19 Homology Spanning Tree While the Cyclomatic Number Is Independent of That Choice

Home , Boundary (topology)

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 geometric object consisting of geometric simplices. In the latter 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 boundary 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 homomorphisms. 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.  1  ∂3 = .  1  For each recursive call, we have at most a linear number    1  of row and column operations. The total running time is therefore at most cubic in the number of simplices. Figure Given a p-chain as a column vector, v, its boundary is 90 shows how we interpret the result. Specifically, the computed by matrix multiplication, ∂pv. The result is a number of zero columns is the rank of the cycle group, combination of columns in the p-th boundary matrix, as Zp, and the number of 1s in the diagonal is the rank of the v v v v specified by . Thus, is a p-cycle iff ∂p =0 and is a boundary group, Bp−1. The Betti numberis the difference, p-boundary iff there is u such that ∂ +1u = v. p Z B βp = rank p − rank p, taking the rank of the boundary group from the reduced Matrix reduction. Letting n be the number of p- p matrix one dimension up. Working on our example, we simplices in K, we note that it is also the rank of the p-th get the following reduced matrices. chain group, np = rank Cp. The p-th boundary matrix thus has np−1 rows and np columns. To figure the sizes of ∂0 = 0 00 0 ; the cycle and boundary groups, and thus of the homology 100000 groups, we reduce the matrix to normal form, as shown  010000  in Figure 90. The algorithm of choice uses column and ∂1 = ;  001000     000000  rankC p 1 00 0 rankZ p  0 10 0   0 01 0  ∂2 =   ; rankBp −1  0 00 0     0 00 0  rankCp −1    0 00 0  1  0  ∂3 = .  0    Figure 90: The p-th boundary matrix in normal form. The entries  0  in the shaded portion of the diagonal are 1 and all other entries are 0. Writing zp = rank Zp and bp = rank Bp, we get z0 = 4 from the zeroth and b0 = 3 from the first reduced bound- row operations similar to Gaussian elimination for solv- ary matrix. Hence β0 = z0 = b0 = 1. Furthermore,

70 z1 = 3 and b1 = 3 giving β1 = 0, z2 = 1 and b2 = 1 giving β2 = 0, and z3 = 0 giving β3 = 0. These are the Betti numbers of the closed ball.

Euler-Poincare´ Theorem. The Euler characteristic ofa simplicial complex is the alternating sum of simplex numbers,

χ = (−1)pn . X p p≥0

Recalling that np is the rank of the p-th chain group and that it equals the rank of the p-th cycle group plus the rank of the (p − 1)-st boundary group, we get

p χ = (−1) (z + b 1) X p p− p≥0 = (−1)p(z − b ), X p p p≥0 which is the same as the alternating sum of Betti numbers. To appreciate the beauty of this result, we need to know that the Betti numbers do not depend on the trian- gulation chosen for the space. The proof of this property is technical and omitted. This now implies that the Euler characteristic is an invariant of the space, same as the Betti numbers.

p EULER-POINCARE´ THEOREM. χ = (−1) βp. P