Singular Homology

Overview

Singular homology is a functor from topological spaces to graded abelian groups that measures the presence of 'holes' of various dimensions. The n-th homology group H_n(X) detects n-dimensional holes: H_0 counts connected components, H_1 detects loops not bounding disks, H_2 detects enclosed voids, and so on. It is one of the central tools of algebraic topology, providing invariants that distinguish spaces up to homotopy equivalence.

Intuition

Think of singular homology as a systematic way to detect holes by probing a space with triangles of all dimensions. A singular n-simplex is a continuous map from the standard n-simplex into X. We form a chain complex from formal sums of these maps and then compute cycles (chains with no boundary) modulo boundaries (chains that are the boundary of something). What remains — cycles that are 'stuck' because they bound nothing — represents genuine topological holes.

Formal Definition

Definition

The standard n-simplex is Delta_n = {(t_0,...,t_n) in R^{n+1} : t_i >= 0, sum t_i = 1}. A singular n-simplex in X is a continuous map sigma: Delta_n -> X. The singular chain group C_n(X) is the free abelian group on all singular n-simplices. The boundary map partial_n: C_n(X) -> C_{n-1}(X) is defined by partial_n(sigma) = sum_{i=0}^{n} (-1)^i sigma|_{Delta_n^i} where Delta_n^i is the i-th face. The n-th singular homology group is H_n(X) = ker(partial_n) / im(partial_{n+1}).

\Delta_n = \left\{(t_0,\ldots,t_n)\in\mathbb{R}^{n+1} : t_i\ge 0,\, \sum_{i=0}^n t_i = 1\right\}

Standard n-simplex

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

where F_i^n is the i-th face inclusion

Boundary map

\partial_{n-1}\circ\partial_n = 0

Chain complex condition

H_n(X) = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})

n-th singular homology group

Theorems

Theorem 1: Homotopy Invariance

If f, g: X \to Y are homotopic maps then H_n(f) = H_n(g) for all n. In particular, homotopy equivalent spaces have isomorphic homology groups.

Theorem 2: Mayer-Vietoris Sequence

If X = A \cup B with A, B open, there is a long exact sequence \cdots \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 \cdots

Theorem 3: Dimension Axiom

For a point \{*\}, H_n(\{*\}) = \mathbb{Z} if n=0 and 0 otherwise.

Worked Examples

Decompose S^1 = U union V where U and V are open arcs, each contractible, with U ∩ V homotopy equivalent to two disjoint points.
$U \simeq \{*\},\quad V \simeq \{*\},\quad U\cap V \simeq \{*\}\sqcup\{*\}$
Apply the Mayer-Vietoris sequence.
$\cdots \to H_1(U)\oplus H_1(V) \to H_1(S^1) \xrightarrow{\partial} H_0(U\cap V) \to H_0(U)\oplus H_0(V) \to H_0(S^1) \to 0$
Since U and V are contractible, H_1(U) = H_1(V) = 0 and H_0(U) = H_0(V) = Z. Also H_0(U ∩ V) = Z^2.
$0 \to H_1(S^1) \xrightarrow{\partial} \mathbb{Z}^2 \xrightarrow{\phi} \mathbb{Z}^2 \to H_0(S^1) \to 0$
The map phi sends each generator to (1,1) minus (1,1), so its kernel has rank 1 and image has rank 1. Hence H_1(S^1) ≅ Z and H_0(S^1) ≅ Z.

Answer: H_0(S^1) = Z, H_1(S^1) = Z, H_n(S^1) = 0 for n >= 2.

Practice Problems

Difficulty 7/10

Compute H_n(T^2) for all n, where T^2 = S^1 x S^1 is the torus.

Difficulty 8/10

Prove that H_0(X) is a free abelian group whose rank equals the number of path-connected components of X.

Difficulty 9/10

Use the long exact sequence of the pair (D^2, S^1) to compute the reduced homology of S^1.

Common Mistakes

Common Mistake

Homology groups are the same as homotopy groups.

They are related but different. H_1(X) ≅ pi_1(X)^{ab} by the Hurewicz theorem, but higher homology and homotopy groups can differ drastically.

Common Mistake

The boundary of a boundary is some nontrivial cycle.

By the chain complex condition ∂∘∂ = 0, the boundary of any boundary is zero. This is why boundaries are always cycles.

Quiz

Summary

Singular homology groups H_n(X) are algebraic invariants that detect n-dimensional holes in a space X.
They are defined via chain complexes of formal sums of continuous maps from standard simplices into X.
The key tools for computation are Mayer-Vietoris, excision, and the long exact sequence of a pair.
Homotopy equivalent spaces have isomorphic homology, making homology a homotopy invariant.
H_0(X) counts path components; H_1(X) is the abelianization of the fundamental group for path-connected X.