Singular homology

Intuition [Beginner]

A singular simplex in a space $X$ is any continuous way of placing a triangle (or tetrahedron, or higher analogue) inside $X$. The standard $n$-simplex is the solid triangle in dimension $n$ with $n+1$ vertices, and a singular $n$-simplex is just a continuous map from that standard piece into $X$. The map need not be injective. It can crumple the triangle, fold it onto itself, or send the whole thing to a single point. Anything continuous counts.

A singular chain is a formal integer combination of these triangles: take three triangles, subtract another, double a fourth. The bookkeeping is what algebraic topology calls a free abelian group on the set of all continuous maps from the standard simplex into $X$. Each triangle has a boundary made of its edges, each edge has a boundary made of its endpoints, and these boundaries fit together so that taking the boundary twice always gives zero.

The reason this construction matters: the boundary operator turns the formal combinations into an algebraic gadget called a chain complex, and the homology of that complex is the same for any two spaces of the same homotopy type. A circle, a coffee cup, and an annulus all share the same first homology. A sphere shares the same homology as a balloon and an apple. The chain complex sees through deformation.

Visual [Beginner]

A schematic showing the standard 2-simplex as a triangle with vertices labelled, three edges each oriented along the boundary, and the alternating sum picture: the boundary of the triangle is edge one, minus edge two, plus edge three.

A continuous map sends this standard triangle into the space $X$; its boundary is the formal alternating sum of the three boundary edges' images.

Worked example [Beginner]

Compute the homology of a single point. Call the space $X=\{p\}$.

Step 1. Identify the singular simplices. For each $n\ge 0$ there is exactly one continuous map from the standard $n$-simplex into $\{p\}$: the constant map sending everything to $p$. So the chain group $C_n$ in dimension $n$ is the free abelian group on one generator, which is just the integers.

Step 2. Identify the boundary. The boundary of the unique $n$-simplex is the alternating sum of its $n+1$ faces. Each face is the unique $(n-1)$-simplex on the point. So the boundary equals the alternating sum $1-1+1-\cdots$ with $n+1$ terms. When $n+1$ is even (so $n$ is odd) this sum is zero. When $n+1$ is odd (so $n$ is even and at least $2$) the sum equals one.

Step 3. Compute kernels and images. For $n$ odd the boundary is zero, so the kernel is all of the integers. For $n$ even the boundary is the identity, so the kernel is zero. The image of the boundary into dimension $n-1$ is the integers when $n$ is even and at least $2$, and zero otherwise.

Step 4. Take quotients. In dimension $0$ the kernel is the integers and the image is zero, giving the integers. In every higher dimension the kernel and the image cancel exactly, giving zero.

What this tells us: a point has one-dimensional homology in degree zero and nothing in higher degrees. The single integer in degree zero counts the connected component.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a topological space. The standard $n$-simplex is

$${\Delta }^n=\{(t_0,\dots ,t_n)\in {\mathbb{R}}^{n+1}:t_i\ge 0,\sum t_i=1\}.$$

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 the set of all singular $n$-simplices in $X$. An element of $C_n(X)$ is a formal finite sum $\sum n_{\sigma }\sigma$ with $n_{\sigma }\in \mathbb{Z}$.

For each $0\le i\le n$ the $i$-th face inclusion $d^i:{\Delta }^{n-1}\to {\Delta }^n$ embeds ${\Delta }^{n-1}$ as the face opposite the $i$-th vertex:

$$d^i(t_0,\dots ,t_{n-1})=(t_0,\dots ,t_{i-1},0,t_i,\dots ,t_{n-1}).$$

The boundary operator ${\partial }_n:C_n(X)\to C_{n-1}(X)$ is defined on a generator $\sigma$ by

$${\partial }_n\sigma =\sum _{i=0}^n(-1{)}^i\sigma \circ d^i,$$

extended by linearity. The fundamental relation ${\partial }_{n-1}\circ {\partial }_n=0$ holds and is checked by a direct computation on faces (a face omitting vertex $i$ then vertex $j$ equals the face omitting vertex $j$ then vertex $i$, with opposite signs).

