Cohomology — Simplicial and Singular

⚡ Key Takeaways

Cohomology assigns to a topological space $X$ a graded ring $H^*(X;R)$ built by dualizing the chain complex of simplices and taking the cohomology of the resulting cochain complex. The core result: simplicial and singular cohomology agree on triangulable spaces, both are homotopy invariants, and the cup product turns cohomology into a graded-commutative ring — strictly finer than homology. Applies across algebraic topology, geometry, and physics (gauge theory, anomalies). Developed by Alexander, Kolmogorov, Čech, Whitney, and Eilenberg–MacLane in the 1930s–1940s.

💡 The Core Insight

Homology measures cycles modulo boundaries by pushing chains forward; cohomology measures functions on chains that vanish on boundaries, modulo those that are themselves "derivatives." Dualizing flips the boundary map $\partial$ into a coboundary $\delta$ running in the opposite direction, and the resulting contravariant functor $X\mapsto H^*(X)$ acquires a multiplicative structure — the cup product — that makes cohomology a far sharper invariant than its dimensions alone suggest.

DefinitionSingular chain complex

For a space $X$ and the standard $n$ -simplex $\Delta^n=\{(t_0,\dots,t_n): t_i\ge 0,\ \sum t_i=1\}$ , a singular $n$ -simplex is a continuous map $\sigma:\Delta^n\to X$ . The singular chain group $C_n(X)$ is the free abelian group on these. The boundary operator is

1

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

where $d_i:\Delta^{n-1}\to\Delta^n$ is the $i$ -th face inclusion (omitting vertex $i$ ). One has $\partial_{n-1}\partial_n=0$ .

DefinitionSingular cochain complex and cohomology

Fix a commutative ring $R$ (often $\mathbb{Z}$ , a field, or $\mathbb{Z}/p$ ). The group of singular $n$ -cochains with coefficients in $R$ is $C^n(X;R)=\operatorname{Hom}_{\mathbb{Z}}(C_n(X),R)$ . The coboundary $\delta^n:C^n\to C^{n+1}$ is the dual of $\partial_{n+1}$ :

2

\quad (\delta^n\varphi)(\sigma)\;=\;\varphi(\partial_{n+1}\sigma)\;=\;\sum_{i=0}^{n+1}(-1)^i\,\varphi(\sigma\circ d_i),\qquad \varphi\in C^n,\ \sigma\in C_{n+1}.

Since $\partial\partial=0$ we get $\delta\delta=0$ . The $n$ -th singular cohomology group is

3

\quad H^n(X;R)\;=\;\ker\delta^n\big/\operatorname{im}\delta^{n-1}\;=\;Z^n(X;R)\big/B^n(X;R).

Elements of $Z^n=\ker\delta^n$ are cocycles; elements of $B^n=\operatorname{im}\delta^{n-1}$ are coboundaries.

DefinitionSimplicial cohomology

Let $K$ be a simplicial complex (or $\Delta$ -complex) with oriented simplices. Let $\Delta_n(K)$ be the free abelian group on $n$ -simplices and $\partial_n$ the simplicial boundary. Set $\Delta^n(K;R)=\operatorname{Hom}(\Delta_n(K),R)$ with coboundary dual to $\partial$ . The simplicial cohomology is

4

\quad H^n_{\Delta}(K;R)\;=\;\ker(\delta^n)\big/\operatorname{im}(\delta^{n-1}).

Because $K$ has finitely many simplices in each dimension when it is finite, $\Delta^n(K;R)$ is finitely generated and $H^n_\Delta$ is directly computable by linear algebra over $R$ .

TheoremSimplicial = Singular

If $X$ is the geometric realization $|K|$ of a simplicial (or $\Delta$ -) complex $K$ , the canonical chain map inducing $\Delta_*(K)\hookrightarrow C_*(X)$ induces, after dualizing, a natural isomorphism

5

\quad H^n_{\Delta}(K;R)\;\xrightarrow{\ \cong\ }\;H^n(|K|;R)\qquad\text{for all }n,\ R.

TheoremUniversal Coefficient Theorem for cohomology

For any space $X$ and abelian group $G$ there is a natural short exact sequence, split (non-naturally),

6

\quad 0\to \operatorname{Ext}^1_{\mathbb{Z}}\!\big(H_{n-1}(X;\mathbb{Z}),\,G\big)\to H^n(X;G)\xrightarrow{\ h\ }\operatorname{Hom}\!\big(H_n(X;\mathbb{Z}),\,G\big)\to 0.

The map $h$ is the evaluation (Kronecker) homomorphism $[\varphi]\mapsto\big([z]\mapsto\varphi(z)\big)$ .

TheoremCup product / ring structure

For cochains $\varphi\in C^p(X;R),\ \psi\in C^q(X;R)$ define, for a singular $(p+q)$ -simplex $\sigma:\Delta^{p+q}\to X$ ,

7

\quad (\varphi\smile\psi)(\sigma)\;=\;\varphi\big(\sigma|_{[v_0,\dots,v_p]}\big)\cdot\psi\big(\sigma|_{[v_p,\dots,v_{p+q}]}\big),

using front and back faces. Then $\delta(\varphi\smile\psi)=\delta\varphi\smile\psi+(-1)^p\varphi\smile\delta\psi$ , so $\smile$ descends to cohomology, making $H^*(X;R)=\bigoplus_n H^n(X;R)$ a graded-commutative ring:

8

\quad \alpha\smile\beta\;=\;(-1)^{|\alpha||\beta|}\,\beta\smile\alpha.

ProofCohomology is a homotopy-invariant functor

Step 1 — Contravariant functoriality. A continuous $f:X\to Y$ induces $f_\#:C_n(X)\to C_n(Y)$ , $\sigma\mapsto f\circ\sigma$ , a chain map. Dualizing reverses the arrow: $f^\#=\operatorname{Hom}(f_\#,R):C^n(Y;R)\to C^n(X;R)$ commutes with $\delta$ , hence induces $f^*:H^n(Y;R)\to H^n(X;R)$ with $(g\circ f)^*=f^*\circ g^*$ .

Step 2 — Chain homotopy from a homotopy. If $F:X\times[0,1]\to Y$ is a homotopy $f_0\simeq f_1$ , the prism (subdivision of $\Delta^n\times[0,1]$ ) gives a chain homotopy $P:C_n(X)\to C_{n+1}(Y)$ with $\partial P+P\partial=f_{1\#}-f_{0\#}$ .

Step 3 — Dualize the homotopy. Apply $\operatorname{Hom}(-,R)$ : $P^\#$ satisfies $\delta P^\#+P^\#\delta = f_1^\#-f_0^\#$ , i.e. $f_0^\#$ and $f_1^\#$ are cochain-homotopic.

Step 4 — Conclusion. Cochain-homotopic maps induce equal maps on cohomology: if $\varphi$ is a cocycle, $(f_1^\#-f_0^\#)\varphi=\delta(P^\#\varphi)\in B^n$ . Hence $f_0^*=f_1^*$ , and homotopy equivalences induce isomorphisms. $\square$

∎

ProofMayer–Vietoris in cohomology

Step 1. For an open cover $X=A\cup B$ , the short exact sequence of cochain complexes $0\to C^*(A+B)\to C^*(A)\oplus C^*(B)\to C^*(A\cap B)\to 0$ arises from restriction maps; exactness uses that small-simplex chains $C_*(A+B)\hookrightarrow C_*(X)$ are a quasi-isomorphism (excision/barycentric subdivision).

Step 2. The snake lemma yields the long exact sequence

9

\quad \cdots\to H^n(X)\to H^n(A)\oplus H^n(B)\to H^n(A\cap B)\xrightarrow{\ \delta^*\ } H^{n+1}(X)\to\cdots

Step 3. Naturality of the connecting map $\delta^*$ follows from naturality of the snake lemma; this gives a computational engine reducing $H^*(X)$ to pieces. $\square$

∎

mathematician · Samuel Eilenberg & Saunders MacLane | 1940s

Eilenberg (1913–1998) and MacLane (1909–2005) axiomatized homology and cohomology, introducing the Eilenberg–Steenrod axioms (with Steenrod) and the spaces $K(\pi,n)$ that represent cohomology: $H^n(X;G)\cong[X,K(G,n)]$ . Their invention of category theory (1945) was motivated precisely by the need to make "natural" transformations like the cup product and the UCT splitting precise. MacLane's work on cohomology of groups and Eilenberg's on singular theory unified the competing 1930s definitions into the modern functorial framework.

