reverse, the singular chain complex in degree one, after quotienting by boundaries, is generated by loops modulo the relation that two loops homotopic through a "filled-in disk" are identified. Identifying each generator with its image in the abelianisation of $π_{1}$ gives a well-defined isomorphism $H_{1} (X; Z) \sim π_{1} (X, x_{0})_{ab}$ , inverse to $h$ post-composed with abelianisation. So $ker h = [π_{1}, π_{1}]$ . $□$

Connections Master

Homotopy and the fundamental group 03.12.01 — the prerequisite invariant: the Hurewicz theorem identifies $H_{1}$ with the abelianisation of $π_{1}$ , and identifies $π_{n}$ with $H_{n}$ for the first non-vanishing dimension. Homology is the abelian, computable shadow of homotopy. By conn:hurewicz.bridge, $H_{1}$ is the abelianisation of $π_{1}$ (the central bridge from homotopy to homology, expanded in 03.12.19).
Singular, simplicial, and cellular homology [03.12.11, 03.12.12, 03.12.13] — the three concrete avatars of the single axiomatic theory. The Eilenberg-Steenrod axioms guarantee these agree on CW-pairs; cellular homology is the one used for hand computation. The unit here is the axiomatic unification; the per-theory units carry the chain-level details.
Eilenberg-Steenrod axioms (dedicated unit) 03.12.15 — the full axiomatic treatment, including the uniqueness theorem and the extension to generalised cohomology theories. The present unit states the axioms and proves the exactness axiom in detail; 03.12.15 covers the remaining four and the uniqueness proof.
Spectral sequences 03.13.01 — the long exact sequence of a pair, iterated through the exact-couple machinery of Massey, becomes the spectral sequence of a filtered complex. The Leray–Serre spectral sequence 03.13.02 computes the homology of a total space from base and fibre, exactly the propagation pattern of the exactness axiom at higher bookkeeping resolution. By conn:les.iterates, every spectral sequence is an iterated long exact sequence (specialisation).
Poincaré duality and the cap product [03.12.16, 03.12.17] — the duality between homology and cohomology on a closed orientable manifold, implemented by cap product with the fundamental class. Stated here as a theorem; the dedicated unit carries the proof via the dual cell decomposition.
De Rham cohomology and the Čech–de Rham complex [03.04.06, 03.04.11] — de Rham's theorem identifies the analytic theory of differential forms with the topological theory of singular cochains. The Čech–de Rham double complex of 03.04.11 is the prototype filtered complex whose spectral sequence gives the de Rham isomorphism; the hypercohomology framework of 03.04.14 is the sheaf-theoretic generalisation.
Sheaf cohomology 04.03.01 — the cohomology of a space with coefficients in a sheaf is the natural generalisation of singular cohomology beyond constant coefficients. The Leray spectral sequence computes it from the higher direct image sheaves, unifying the topological and analytic theories.
Riemannian geometry and Hodge theory 48.01.01 — a Riemannian metric selects a harmonic representative in each de Rham class (Hodge theorem), refining the bare de Rham isomorphism into the Hodge decomposition on Kähler manifolds. This is the gateway from algebraic topology to complex and algebraic geometry.

Historical & philosophical context Master

Homology begins with Henri Poincaré's 1895 memoir Analysis Situs ^{[Poincare1895]}, where he introduced the Betti numbers (named after Enrico Betti) and the incidence matrices that encoded how $n$ -dimensional cells meet $(n - 1)$ -dimensional cells — the precursor of the boundary operator. Poincaré's motivating problem was the classification of manifolds up to homeomorphism, and his central observation was the Euler–Poincaré formula $χ = \sum_{n} (- 1)^{n} β_{n}$ , identifying the alternating sum of the Betti numbers with the classical Euler characteristic. The memoir did not have the modern chain-complex formalism; that emerged only in the 1920s through the work of Emmy Noether and her school, who recognised that "homology" should take values in abelian groups rather than merely in integer Betti numbers. The shift from numbers to groups is the decisive abstraction that made modern algebraic topology possible.

The axiomatic foundation came with Samuel Eilenberg and Norman Steenrod's 1952 book Foundations of Algebraic Topology ^{[EilenbergSteenrod1952]}, which proposed that a homology theory is anything satisfying the five axioms — homotopy, exactness, excision, dimension, additivity. The axioms isolate the properties that make the theory computable, and the uniqueness theorem shows that on CW-pairs there is only one such theory with the prescribed coefficient group. The axiomatic viewpoint made it possible to define generalised homology theories — K-theory, cobordism, stable homotopy — by relaxing the dimension axiom. Every cohomology theory in modern topology is, by design, a variation on the Eilenberg-Steenrod axioms.

Georges de Rham's 1931 theorem ^[DeRham1931] settled a question going back to Poincaré and Élie Cartan: the topological obstructions measured by singular cohomology with real coefficients are exactly the obstructions to solving $d ω = η$ on a smooth manifold, i.e. the closed differential forms modulo the exact ones. De Rham's proof used a double-complex argument that anticipates the sheaf-theoretic machinery later codified by Leray, Cartan, and Serre. The theorem is the cornerstone of differential topology: it is what licenses the replacement of topological data (singular cochains) with analytic data (differential forms) whenever a smooth structure is present.

The Hurewicz theorem — published by Witold Hurewicz in 1935–1936 in his four-part Beiträge zur Topologie der Deformationen — established the precise bridge from homotopy to homology: the first non-vanishing homotopy and homology groups of a simply connected space agree. This result, together with the higher homotopy groups Hurewicz introduced in the same papers, completed the conceptual triangle linking homotopy, homology, and the geometry of manifolds that has organised algebraic topology ever since.

