Singular homology is a homology theory defined for every topological space. It needs no subdivision into simplices, and spaces that are topologically the same get the same homology groups.
Simplicial homology needs a triangulation of the space. Singular homology removes that requirement by using every continuous map from a simplex into the space. The chain groups become huge, but topological invariance is almost built into the definition, which makes the theory a convenient foundation [1,2].
The model shapes are the standard simplices. The standard n-simplex is \Delta^n = \{ (t_0, \ldots, t_n) \in \mathbb{R}^{n+1} \mid t_i \geq 0, \sum_i t_i = 1 \}.
\Delta^0 is a point, \Delta^1 a segment, \Delta^2 a triangle and \Delta^3 a tetrahedron, with vertices e_0, \ldots, e_n.
A singular n-simplex in a space X is a continuous map \sigma : \Delta^n \to X from the standard simplex.
The word singular matters here. The map \sigma need not be injective, and its image need not look like an n-dimensional shape. A map that folds the triangle, or one that sends everything to a single point, is a singular simplex too, and this tolerance is what frees the theory from triangulations.
In low dimensions, a singular 0-simplex is a point of X, a singular 1-simplex is a path in X, and a singular 2-simplex is a triangle mapped into X like a membrane.
Singular homology is built in the same four steps as simplicial homology: collect the continuous maps from standard simplices, form chain groups on them, take boundaries with the boundary operator, and divide cycles by boundaries. Only the first step differs.
The singular chain group S_n(X) is the free abelian group with all singular n-simplices of X as a basis.
S_n(X) = \bigoplus_{\sigma : \Delta^n \to X} \mathbb{Z} \sigma
Its elements are finite formal sums \sum_i n_i \sigma_i with integer coefficients. Since every continuous map is a basis element, the basis is usually uncountable, yet after taking homology the result is often a small finitely generated group.
The i-th face map d_i : \Delta^{n-1} \to \Delta^n is the linear map that skips the vertex e_i and sends the vertices of \Delta^{n-1} in order to the remaining ones. The composite \sigma \circ d_i is the i-th face of \sigma.
The boundary operator \partial_n : S_n(X) \to S_{n-1}(X) adds up the faces with alternating signs.
\partial_n \sigma = \sum_{i=0}^{n} (-1)^i \sigma \circ d_i
For a path \sigma : \Delta^1 \to X this gives \partial_1 \sigma = \sigma(e_1) - \sigma(e_0), the end point minus the starting point. For a triangle, the three edges appear with signs +, -, +, and together they run once around its boundary.
Because of the alternating signs, the boundary of a boundary always vanishes: \partial_{n-1} \circ \partial_n = 0. This follows from the identities d_i \circ d_j = d_j \circ d_{i-1} for i > j, and the whole theory rests on it.
The kernel \ker \partial_n consists of the cycles and the image \mathrm{im} \partial_{n+1} of the boundaries. Since \partial^2 = 0, every boundary is a cycle, and the quotient can be taken.
H_n(X) = \ker \partial_n / \mathrm{im} \partial_{n+1}
A cycle that is closed but is not the boundary of anything survives as a nontrivial homology class and detects an n-dimensional hole of the space. The group H_n(X) is a topological invariant: homeomorphic spaces have isomorphic homology groups.
Let us run the definition to the end on the smallest space, a point X = \{*\}. In each dimension there is exactly one continuous map \Delta^n \to \{*\}, the constant map \sigma_n, so S_n(\{*\}) \cong \mathbb{Z} for every n \geq 0.
All faces of \sigma_n equal \sigma_{n-1}, so \partial_n \sigma_n = \sum_{i=0}^{n} (-1)^i \sigma_{n-1}. The sum of the signs is 0 for odd n and 1 for even n > 0, so the boundary maps alternate between zero and the identity.
H_0(\{*\}) \cong \mathbb{Z}, \quad H_n(\{*\}) = 0 \quad (n \geq 1)
The obvious fact that a point has no holes of higher dimension comes out of the definition, and this computation is the reference point for every other space.
For any space X, the group H_0(X) is free abelian with one \mathbb{Z} for each path component, so H_0(X) \cong \mathbb{Z}^k when there are k components. The reason is the boundary of a path, the end point minus the starting point: two points become equal in homology exactly when a path joins them.
For the discrete space X = \{p, q\} of two points no path joins p and q, so H_0(X) \cong \mathbb{Z}^2. In short, the rank of H_0 counts the pieces of the space, while H_1 detects loops that cannot be filled in by a membrane.
The \mathbb{Z} that remains in H_0 of a point often gets in the way. The reduced homology \tilde H_n(X) removes one copy of it, so that H_0(X) \cong \tilde H_0(X) \oplus \mathbb{Z} for nonempty X and \tilde H_n(X) = H_n(X) for n \geq 1. A point then has \tilde H_n = 0 in every degree.
Singular homology is even homotopy invariant: if X \simeq Y, then H_n(X) \cong H_n(Y) for all n [1]. In particular a contractible space such as the disk D^n or \mathbb{R}^n has the homology of a point.
A continuous map f : X \to Y sends each singular simplex \sigma to f \circ \sigma. This commutes with the boundary operator and induces homomorphisms f_* : H_n(X) \to H_n(Y).
These satisfy (g \circ f)_* = g_* \circ f_* and (\mathrm{id}_X)_* = \mathrm{id}, so H_n is a functor from topological spaces to abelian groups. That a homeomorphism induces isomorphisms is the simplest consequence.
The main tool for computing homology is the Mayer-Vietoris sequence. If X = U \cup V with U and V open, their homology groups are joined by 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
It is the homological counterpart of van Kampen's theorem: the whole space is recovered from two overlapping pieces and their overlap, now in the linear form of an exact sequence.