🧠 Intuition

Think of a $p$ -cochain as a "measuring device" that eats a $p$ -dimensional piece of the space and returns a number. The cocycle condition $\delta\varphi=0$ says the measurement is conservative: integrating $\varphi$ over the boundary of any $(p+1)$ -cell gives zero — a discrete Stokes' theorem. A coboundary $\varphi=\delta\psi$ is a measurement that is the "total change" of a lower-dimensional potential, hence trivially conservative. Cohomology = conservative measurements modulo exact ones = the genuinely "circulating" fields the space supports. The de Rham analogy is exact: differential forms are continuous cochains, $d$ is $\delta$ , closed/exact forms are cocycles/coboundaries, and the cup product is the wedge product $\wedge$ .

WA

Why a physicist cares

In gauge theory a connection is a 1-form and its field strength a 2-form; the magnetic monopole's charge is the integral of $F/2\pi$ over a sphere, an element of $H^2(S^2;\mathbb{Z})\cong\mathbb{Z}$ . Dirac quantization is the statement that this cohomology class is integral. Cup products encode anomalies and the ring structure of characteristic classes (Chern, Stiefel–Whitney). Discrete versions (lattice gauge theory, topological order) literally use simplicial cochains on a triangulated spacetime.

ExampleCohomology of the circle

S^1

Given: $S^1$ as a $\Delta$ -complex with one vertex $v$ and one edge $e$ (a loop). Coefficients $R=\mathbb{Z}$ .

Step 1 — Chain groups. $\Delta_0=\mathbb{Z}\langle v\rangle,\ \Delta_1=\mathbb{Z}\langle e\rangle$ , higher zero. Boundary $\partial_1 e = v - v = 0$ .

Step 2 — Cochains. $\Delta^0=\mathbb{Z},\ \Delta^1=\mathbb{Z}$ . Coboundary $\delta^0:\Delta^0\to\Delta^1$ is the transpose of $\partial_1=0$ , so $\delta^0=0$ .

Step 3 — Cohomology. $H^0=\ker\delta^0/0=\mathbb{Z}$ (since $\delta^0=0$ , all of $\Delta^0$ is cocycles). $H^1=\Delta^1/\operatorname{im}\delta^0=\mathbb{Z}/0=\mathbb{Z}$ .

Step 4 — Check via UCT. $H_0=\mathbb{Z},\ H_1=\mathbb{Z}$ , both free, so $\operatorname{Ext}=0$ and $H^n\cong\operatorname{Hom}(H_n,\mathbb{Z})$ , confirming the result.

\boxed{H^0(S^1;\mathbb{Z})=\mathbb{Z},\quad H^1(S^1;\mathbb{Z})=\mathbb{Z},\quad H^n=0\ (n\ge2).}

ExampleCohomology of

\mathbb{RP}^2

— torsion appears

Given: Real projective plane, with known integral homology $H_0=\mathbb{Z},\ H_1=\mathbb{Z}/2,\ H_2=0$ .

Step 1 — Apply UCT in degree 1. $H^1=\operatorname{Hom}(\mathbb{Z}/2,\mathbb{Z})\oplus\operatorname{Ext}(H_0,\mathbb{Z})=0\oplus 0=0.$

Step 2 — Degree 2 (the torsion shift). $H^2=\operatorname{Hom}(H_2,\mathbb{Z})\oplus\operatorname{Ext}^1(H_1,\mathbb{Z})=\operatorname{Hom}(0,\mathbb{Z})\oplus\operatorname{Ext}^1(\mathbb{Z}/2,\mathbb{Z}).$ Since $\operatorname{Ext}^1(\mathbb{Z}/2,\mathbb{Z})=\mathbb{Z}/2$ , we get $H^2=\mathbb{Z}/2$ .

Step 3 — With $\mathbb{Z}/2$ coefficients (ring structure). $H^*(\mathbb{RP}^2;\mathbb{Z}/2)=\mathbb{Z}/2[\alpha]/(\alpha^3)$ with $|\alpha|=1$ , so $\alpha\smile\alpha=\alpha^2\ne 0$ generates $H^2$ .

\boxed{H^0=\mathbb{Z},\ H^1=0,\ H^2=\mathbb{Z}/2\quad(\mathbb{Z}\text{ coeffs});\quad H^*( \cdot;\mathbb{Z}/2)=\mathbb{Z}/2[\alpha]/(\alpha^3).}

Note the contrast with homology: torsion lives in $H_1$ but in $H^2$ — UCT shifts it up by one. The nonzero cup square distinguishes $\mathbb{RP}^2$ from $S^1\vee S^2$ , which have the same homology groups but different rings.

