User:Mark viking/Homology

Informal examples
Informally, the homology of a topological space $$X$$ is a set of topological invariants of $$X$$ represented by its homology groups
 * $$H_0(X), H_1(X), H_2(X), \ldots $$

where the $$k^{\rm th}$$ homology group $$H_k(X)$$ describes the $$k$$-dimensional holes in$$X$$. A 0-dimensional hole is simply a gap between two components, consequently $$H_0(X)$$ describes the path-connected components of $$X$$.

A one-dimensional sphere $$S^1$$ is a circle. It has a single connected component and a one-dimensional hole, but no higher dimensional holes. The corresponding homology groups are given as
 * $$H_k(S^1) = \left\{ \begin{array}{l} \mathbb Z {\rm\ for\ } k=0 {\rm\ and\ }k=1, \\ \{0\} {\rm\ otherwise}.\,\!\end{array}\right.$$

where $$\mathbb Z$$ is the group of integers and $$\{0\}$$ is the trivial group. The group $$H_1(S^1) = \mathbb Z$$ represents a kind of one-dimensional vector space, with integer coefficients, that represents the single one-dimensional hole contained in a circle.

A two-dimensional sphere $$S^2$$ has a single connected component, no one-dimensional holes, a two-dimensional hole, and no higher dimensional holes. The corresponding homology groups are
 * $$H_k(S^2) = \left\{ \begin{array}{l} \mathbb Z {\rm\ for\ } k=0 {\rm\ and\ }k=2, \\ \{0\} {\rm\ otherwise}.\,\!\end{array}\right.$$

In general for an $$n$$-dimensional sphere $$S^n$$, the homology groups are
 * $$H_k(S^n) = \left\{ \begin{array}{l} \mathbb Z {\rm\ for\ } k=0 {\rm\ and\ }k=n, \\ \{0\} {\rm\ otherwise}.\,\!\end{array}\right.$$

A one-dimensional ball $$B^1$$ is a solid disc. It has a single connected component, but in contrast to the circle, has no one-dimensional or higher-dimensional holes. The corresponding homology groups are all trivial except for $$H_0(B^1) = \mathbb Z$$. In general, for an $$n$$-dimensional ball $$B^n$$,
 * $$H_k(B^n) = \left\{ \begin{array}{l} \mathbb Z {\rm\ for\ } k=0, \\ \{0\} {\rm\ otherwise}.\,\!\end{array}\right.$$

The torus is defined as a Cartesian product of two circles $$T = S^1 \times S^1$$. The torus has a single connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are
 * $$H_k(T) = \left\{ \begin{array}{l} \mathbb Z {\rm\ for\ } k=0,2 \\

\mathbb Z\times \mathbb Z {\rm\ for\ } k=1 \\ \{0\} {\rm\ otherwise}.\,\!\end{array}\right.$$ The two independent 1D holes form a two-dimensional vector space with integer coefficients, producing the Cartesian product group $$\mathbb Z\times \mathbb Z$$.