19 Homology spanning tree while the cyclomatic number is independent of that choice. In topology, the main focus is not on geometric size but rather on how a space is connected. The most elementary Simplicial complexes. We begin with a combinatorial notion distinguishes whether we can go from one place representation of a topological space. Using a finite to another. If not then there is a gap we cannot bridge. ground set of vertices, V , we call a subset σ ⊆ V an Next we would ask whether there is a loop going around abstract simplex. Its dimension is one less than the car- an obstacle, or whetherthere is a void missing in the space. dinality, dim σ = |σ|− 1.A face is a subset τ ⊆ σ. Homology is a formalization of these ideas. It gives a way to define and count holes using algebra. DEFINITION. An abstract simplicial complex over V is a system K ⊆ 2V such that σ ∈ K and τ ⊆ σ implies The cyclomatic number of a graph. To motivate the τ ∈ K. more general concepts, consider a connected graph, G, with n vertices and m edges. A spanning tree has n − 1 The dimension of K is the largest dimension of any sim- edges and every additional edge forms a unique cycle to- plex in K. A graph is thus a 1-dimensional abstract sim- gether with edges in this tree; see Figure 86. Every other plicial complex. Just like for graphs, we sometimes think of K as an abstract structure and at other times as a geo- metric object consisting of geometric simplices. In the lat- ter interpretation, we glue the simplices along shared faces to form a geometric realization of K, denoted as |K|. We say K triangulates a space X if there is a homeomorphism h : X → |K|. We have seen 1- and 2-dimensional exam- ples in the preceding sections. The boundary of a simplex σ is the collection of co-dimension one faces, ∂σ = {τ ⊆ σ | dim τ = dim σ − 1}. Figure 86: A tree with three additional edges defining the same If dim σ = p then the boundary consists of p + 1 (p − 1)- number of cycles. simplices. Every (p − 1)-simplex has p (p − 2)-simplices in its own boundary. This way we get (p + 1)p (p − 2)- p+1 p+1 cycle in can be written as a sum of these simplices, counting each of the 1 = 2 (p − 2)- G m − (n − 1) p− cycles. To make this concrete, we define a cycle as a sub- dimensional faces of σ twice. set of the edges such that every vertex belongs to an even number of these edges. A cycle does not need to be con- Chain complexes. We now generalize the cycles in nected. The sum of two cycles is the symmetric difference graphs to cycles of different dimensions in simplicial com- of the two sets such that multiple edges erase each other plexes. A p-chain is a set of p-simplices in K. The sum in pairs. Clearly, the sum of two cycles is again a cy- of two p-chains is their symmetric difference. We usually cle. Every cycle, , in contains some positive number γ G write the sets as formal sums, of edges that do not belong to the spanning tree. Call- ing these edges e1,e2,...,ek and the cycles they define c = a1σ1 + a2σ2 + . + anσn; γ1,γ2,...,γk, we claim that d = b1σ1 + b2σ2 + . + bnσn, γ = γ1 + γ2 + . + γk. where the ai and bi are either 0 or 1. Addition can then be done using modulo 2 arithmetic, To see this assume that δ = γ1 + γ2 + . + γk is different from γ. Then γ+δ is againa cycle but it contains no edges c +2 d = (a1 +2 b1)σ1 + . + (an +2 bn)σn, that do not belong to the spanning tree. Hence γ + δ = ∅ and therefore γ = δ, as claimed. This implies that the where ai +2 bi is the exclusive or operation. We simplify m−n+1 cycles form a basis of the group of cycles which notation by dropping the subscript but note that the two motivates us to call m − n +1 the cyclomatic number of plus signs are different, one modulo two and the other a the graph. Note that the basis depends on the choice of formal notation separating elements in a set. The p-chains 68 form a group, which we denote as (Cp, +) or simply Cp. Note that the boundary of a p-simplex is a (p − 1)-chain, an element of Cp−1. Extending this concept linearly, we δ define the boundary of a p-chain as the sum of boundaries of its simplices, ∂c = a1∂σ1 +. .