⚠️ Cohomology is not just the dual of homology

A pervasive error is to assume $H^n(X;\mathbb{Z})\cong\operatorname{Hom}(H_n(X),\mathbb{Z})$ . This fails whenever $H_{n-1}$ has torsion: the $\operatorname{Ext}$ term in [6] contributes. For $\mathbb{RP}^2$ , homology has $\mathbb{Z}/2$ in degree 1, but cohomology has $\mathbb{Z}/2$ in degree 2, not degree 1. Over a field the $\operatorname{Ext}$ term vanishes and $H^n\cong\operatorname{Hom}(H_n,F)$ does hold — which is why people forget the correction.

⚠️ The cup product is not commutative on cochains

Formula [7] is not symmetric: $\varphi\smile\psi\ne\pm\psi\smile\varphi$ as cochains in general. Graded-commutativity [8] holds only after passing to cohomology, via a non-trivial chain homotopy (whose obstruction to vanishing yields the Steenrod squares $Sq^i$ ). Confusing cochain-level and cohomology-level identities is a classic source of false "proofs."

CounterexampleSame groups, different rings

$X = \mathbb{CP}^2$ and $Y=S^2\vee S^4$ have identical cohomology groups: $\mathbb{Z}$ in degrees $0,2,4$ . But in $\mathbb{CP}^2$ the generator $a\in H^2$ satisfies $a^2\ne 0$ (it generates $H^4$ ), while in $S^2\vee S^4$ the wedge kills all products, so $a^2=0$ . The cup product detects that $\mathbb{CP}^2$ is "twisted" while the wedge is not — these spaces are not homotopy equivalent, a fact invisible to homology or cohomology groups alone.

Figure

The coboundary

\delta

runs opposite to the boundary

\partial

, dualizing the chain complex

```svg

```

connection

Cohomology sits at a crossroads. Via the de Rham theorem, de Rham Cohomology of a smooth manifold is isomorphic (as rings, by the wedge/cup correspondence) to singular cohomology with $\mathbb{R}$ coefficients. Čech Cohomology agrees with singular on paracompact spaces and is the natural home for sheaf-theoretic generalizations, leading to Sheaf Cohomology and Derived Functors. The representability $H^n(X;G)\cong[X,K(G,n)]$ links cohomology to Homotopy Theory and Eilenberg–MacLane Spaces. Generalizing the Eilenberg–Steenrod axioms by dropping the dimension axiom yields Generalized Cohomology Theories such as K-theory and cobordism. The cup product is the degree-zero shadow of richer structure in Spectral Sequences and Steenrod Algebra.

🔬 Where this appears

⏳ History

- 1895–1904: Poincaré introduces homology and duality (Analysis Situs), implicitly the seed of cohomological duality but with no cochain language.

- 1925–1928: Emmy Noether reframes homology in terms of abelian groups, enabling the dual (Hom) construction.

- 1934–1936: J. W. Alexander and A. N. Kolmogorov independently introduce cochains and the cup/coboundary operations at the Moscow conference (1935); priority and sign conventions are debated.

- 1935: E. Čech defines cohomology via open covers and nerves — Čech Cohomology.

- 1937: H. Whitney coins the term "cup product" and clarifies its formula and properties.

- 1944: Eilenberg gives the modern, rigorous definition of singular homology and cohomology, with continuous maps of simplices.

- 1945: Eilenberg & Steenrod state the axioms characterizing (co)homology; Eilenberg & MacLane found category theory to express naturality.

- 1947–1952: Steenrod constructs the squares

Sq^i

; the failure of cochain-level commutativity becomes a powerful invariant. de Rham's theorem (1931, sharpened later) ties forms to cohomology.

- 1950s: Serre's spectral sequence and Leray's sheaf cohomology integrate singular cohomology into the broader machinery of modern algebraic topology and geometry.

open-problem

The cup product is only the first layer: the full Steenrod operations and the failure of strict commutativity at the cochain level encode deep secondary information, and computing the action of the Steenrod algebra on $H^*(X;\mathbb{Z}/p)$ for complicated $X$ remains hard. More broadly, the Adams spectral sequence built from $\operatorname{Ext}$ over the Steenrod algebra is the gateway to stable homotopy groups of spheres — still unknown in general. On the foundational side, while singular and Čech cohomology agree for nice (paracompact, locally contractible) spaces, they diverge for pathological spaces (e.g. the Warsaw circle), and characterizing exactly when sheaf, Čech, and singular theories coincide leads into subtle point-set and topos-theoretic questions still studied today.

Cohomology — Simplicial and Singular

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also