The chain groups and boundaries together form the singular chain complex $C_{\bullet }(X)=(C_n(X),{\partial }_n{)}_{n\ge 0}$. The $n$-th singular homology of $X$ is

$$H_n(X)=\ker ({\partial }_n)/\mathrm{im}({\partial }_{n+1}).$$

A continuous map $f:X\to Y$ induces a chain map $f_{♯}:C_{\bullet }(X)\to C_{\bullet }(Y)$ by $f_{♯}\sigma =f\circ \sigma$, and hence a homomorphism $f_{\ast }:H_n(X)\to H_n(Y)$ for each $n$. The assignment $X\mapsto H_n(X)$ is a covariant functor from $\mathbf{Top}$ to $\mathbf{Ab}$.

The reduced singular homology ${\tilde{H}}_n(X)$ is defined for $n\ge 1$ as $H_n(X)$ and for $n=0$ as the kernel of the augmentation $\epsilon :C_0(X)\to \mathbb{Z}$, $\epsilon (\sum n_{x}x)=\sum n_x$, modulo the image of ${\partial }_1$. Reduced homology of a non-empty space differs from unreduced homology only in dimension zero, where ${\tilde{H}}_0(X)={\mathbb{Z}}^{({\pi }_0(X)-1)}$ counts components beyond the first.

For an abelian group $G$ the homology with coefficients in $G$ is the homology of the chain complex $C_{\bullet }(X){\otimes }_{\mathbb{Z}}G$, written $H_n(X;G)$. The case $G=\mathbb{Z}$ recovers integral homology.

Counterexamples to common slips

• A non-zero element of $C_n(X)$ need not be the image of any single singular simplex; it is a formal integer combination, which is essential because cancellation between simplices is what produces cycles.
• The boundary operator depends on the choice of orientation of ${\Delta }^n$ (the ordering of the vertices); changing the ordering changes signs, and the standard convention takes the vertices in their natural order.
• The chain group $C_n(X)$ is enormous (free on a set of cardinality at least the continuum for any non-discrete $X$), but the homology is computable by replacing $C_{\bullet }$ with smaller chain-equivalent subcomplexes (small simplices, simplicial chains, cellular chains).

Key theorem with proof [Intermediate+]

Theorem (homotopy invariance). Let $f,g:X\to Y$ be continuous maps. If $f$ is homotopic to $g$, then the induced maps $f_, g_*: H_n(X) \to H_n(Y)$areequalforevery$n$.*

Proof. The strategy is to construct a chain homotopy $P:C_n(X)\to C_{n+1}(Y)$ between the chain maps $f_{♯}$ and $g_{♯}$, which is to say a sequence of homomorphisms satisfying

$${\partial }_{n+1}P+P{\partial }_n=g_{♯}-f_{♯}.$$

Once we have such a $P$, every cycle $z\in \ker {\partial }_n$ satisfies $g_{♯}z-f_{♯}z={\partial }_{n+1}Pz$, which is a boundary, so $f_{♯}z$ and $g_{♯}z$ represent the same class in homology, giving $f_{\ast }=g_{\ast }$.

Let $h:X\times [0,1]\to Y$ be a homotopy from $f$ to $g$, so $h_0=f$ and $h_1=g$. The geometric idea is the prism construction: for each singular simplex $\sigma :{\Delta }^n\to X$, the composite

$$\sigma \times \mathrm{id}:{\Delta }^n\times [0,1]\to X\times [0,1]\stackrel{h}{\to }Y$$

defines a map from the prism ${\Delta }^n\times [0,1]$ into $Y$. Triangulate the prism by $n+1$ copies of ${\Delta }^{n+1}$. Concretely, label the bottom face of the prism with vertices $v_0,\dots ,v_n$ and the top with $w_0,\dots ,w_n$, and decompose ${\Delta }^n\times [0,1]$ as the union of the $(n+1)$-simplices $[v_0,\dots ,v_i,w_i,\dots ,w_n]$ for $i=0,\dots ,n$.

Define

$$P\sigma =\sum _{i=0}^n(-1{)}^{i}h\circ (\sigma \times \mathrm{id})\circ {\tau }_i,$$

where ${\tau }_i:{\Delta }^{n+1}\to {\Delta }^n\times [0,1]$ is the affine homeomorphism onto the $i$-th piece $[v_0,\dots ,v_i,w_i,\dots ,w_n]$. Extend by linearity to all of $C_n(X)$.

A direct computation on the faces of ${\Delta }^{n+1}$ gives

$$\partial P\sigma =(g\circ \sigma )-(f\circ \sigma )-P(\partial \sigma ).$$

The key step: the faces of $[v_0,\dots ,v_i,w_i,\dots ,w_n]$ split into the top face (which contributes $g\circ \sigma$ for $i=0$ in the alternating sum), the bottom face (which contributes $-f\circ \sigma$ for $i=n$), the side faces between adjacent prism slices (which cancel pairwise because two consecutive slices share a common face with opposite orientation), and the side faces coming from the boundary of ${\Delta }^n$ (which assemble into $-P(\partial \sigma )$). The signed combination is exactly the chain-homotopy identity. Hence $\partial P+P\partial =g_{♯}-f_{♯}$ as required. $\square$

Bridge. This builds toward the entire functorial machinery of singular homology: every theorem that follows — Mayer-Vietoris, excision, the Eilenberg-Steenrod axioms — depends on homotopy invariance because it lets one replace a space by any homotopy-equivalent simpler space before computing. The foundational reason the prism construction works is exactly the geometric content of homotopy: a homotopy provides a continuous interpolation, and triangulating the cylinder produces the chain-level interpolation that translates the geometric picture into algebra. This is exactly the same chain-homotopy pattern that appears again in 03.12.13 (cellular homology), where the cellular and singular complexes are shown chain-homotopy-equivalent, and it generalises directly to any homotopy-invariant functor on chain complexes. The bridge is the recognition that homotopy invariance and chain homotopy are not two facts but one fact viewed from two sides: the geometric cylinder $X\times [0,1]$ on the topology side and the prism operator $P$ on the chain-complex side. Putting these together, the singular chain complex carries enough homotopical data to be a fully faithful invariant of the homotopy type at the level of homology, and the same machine appears again in 03.12.14 (excision) and in 03.12.15 (the Eilenberg-Steenrod axiomatisation) as the homotopy axiom that any homology theory must satisfy.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib has the abstract singular set construction in AlgebraicTopology.SingularSet and the homology of a chain complex in Algebra.Homology.HomologicalComplex, but does not yet ship a unified named singular-homology functor. The intended definition would read schematically:

[object Promise]

The proof gap is real and substantive. The boundary's ${\partial }^2=0$ requires the alternating-face-sum identity on the simplicial-set side; the homotopy invariance requires the prism operator and its chain-homotopy identity. Both are formalisable from the existing SimplicialObject machinery but have not yet been packaged as a named theorem.

Advanced results [Master]

Theorem (Mayer-Vietoris). Let $X=U\cup V$ with $U,V\subseteq X$ open. There is a long exact sequence

$$\cdots \to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(X)\to H_{n-1}(U\cap V)\to \cdots$$

natural in the triple $(X;U,V)$.

The connecting homomorphism is the boundary morphism of the short exact sequence of chain complexes $0\to C_{\bullet }(U\cap V)\to C_{\bullet }(U)\oplus C_{\bullet }(V)\to C_{\bullet }^{U+V}(X)\to 0$, combined with the isomorphism $H_n(C_{\bullet }^{U+V}(X))\cong H_n(X)$ furnished by barycentric subdivision (the small-simplices theorem). Mayer-Vietoris is the singular-homology incarnation of the Brown-sheaf-cohomology Mayer-Vietoris sequence; both descend from the same descent property of the underlying functor.

