Simplicial homology

In the area of algebraic topology, simplicial homology is a process that provides information about the simplicial complex by the number of cycles (a type of hole) it contains, including as the 0-dimensional cycles the number of pieces it is composed of. It also refers to the information produced by that process.

Simplicial homology concerns topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-dimensional simplex), a line segment (1-dimensional simplex), a triangle (2-dimensional simplex) and a tetrahedron (3-dimensional simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Replacing n-simplices by their continuous images in a given topological space gives singular homology. The simplicial homology of a simplicial complex is naturally isomorphic to the singular homology of its geometric realization. This implies, in particular, that the simplicial homology of a space does not depend on the triangulation chosen for the space.

It can be shown that all smooth manifolds allow a triangulation (via e.g. Morse theory). This, together with the fact that it is possible to resolve the simplicial homology of a simplicial complex automatically and efficiently, make this theory feasible for application to real life situations, such as image analysis, medical imaging, and data analysis in general.

Definition

The boundary of a boundary of a 2-simplex (left) and the boundary of a 1-chain (right) are taken. Both are 0, being sums in which both the positive and negative of a 0-simplex occur once. The boundary of a boundary is always 0. A nontrivial cycle is something that closes up like the boundary of a simplex, in that its boundary sums to 0, but which isn't actually the boundary of a simplex or chain. Because trivial 1-cycles are equivalent to 0 in

H_1

, the 1-cycle at right-middle is homologous to its sum with the boundary of the 2-simplex at left.

Let S be a simplicial complex. A simplicial k-chain is a formal sum of k-simplices

\sum_{i=1}^N c_i \sigma^i \,

, where

c_i \in \mathbb{Z}, \sigma^i \in S

is the i-th k-simplex.

The group of k-chains on S, the free abelian group with basis the set of k-simplices in S, is denoted C_k.

Consider that a basis element of C_k, a k-simplex, is given by a tuple of 0-simplices, or vertices

\sigma = \left \langle v^0 , v^1 , \dots ,v^k\right \rangle.

The boundary operator

\partial_k: C_k \rightarrow C_{k-1}

is a homomorphism defined by:

\partial_k(\sigma)=\sum_{i=0}^k (-1)^i \left \langle v^0 , \dots , \widehat{v^i} , \dots ,v^k\right \rangle ,

where the simplex

\left \langle v^0 , \dots , \widehat{v^i} , \dots ,v^k\right \rangle

is the i^th face of σ obtained by deleting its i^th vertex.

In C_k, elements of the subgroup

Z_k = \ker \partial_k

are referred to as cycles, and the subgroup

B_k = \operatorname{im} \partial_{k+1}

is said to consist of boundaries.

Direct computation shows that B_k lies in Z_k, that is, B_k ⊆ Z_k. The boundary of a boundary must be zero. In other words,

(C_k, \partial_k)

form a simplicial chain complex.

A simplicial complex with 2 1-holes

The k^th homology group H_k of S is defined to be the quotient

H_k(S) = Z_k/B_k\, .

A homology group H_k is not trivial if the complex at hand contains k-cycles which are not boundaries. This indicates that there are k-dimensional holes in the complex. For example consider the complex obtained by gluing two triangles (with no interior) along one edge, shown in the image. This is a triangulation of the figure eight. The edges of each triangle form a cycle. These two cycles are by construction not boundaries (there are no 2-chains). Therefore the figure has two "1-holes".

Holes can be of different dimensions. The rank of the homology groups, the numbers

\beta_k = {\rm rank} (H_k(S))\,

are referred to as the Betti numbers of the space S, and gives a measure of the number of k-dimensional holes in S.

Example

In order to compute the homology groups of the triangle, one should compute the different groups $\mathrm{ker}(\partial_0),\mathrm{Im}(\partial_1)$ etc. Here, by the definition of the boundary operator, we have $\partial_0([v_i]) = 0$ , therefore the kernel is:

\mathrm{ker}(\partial_0) = C_0 = \{a_1[v_1] + a_2[v_2] + a_3[v_3] | a_1,a_2,a_3 \in \mathbb{Z}\} \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}

that is every 0-chain is in the kernel. Next, given a 1-chain $c_1 = b_1[v_1,v_2] + b_2[v_2,v_3] + b_3[v_3,v_1]$ there exists:

\partial_1(c_1) = (b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3]

That is,

\mathrm{Im}(\partial_1) = \{(b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3] | b_1,b_2,b_3\in \mathbb{Z}\}

which means that a 0-chain $c_0 = a_1[v_1] + a_2[v_2] + a_3[v_3]$ is in the image of $\partial_1$ if and only if

a_1 = b_3-b_1

a_2 = b_1-b_2

a_3 = b_2-b_3

This implies that we have only two degrees of freedom for choosing $a_i$ , or in other words:

\mathrm{Im}(\partial_1) \cong \mathbb{Z} \oplus \mathbb{Z}

Now we can use the definition:

H_0(S) \cong (\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})/ (\mathbb{Z} \oplus \mathbb{Z}) \cong \mathbb{Z}

As for the other homology groups, computations are easier. $\partial_1(c_1) = 0$ if and only if $b_1=b_2=b_3$ , therefore

\mathrm{ker}(\partial_1) = \{b[v_1,v_2] + b[v_2,v_3] + b[v_3,v_1] | b\in \mathbb{Z}\} \cong \mathbb{Z}.

Now, since there are no 2-chains, the kernel and image of $\partial_2$ are trivial, that is $\mathrm{ker}(\partial_2) = \mathrm{Im}(\partial_2) = 0$ . This yields:

H_1(S) = \mathrm{ker}(\partial_1) / \mathrm{Im}(\partial_2) = \mathrm{ker}(\partial_1) \cong \mathbb{Z}

H_2(S) = \mathrm{ker}(\partial_2) / \mathrm{Im}(\partial_3) \cong 0

Applications

A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) 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 (Edelsbrunner et al.2002 Robins, 1999) involves analysis of homology at different resolutions, registering homology classes (holes) 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. A MATLAB toolbox for computing persistent homology, Plex (Vin de Silva, Gunnar Carlsson), is available at this site Stand-alone implementations in C++ are available as part of the Perseus and Dionysus software projects. More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.

References

Lee, J.M., Introduction to Topological Manifolds, Springer-Verlag, Graduate Texts in Mathematics, Vol. 202 (2000) ISBN 0-387-98759-2
Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
Moise, E.E., Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. Math. 96-114 (1952).

Simplicial homology

Definition

Example

Applications

See also

References

External links