Cover the circle S^1 by an upper arc U and a lower arc V with some room to spare. Both are contractible, and U \cap V is two disjoint arcs, homotopy equivalent to two points. In reduced homology the sequence becomes 0 \to \tilde H_1(S^1) \to \tilde H_0(U \cap V) \to 0 with \tilde H_0(U \cap V) \cong \mathbb{Z}, so H_1(S^1) \cong \mathbb{Z}, generated by the loop that goes around once.
The same computation works for the sphere S^n. Cover it by slightly enlarged upper and lower hemispheres. Both are contractible, and their intersection is a band around the equator, homotopy equivalent to S^{n-1}.
The Mayer-Vietoris sequence in reduced homology then gives \tilde H_k(S^n) \cong \tilde H_{k-1}(S^{n-1}) for every k, which raises the dimension by one at each step. Starting from the circle we obtain the following for n \geq 1.
H_0(S^n) \cong \mathbb{Z}, \quad H_n(S^n) \cong \mathbb{Z}, \quad H_k(S^n) = 0 \quad (k \neq 0, n)
A \mathbb{Z} appears only in the two extreme degrees. The class in degree n is the whole sphere, an n-dimensional cycle that bounds nothing.
The torus T^2 = S^1 \times S^1 has two independent loops, so H_1 is \mathbb{Z}^2, and the whole surface is a 2-cycle that gives H_2 \cong \mathbb{Z}.
H_0(T^2) \cong \mathbb{Z}, \quad H_1(T^2) \cong \mathbb{Z}^2, \quad H_2(T^2) \cong \mathbb{Z}
The real projective plane \mathbb{RP}^2 behaves differently. It has a loop a that is not a boundary after one turn but is one after two turns, because the boundary of the disk is 2a. This shows up as torsion, an element of finite order.
H_0(\mathbb{RP}^2) \cong \mathbb{Z}, \quad H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}, \quad H_2(\mathbb{RP}^2) = 0
The \mathbb{Z}/2\mathbb{Z} in H_1 reflects the fact that \mathbb{RP}^2 is not orientable. With integer coefficients, homology records not only how many holes there are but also such twisting.
For a path-connected space X, the first homology group is the abelianization of the fundamental group [1].
H_1(X) \cong \pi_1(X)^{\mathrm{ab}}
Composing loops in the fundamental group is not commutative, but homology adds loops as oriented cycles and so forces commutativity. For the figure eight S^1 \vee S^1, \pi_1 is the free group F_2 with ab \neq ba, yet both loops give the same cycle a + b, and H_1 \cong \mathbb{Z}^2. For the torus, \pi_1 \cong \mathbb{Z}^2 is already abelian and H_1 \cong \mathbb{Z}^2.
For a pair (X, A) with A \subseteq X, the relative chain groups are S_n(X, A) = S_n(X) / S_n(A), and the relative homology groups H_n(X, A) are built from them in the same way. A relative cycle is a chain whose boundary lies in A.
For a good pair, where A is closed and a deformation retract of a neighbourhood, H_n(X, A) \cong \tilde H_n(X / A): relative homology is the homology of the space with A collapsed to a point.
The pair has a long exact sequence joining absolute and relative homology.
\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \to H_{n-1}(A) \to \cdots
In an exact sequence the image coming into each term equals the kernel going out, so known terms determine unknown ones. For the disk and its boundary it gives H_n(D^n, S^{n-1}) \cong \tilde H_{n-1}(S^{n-1}), and since D^n / S^{n-1} is the sphere S^n, this is another way to compute spheres by induction.
The second basic theorem next to the long exact sequence is excision. Let U \subseteq A \subseteq X with the closure of U contained in the interior of A. Then cutting U out of both spaces leaves relative homology unchanged.
H_n(X \setminus U, A \setminus U) \cong H_n(X, A)
So H_n(X, A) does not see what happens deep inside A, and that part can be cut away. Excision is also the key step in deriving the Mayer-Vietoris sequence.
Homology also classifies maps. A continuous map f : S^n \to S^n induces f_* : H_n(S^n) \to H_n(S^n), and since H_n(S^n) \cong \mathbb{Z}, this homomorphism is multiplication by an integer, the degree \deg f.
f_*(m) = (\deg f) m
The degree counts how many times f wraps the sphere around itself; on the circle, z \mapsto z^k has degree k. The identity has degree 1, a constant map degree 0, and the antipodal map x \mapsto -x degree (-1)^{n+1}. In the figure a point y moves around the circle and the points that z \mapsto z^k sends to y are marked; counting them with the sign of the orientation gives k, the degree.
Homotopic maps have the same degree, so the degree distinguishes homotopy classes of maps. It is used, for example, in proofs of the fundamental theorem of algebra and of the hairy ball theorem.
For a simplicial complex K, simplicial homology and the singular homology of its geometric realization always agree: H_n^{\mathrm{simp}}(K) \cong H_n(|K|). So the two share the work. Concrete numbers come from simplicial homology with finitely many simplices and integer matrices, and theoretical arguments use singular homology with its good properties.
The figure lists the groups computed above. All these spaces are path-connected, so every H_0 is \mathbb{Z}. The loop of S^1, the cavity of S^2, the two loops and the surface of T^2 and the torsion of \mathbb{RP}^2 all show up in the groups.
Functoriality, homotopy invariance, the long exact sequence of a pair and excision, together with the dimension axiom that a point has homology \mathbb{Z} in degree 0 only, form the Eilenberg-Steenrod axioms for a homology theory [3]. Singular homology is the standard model that satisfies them, and its agreement with simplicial homology on finite complexes follows from them.