Simplicial homology


In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components.
Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point, a line segment, a triangle and a tetrahedron. By definition, such a space is homeomorphic to a simplicial complex. Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold.
Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.

Definitions

Orientations

A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as, with the rule that two orderings define the same orientation if and only if they differ by an even permutation. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.

Chains

Let S be a simplicial complex. A simplicial k-chain is a finite formal sum
where each ci is an integer and σi is an oriented k-simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example,
The group of k-chains on S is written Ck. This is a free abelian group which has a basis in one-to-one correspondence with the set of k-simplices in S. To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.

Boundaries and cycles

Let σ = be an oriented k-simplex, viewed as a basis element of Ck. The boundary operator
is the homomorphism defined by:
where the oriented simplex
is the ith face of σ, obtained by deleting its ith vertex.
In Ck, elements of the subgroup
are referred to as cycles, and the subgroup
is said to consist of boundaries.

Boundaries of boundaries

A direct computation shows that ∂2 = 0. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups
form a chain complex. Another equivalent statement is that Bk is contained in Zk.
As an example, consider a tetrahedron with vertices oriented as w,x,y,z. By definition, its boundary is given by: xyz - wyz + wxz - wxy. The boundary of the boundary is given by: -+- = 0.

Homology groups

The kth homology group Hk of S is defined to be the quotient abelian group
It follows that the homology group Hk is nonzero exactly when there are k-cycles on S which are not boundaries. In a sense, this means that there are k-dimensional holes in the complex. For example, consider the complex S obtained by gluing two triangles along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries. One can compute that the homology group H1 is isomorphic to Z2, with a basis given by the two cycles mentioned. This makes precise the informal idea that S has two "1-dimensional holes".
Holes can be of different dimensions. The rank of the kth homology group, the number
is called the kth Betti number of S. It gives a measure of the number of k-dimensional holes in S.

Example

Homology groups of a triangle

Let S be a triangle, viewed as a simplicial complex. Thus S has three vertices, which we call v0, v1, v2, and three edges, which are 1-dimensional simplices. To compute the homology groups of S, we start by describing the chain groups Ck:
The boundary homomorphism ∂: C1 → C0 is given by:
Since C−1 = 0, every 0-chain is a cycle ; moreover, the group B0 of the 0-boundaries is generated by the three elements on the right of these equations, creating a two-dimensional subgroup of C0. So the 0th homology group H0 = Z0/B0 is isomorphic to Z, with a basis given by the image of the 0-cycle. Indeed, all three vertices become equal in the quotient group; this expresses the fact that S is connected.
Next, the group of 1-cycles is the kernel of the homomorphism ∂ above, which is isomorphic to Z, with a basis given by − +. Since C2 = 0, the group of 1-boundaries is zero, and so the 1st homology group H1 is isomorphic to Z/0 ≅ Z. This makes precise the idea that the triangle has one 1-dimensional hole.
Next, since by definition there are no 2-cycles, C2 = 0. Therefore the 2nd homology group H2 is zero. The same is true for Hi for all i not equal to 0 or 1.

Homology groups of a higher-dimensional simplices

Let S be a tetrahedron, viewed as a simplicial complex. Thus S has four 0-dimensional vertices, six 1-dimensional edges, and four 2-dimensional faces. The construction of the homology groups of a tetrahedron is described in detail here. It turns out that H0 is isomorphic to Z, H2 is isomorphic to Z too, and all other groups are trivial.
If the tetrahedron contains its interior, then H2 is trivial too.
In general, if S is a d-dimensional simplex, the following holds:
Let S and T be simplicial complexes. A simplicial map f from S to T is a function from the vertex set of S to the vertex set of T such that the image of each simplex in S is a simplex in T. A simplicial map f: S → T determines a homomorphism of homology groups Hk → Hk for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by
if f,..., f are all distinct, and otherwise f) = 0.
This construction makes simplicial homology a functor from simplicial complexes to abelian groups. This is essential to applications of the theory, including the Brouwer fixed point theorem and the topological invariance of simplicial homology.

Related homologies

is a related theory which is better adapted to theory rather than computation. Singular homology is defined for all topological spaces and obviously depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as image analysis, medical imaging, and data analysis in general.
Another related theory is Cellular homology.

Applications

A standard scenario in many computer applications is a collection of points in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology involves analysis of homology at different resolutions, registering homology classes that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data.
More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.

Implementations