+an∂σn. The boundary ε is thus a map between chain groups and we sometimes γ write the dimension as index for clarity, ∂p : Cp → Cp−1. Figure 88: The 1-cycles γ and δ are not 1-boundaries. Adding the 1-boundary ε to δ gives a 1-cycle homologous to δ. It is a homomorphism since ∂p(c + d)= ∂pc + ∂pd. The infinite sequence of chain groups connected by boundary homomorphisms is called the chain complex of K. All as c ∼ c′. Note that [c] = [c′] whenever c ∼ c′. Also note groups of dimension smaller than 0 and larger than the di- that [c + d] = [c′ + d′] whenever c ∼ c′ and d ∼ d′. We mension of K are trivial. It is convenient to keep them use this as a definition of addition forhomologyclasses, so around to avoid special cases at the ends. A p-cycle is a we again have a group. For example, the 1-st homology p-chain whose boundary is zero. The sum of two p-cycles group of the torus consists of four elements, [0] = B1, Z C is again a p-cycle so we get a subgroup, p ⊆ p. A [γ]= γ + B1, [δ]= δ + B1, and [γ + δ]= γ + δ + B1. We p-boundary is a p-chain that is the boundary of a (p + 1)- often draw the elements as the corners of a cube of some chain. The sum of two p-boundaries is again a p-boundary dimension; see Figure 89. If the dimension is β then it has so we get another subgroup, Bp ⊆ Cp, Taking the bound- ary twice in a row gives zero for every simplex and thus [γ] [γ+δ] for every chain, that is, (∂p(∂p+1d)=0. It follows that Bp is a subgroup of Zp. We can therefore draw the chain complex as in Figure 87. [0] [γ] Cp+1 C p Cp−1 Figure 89: The four homology classes of H1 are generated by Z p+1 Z p Z p−1 two classes, [γ] and [δ]. B p+1 B p B p−1 β p+2 p+1 p p−1 2 corners. The dimension is also the number of classes needed to generate the group, the size of the basis. For 0 0 0 the p-th homology group, this number is βp = rank Hp = Figure 87: The chain complex consisting of a linear sequence log2 |Hp|, the p-th Betti number. For the torus we have of chain, cycle, and boundary groups connected by homomor- phisms. β0 = 1; β1 = 2; β2 = 1, Homology groups. We would like to talk about cycles but ignore the boundaries since they do not go around a and βp = 0 for all p 6= 0, 1, 2. Every 0-chain is a 0- hole. At the same time, we would like to consider two cycle. Two 0-cycles are homologous if they are both the cycles the same if they differ by a boundary. See Figure sum of an even number or both of an odd number of ver- 88 fora few 1-cycles, some of which are 1-boundaries and tices. Hence β0 = log2 2=1. We have seen the reason some of which are not. This is achieved by taking the for β1 = 2 before. Finally, there are only two 2-cycles, quotient of the cycle group and the boundary group. The namely 0 and the set of all triangles. The latter is not a result is the p-th homology group, boundary, hence β2 = log2 2=1. Hp = Zp/Bp. Boundary matrices. To compute homology groups and Its elements are of the form [c] = c + Bp, where c is a p- Betti numbers, we use a matrix representation of the sim- cycle. [c] is called a homology class, c is a representative plicial complex. Specifically, we store the boundary ho- of [c], and any two cycles in [c] are homologous denoted momorphism for each dimension, setting ∂p[i, j]=1 if 69 the i-th (p − 1)-simplex is in the boundary of the j-th p- ing a linear system. We write it recursively, calling it with simplex, and ∂p[i, j]=0, otherwise. For example, if the m =1. complex consists of all faces of the tetrahedron, then the boundary matrices are void REDUCE(m) if ∃k,l ≥ m with ∂p[k,l]=1 then ∂0 = 0 0 0 0 ; exchange rows m and k and columns m and l; 111000 for i = m +1 to np−1 do 100110 if ∂p[i,m]=1 then ∂1 = ; 010101 add row m to row i endif 001011 endfor; 1 10 0 for j = m +1 to np do 1 01 0 if ∂p[m, j]=1 then 0 11 0 ∂2 = ; add column m to column j 1 00 1 endif 0 10 1 endfor; 0 01 1 REDUCE(m + 1) 1 endif.