Bibliography Master

Hatcher, A., Algebraic Topology, Cambridge University Press, 2002.
Spanier, E. H., Algebraic Topology, McGraw-Hill, 1966.
Bredon, G. E., Topology and Geometry, Graduate Texts in Mathematics 139, Springer, 1993.
Eilenberg, S. & Steenrod, N., Foundations of Algebraic Topology, Princeton University Press, 1952.
de Rham, G., "Sur l'analysis situs des variétés à $n$ dimensions", Journal de Mathématiques Pures et Appliquées 10 (1931), 115–200.
Poincaré, H., "Analysis Situs", Journal de l'École Polytechnique 1 (1895), 1–121.
Hurewicz, W., "Beiträge zur Topologie der Deformationen I–IV", Proc. Akad. Wet. Amsterdam 38–39 (1935–1936).
Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer, 1982.
Milnor, J. W. & Stasheff, J. D., Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press, 1974.
Weibel, C. A., An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994.

@book{Hatcher2002,
  author = {Hatcher, Allen},
  title  = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year   = {2002}
}
@book{Spanier1966,
  author = {Spanier, Edwin H.},
  title  = {Algebraic Topology},
  publisher = {McGraw-Hill},
  year   = {1966}
}
@book{Bredon1993,
  author = {Bredon, Glen E.},
  title  = {Topology and Geometry},
  series = {Graduate Texts in Mathematics},
  volume = {139},
  publisher = {Springer},
  year   = {1993}
}
@book{EilenbergSteenrod1952,
  author = {Eilenberg, Samuel and Steenrod, Norman},
  title  = {Foundations of Algebraic Topology},
  publisher = {Princeton University Press},
  year   = {1952}
}
@article{DeRham1931,
  author = {de Rham, Georges},
  title  = {Sur l'analysis situs des vari\'et\'es \`a $n$ dimensions},
  journal = {Journal de Math\'ematiques Pures et Appliqu\'ees},
  volume = {10},
  pages  = {115--200},
  year   = {1931}
}
@article{Poincare1895,
  author = {Poincar\'e, Henri},
  title  = {Analysis Situs},
  journal = {Journal de l'\'Ecole Polytechnique},
  volume = {1},
  pages  = {1--121},
  year   = {1895}
}
@article{Hurewicz1935,
  author = {Hurewicz, Witold},
  title  = {Beitr\"age zur Topologie der Deformationen I--IV},
  journal = {Proc. Akad. Wet. Amsterdam},
  volume = {38--39},
  year   = {1935--1936}
}
@book{BottTu1982,
  author = {Bott, Raoul and Tu, Loring W.},
  title  = {Differential Forms in Algebraic Topology},
  series = {Graduate Texts in Mathematics},
  volume = {82},
  publisher = {Springer},
  year   = {1982}
}
@book{MilnorStasheff1974,
  author = {Milnor, John W. and Stasheff, James D.},
  title  = {Characteristic Classes},
  series = {Annals of Mathematics Studies},
  number = {76},
  publisher = {Princeton University Press},
  year   = {1974}
}
@book{Weibel1994,
  author = {Weibel, Charles A.},
  title  = {An Introduction to Homological Algebra},
  series = {Cambridge Studies in Advanced Mathematics},
  volume = {38},
  publisher = {Cambridge University Press},
  year   = {1994}
}

Prerequisites

03.12.01
48.01.01

Tier anchors

beginner: Hatcher — Algebraic Topology (Cambridge 2002) §2.1 opening intuition: homology as counting n-dimensional holes; Richeson — Euler's Gem (Princeton 2008) Ch. 15–18 for plain-English holes and Euler characteristic
intermediate: Hatcher — Algebraic Topology (Cambridge 2002) Ch. 2 (homology) and Ch. 3 (cohomology); formal chain complexes, the Eilenberg-Steenrod axioms, cellular homology
master: Hatcher Ch. 2–3; Spanier — Algebraic Topology (McGraw-Hill 1966) Ch. 4–6; Bredon — Topology and Geometry (Springer GTM 139, 1993) Ch. III–V; full proofs of Poincaré duality and de Rham's theorem

References

Hatcher — Algebraic Topology · Cambridge University Press (2002). §2.1 singular homology and the boundary operator; §2.2 computations and cellular homology; §3.1–3.2 cohomology groups and the cup product; the Eilenberg-Steenrod axioms are used throughout as the organising scaffolding, with the long exact sequence of a pair proved from the short exact sequence of singular chain complexes $0 o C_{*} (A) o C_{*} (X) o C_{*} (X, A) o 0$ .
Eilenberg and Steenrod — Foundations of Algebraic Topology · Princeton University Press (1952). The axiomatic treatment: the dimension, exactness, excision, homotopy, and additivity axioms; the uniqueness theorem that these axioms determine ordinary homology on finite CW-pairs.
de Rham — Sur l'analysis situs des variétés à n dimensions · Journal de Mathématiques Pures et Appliquées (1931). The original proof that de Rham cohomology (closed modulo exact differential forms) is isomorphic to singular cohomology with real coefficients.
Poincaré — Analysis Situs · Journal de l'École Polytechnique 1 (1895), 1–121. The founding memoir: the Betti numbers, the incidence matrices (the precursor of the boundary operator), and the Euler–Poincaré relation.
Bott and Tu — Differential Forms in Algebraic Topology · Springer GTM 82 (1982). §1–5 de Rham cohomology; the Mayer–Vietoris argument; Poincaré duality on orientable manifolds via the wedge product pairing.

Estimated time

beginner: 20 min
intermediate: 55 min
master: 100 min