Theorem (computation of spheres). For $k\ge 1$,

$$H_n(S^k)=\{\begin{array}{ll}\mathbb{Z}& n=0\text{ or }n=k,\\ 0& \text{otherwise.}\end{array}$$

The argument is by induction on $k$ using Mayer-Vietoris on $S^k=U\cup V$ where $U$ and $V$ are slightly enlarged hemispheres (each contractible) with intersection homotopy-equivalent to $S^{k-1}$. The base case $S^0$ is two points, where $H_0(S^0)={\mathbb{Z}}^2$, equivalently ${\tilde{H}}_0(S^0)=\mathbb{Z}$.

Theorem (relative homology and the long exact sequence of a pair). For a pair $(X,A)$ with $A\subseteq X$, the relative chain complex is $C_{\bullet }(X,A)=C_{\bullet }(X)/C_{\bullet }(A)$, and its homology $H_n(X,A)$ fits into a long exact sequence

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

This is the standard long exact sequence associated to the short exact sequence $0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X,A)\to 0$ of chain complexes. Relative homology is the technical workhorse for excision and for the Eilenberg-Steenrod axioms.

Theorem (functoriality and naturality). The assignment $X\mapsto C_{\bullet }(X)$ is a functor from $\mathbf{Top}$ to the category of non-negatively graded chain complexes of abelian groups. Composition with the homology functor gives a sequence of functors $H_n:\mathbf{Top}\to \mathbf{Ab}$, and the homotopy-invariance theorem upgrades this to functors on the homotopy category $\mathrm{Ho}(\mathbf{Top})$.

Theorem (universal coefficients, mention). For any abelian group $G$ there is a natural short exact sequence

$$0\to H_n(X)\otimes G\to H_n(X;G)\to \mathrm{Tor}(H_{n-1}(X),G)\to 0$$

which splits unnaturally. The full proof and discussion belong to a future unit on the universal coefficient theorem; the statement is recorded here because most computations with coefficients reduce to this exact sequence.

Synthesis. The singular chain complex is the foundational reason that homology behaves as well as it does: it is defined by a free-abelian-group functor on a set of continuous maps, the boundary is universal in the sense that it is the alternating sum forced by the simplicial structure, and homotopy invariance is exactly the chain-homotopy translation of the cylinder. The central insight is that the same data — singular simplices in $X$ — produce three faces of one structure: the chain complex (from which homology is computed), the simplicial set $\mathrm{Sing}(X)$ (which is the formal homotopical shadow of $X$), and the singular cubical / simplicial functor (which encodes the Eilenberg-Zilber structure on products). This is dual to the singular cochain construction, where one passes to homomorphisms into a coefficient group; the duality is sharpened by the universal coefficient theorem. The bridge is the identification of singular homology with the homotopy groups of the chain complex viewed as a stable invariant: putting these together, the Dold-Kan correspondence identifies $C_{\bullet }$ with the normalised Moore complex of $\mathrm{Sing}(X)$, and the Eilenberg-Mac Lane spectrum $H\mathbb{Z}$ realises singular homology as the generalised homology theory associated to a spectrum. The same colimit-and-chain-complex pattern appears again in cellular homology 03.12.13, where the singular complex is replaced by a much smaller chain-equivalent cellular complex, and in cohomology theories generally, where singular cohomology is the prototype of an Eilenberg-Steenrod cohomology theory and generalises to $K$-theory and bordism by replacing $H\mathbb{Z}$ with another spectrum.

Full proof set [Master]

Theorem (Mayer-Vietoris), proof. Let $X=U\cup V$ with $U,V$ open. Define $C_{\bullet }^{U+V}(X)\subseteq C_{\bullet }(X)$ as the subcomplex generated by singular simplices whose image lies entirely in $U$ or entirely in $V$. Consider the sequence

$$0\to C_{\bullet }(U\cap V)\stackrel{\Phi }{\to }{C}_{\bullet }(U)\oplus C_{\bullet }(V)\stackrel{\Psi }{\to }{C}_{\bullet }^{U+V}(X)\to 0$$

