It formalizes the idea of the number of "holes" in the graph.
[1] The general formula for the 1st homology group of a topological space X is:
The example below explains these symbols and concepts in full detail on a graph.
operator and defined as its target minus its source, so
In other words, the set of cycles in C1 generates the null space (the kernel) of
has two generators: one corresponds to a+b+c and the other to a+b+d (the third cycle, c-d, is a linear combination of the first two).
In a general topological space, we would define higher-dimensional chains.
In particular, C2 would be the free abelian group on the set of 2-dimensional objects.
This corresponds to the intuitive fact that the graph has two "holes".
In a disconnected graph, when C is the set of connected components, a similar computation shows:
The general formula for the 0-th homology group of a topological space X is:
Recall that the group C0 is generated by the set of vertices.
contains an element for each pair of vertices that are boundaries of an edge, i.e., it is generated by the differences {y−x, z−y, x−z}.
To calculate the quotient group, it is convenient to think of all the elements of
Then, every connected component is an equivalence class in the quotient group.
Often, it is convenient to assume that the 0-th homology of a connected graph is trivial (so that, if the graph contains a single point, then all its homologies are trivial).
We can generalize the graph to an abstract simplicial complex by adding elements of a higher dimension.
In the above example graph, we can add a two-dimensional "cell" enclosed between the edges c and d; let's call it A and assume that it is oriented clockwise.
Define C2 as the free abelian group generated by the set of two-dimensional cells, which in this case is a singleton {A}.
The sequence of chains and boundary operators can be presented as follows:[4]
The addition of the 2-dimensional cell A implies that its boundary, c-d, no longer represents a hole (it is homotopic to a single point).
Therefore, the group of "holes" now has a single generator, namely a+b+c (it is homotopic to a+b+d).
is the group of 1-dimensional cycles that are boundaries of 2-dimensional cells, which is isomorphic to Z.
Suppose now that we add another oriented 2-dimensional cell B between the edges c and d, such that
This cycle generates the second homology group, corresponding to the fact that there is a single two-dimensional hole:
We can proceed and add a 3-cell - a solid 3-dimensional object (called C) bounded by A and B.
Define C3 as the free abelian group generated by {C}, and the boundary operator
If the maximum dimension of a chain is k, then we get the following sequence of groups:
It can be proved that any boundary of a (k+1)-dimensional cell is a k-dimensional cycle.
is well-defined, and it is defined as the k-th homology group: