Editing Homology (mathematics) (section)

== Informal examples ==
The homology of a [[topological space]] ''X'' is a set of [[Topological property|topological invariants]] of ''X'' represented by its ''homology groups''
<math display="block">H_0(X), H_1(X), H_2(X), \ldots</math>
where the <math>k^{\rm th}</math> homology group <math>H_k(X)</math> describes, informally, the number of [[Hole (topology)|holes]] in ''X'' with a ''k''-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two [[Connected space#Connected components|components]]. Consequently, <math>H_0(X)</math> describes the path-connected components of ''X''.<ref>{{Harvnb|Spanier|1966|p=155}}</ref>

{{For|the homology groups of a graph|graph homology}}

{{multiple image
| align             = right
| total_width       = 300
| image1            = Circle - black simple.svg
| caption1          = The circle or 1-sphere <math>S^1</math>
| image2            = Sphere wireframe 10deg 4r.svg
| caption2          = The 2-sphere <math>S^2</math> is the outer shell, not the interior, of a ball
}}

A one-dimensional [[sphere]] <math>S^1</math> is a [[circle]]. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as
<math display="block">H_k\left(S^1\right) = \begin{cases}
  \Z & k = 0, 1 \\
 \{0\} & \text{otherwise}
\end{cases}</math>
where <math>\Z</math> is the group of integers and <math>\{0\}</math> is the [[trivial group]]. The group <math>H_1\left(S^1\right) = \Z</math> represents a [[finitely-generated abelian group]], with a single [[Generator (groups)|generator]] representing the one-dimensional hole contained in a circle.<ref name="Gowers 2010 390–391">{{Harvnb|Gowers|Barrow-Green|Leader|2010|pp=390–391}}</ref>

A two-dimensional [[sphere]] <math>S^2</math> has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are<ref name="Gowers 2010 390–391" />
<math display="block">H_k\left(S^2\right) = \begin{cases}
  \Z & k = 0, 2 \\
 \{0\} & \text{otherwise}
\end{cases}</math>

In general for an ''n''-dimensional sphere <math>S^n,</math> the homology groups are
<math display="block">H_k\left(S^n\right) = \begin{cases}
  \Z & k = 0, n \\
 \{0\} & \text{otherwise}
\end{cases}</math>

{{multiple image
| align       = left
| total_width = 325
| image1      = 1-ball.svg
| caption1    = The solid disc or 2-ball <math>B^2</math>
| image2      = Simple torus with cycles.svg
| caption2    = The torus <math>T = S^1 \times S^1</math>
}}
A two-dimensional [[Ball (mathematics)|ball]] <math>B^2</math> is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for <math>H_0\left(B^2\right) = \Z</math>. In general, for an ''n''-dimensional ball <math>B^n,</math><ref name="Gowers 2010 390–391"/>
<math display="block">H_k\left(B^n\right) = \begin{cases}
  \Z & k = 0 \\
      \{0\} & \text{otherwise}
\end{cases}</math>

The [[torus]] is defined as a [[product topology|product]] of two circles <math>T^2 = S^1 \times S^1</math>. The torus has a single path-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<ref name="Hatcher 2002 106">{{Harvnb|Hatcher|2002|p=106}}</ref>
<math display="block">H_k(T^2) = \begin{cases}
 \Z & k = 0, 2 \\
 \Z \times \Z & k = 1 \\
 \{0\} & \text{otherwise}
\end{cases}</math>

If ''n'' products of a topological space ''X'' is written as <math>X^n</math>, then in general, for an ''n''-dimensional torus <math>T^n = (S^1)^n</math>,
<math display="block">H_k(T^n) = \begin{cases}
 \Z^\binom{n}{k} & 0 \le k \le n \\
 \{0\} & \text{otherwise}
\end{cases}</math>
(see ''{{slink|Torus#n-dimensional torus}}'' and ''{{slink|Betti number#More examples}}'' for more details).

The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the [[direct product of groups|product group]] <math>\Z \times \Z.</math> 

For the [[projective plane]] ''P'', a simple computation shows (where <math>\Z_2</math> is the [[cyclic group]] of order 2):
<math display="block">H_k(P) = \begin{cases}
    \Z & k = 0 \\
  \Z_2 & k = 1 \\
 \{0\} & \text{otherwise}
\end{cases}</math>
<math>H_0(P) = \Z</math> corresponds, as in the previous examples, to the fact that there is a single connected component. <math>H_1(P) = \Z_2</math> is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called '''torsion'''.