where $\Phi (c)=(c,c)$ and $\Psi (a,b)=a-b$ (with $a,b$ viewed inside $C_{\bullet }(X)$). Exactness at the left is the observation that a chain in $C_{\bullet }(U\cap V)$ vanishes in $C_{\bullet }(U)\oplus C_{\bullet }(V)$ only if it is zero. Exactness in the middle: $\Psi \Phi (c)=c-c=0$, and conversely if $\Psi (a,b)=0$ then $a=b$ in $C_{\bullet }(X)$, which forces every simplex of $a$ to land in $U\cap V$, giving an element $c=a$ of $C_{\bullet }(U\cap V)$ with $\Phi (c)=(a,b)$. Exactness at the right is by definition: every generator of $C_{\bullet }^{U+V}(X)$ sits in $C_{\bullet }(U)$ or in $C_{\bullet }(V)$, so it lies in the image of $\Psi$.

The associated long exact sequence in homology reads

$$\cdots \to H_n(U\cap V)\to H_n(U)\oplus H_n(V)\to H_n(C_{\bullet }^{U+V}(X))\to H_{n-1}(U\cap V)\to \cdots$$

The small-simplices theorem (proof: barycentric subdivision iterated; see 03.12.14 for the full argument) gives a chain-homotopy equivalence $C_{\bullet }^{U+V}(X)\simeq C_{\bullet }(X)$, hence $H_n(C_{\bullet }^{U+V}(X))\cong H_n(X)$. Substituting yields the Mayer-Vietoris sequence. $\square$

Theorem (computation of $S^k$), proof by induction. Write $S^k=U\cup V$ with $U=S^k\setminus \{N\}$ and $V=S^k\setminus \{S\}$ (the sphere minus the two poles). Both $U$ and $V$ are homeomorphic to ${\mathbb{R}}^k$, hence contractible; $U\cap V$ deformation-retracts onto the equator $S^{k-1}$.

Apply Mayer-Vietoris. For $n\ne 0,1$,

$$H_n(U)\oplus H_n(V)=0\oplus 0=0,H_{n-1}(U)\oplus H_{n-1}(V)=0,$$

so the Mayer-Vietoris boundary $H_n(S^k)\to H_{n-1}(U\cap V)=H_{n-1}(S^{k-1})$ is an isomorphism. Inducting from the base case $S^0=\{N,S\}$, where $H_0(S^0)={\mathbb{Z}}^2$ and $H_n(S^0)=0$ for $n\ge 1$, gives $H_n(S^k)=\mathbb{Z}$ for $n=0$ or $n=k$ and zero otherwise.

The dimensions $n=0$ and $n=1$ are handled by direct examination of the tail of the Mayer-Vietoris sequence, using path-connectedness of $S^k$ for $k\ge 1$ to give $H_0(S^k)=\mathbb{Z}$ and the fact that the connecting map $H_1(S^k)\to H_0(S^{k-1})$ has image the kernel of the sum-of-components map $H_0(U)\oplus H_0(V)\to H_0(S^k)$. $\square$

Theorem (long exact sequence of a pair), proof. The short exact sequence of chain complexes $0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)=C_{\bullet }(X,A)\to 0$ is exact at the chain level by definition (free abelian groups split, so the sequence is split-exact in each degree, which already gives the long exact homology sequence by the snake lemma / zig-zag construction). The connecting map $H_n(X,A)\to H_{n-1}(A)$ sends the class of a relative cycle $z$ (a chain in $X$ with $\partial z\in C_{\bullet }(A)$) to the class of $\partial z$ in $H_{n-1}(A)$. $\square$

Stated without proof — see Hatcher §3.A ^[pending]. The universal coefficient theorem for homology, identifying $H_n(X;G)$ as a direct-sum-and-Tor extension of $H_n(X)$ and $H_{n-1}(X)$ against $G$.

Connections [Master]

• Cellular homology 03.12.13. The cellular chain complex of a CW complex computes the same homology as the singular chain complex; the proof goes via the long exact sequence of CW pairs $(X^{(n)},X^{(n-1)})$ and identifies cellular chains with relative singular chains $H_n(X^{(n)},X^{(n-1)})$. This is the principal computational tool: cellular chain groups are finitely generated when the CW structure is finite, while singular chain groups are uncountable.

• Excision theorem 03.12.14. Excision states that for a good pair $(X,A)$ with $Z\subseteq A^{\circ }$, the inclusion $(X\setminus Z,A\setminus Z)\hookrightarrow (X,A)$ induces an isomorphism on relative homology. Excision is the technical underpinning of the agreement between singular and simplicial / cellular homology and is what makes Mayer-Vietoris a theorem rather than a definition.

• Eilenberg-Steenrod axioms 03.12.15. The seven axioms (homotopy invariance, exactness of the long sequence of a pair, excision, additivity, dimension) characterise singular homology uniquely on the category of CW pairs. Singular homology is the prototype; the axioms abstract away from the chain-complex construction to a list of properties that any homology theory must satisfy.

• CW complex 03.12.10. The cell-by-cell construction of a CW complex meshes with the cellular-homology computation of singular homology, and the homotopy-extension property of CW pairs is the technical witness needed to pass between cellular and singular complexes by acyclic-models arguments.

• Singular cohomology 03.04.13. Singular cohomology is the dual construction: replace chains with $\mathrm{Hom}(C_{\bullet }(X),\mathbb{Z})$, the boundary with its dual, and take homology of the resulting cochain complex. The Eilenberg-Zilber and cup-product structures on the cochain side enrich the additive theory studied here into a graded ring.

• Homotopy and homotopy group 03.12.01. The Hurewicz theorem connects the lowest non-vanishing singular homology of a simply connected space with its lowest non-vanishing homotopy group, identifying $H_n(X)\cong {\pi }_n(X)$ in the first interesting degree. This is the bridge from the abelian invariants of singular homology to the nonabelian homotopy invariants.

Historical & philosophical context [Master]

Singular homology in its modern form was introduced by Samuel Eilenberg in Singular homology theory (Ann. Math. 45 (1944), 407-447) ^[pending]. The paper consolidated and corrected the chain-complex constructions floating around in the work of Vietoris (1927) and Lefschetz (Topology, 1930; Algebraic Topology, 1942). The earlier constructions had each suffered from technical defects: Vietoris's complex was suited to compact metric spaces but lacked good functoriality, and Lefschetz's combinatorial chain-complexes required assumptions on the space that could not always be met. Eilenberg's contribution was to define the chain group as the free abelian group on all continuous maps from the standard simplex, with no regularity restriction, and to prove that the resulting chain complex has all the formal properties one wants — homotopy invariance, functoriality, the Mayer-Vietoris sequence, excision — without any hypothesis on $X$. The construction is the prototype of what would later be called a "homotopy-invariant functor on $\mathbf{Top}$".

The axiomatic culmination came in the 1952 monograph of Eilenberg and Steenrod, Foundations of Algebraic Topology, which proved that singular homology is the unique homology theory on the category of finite CW pairs satisfying the now-standard list of axioms. The axiomatic perspective separated the content of homology theory (a collection of functors, a long exact sequence, excision, dimension, additivity) from its construction (singular chains, simplicial chains, cellular chains, sheaf cohomology, de Rham cohomology). It made rigorous the statement that all the classical homology constructions are computing the same thing.

The chain-complex perspective fed directly into homological algebra, the framework Cartan and Eilenberg systematised in their 1956 textbook. By the early 1960s the singular chain complex had been generalised to the Dold-Kan correspondence (the singular chain complex is the normalised Moore complex of the simplicial abelian group $\mathbb{Z}[\mathrm{Sing}(X)]$) and to the spectrum-level identification $H\mathbb{Z}$, the Eilenberg-Mac Lane spectrum, whose homotopy theory recovers singular homology as a generalised homology theory. The modern synthesis treats singular homology as one specialisation among many: replace $H\mathbb{Z}$ with the $K$-theory spectrum or the bordism spectrum and the same axiomatic machinery yields topological $K$-theory or bordism homology.

Bibliography [Master]

[object Promise]