The Fundamental Group Justification

homology respects homotopy classes and, therefore, classes of homeomorphic spaces.

4.2.1 Chains and Cycles

To define homology groups, we need simplicial analogs of paths and loops. Recalling free Abelian groups from Section 3.3.1, we create the chain group of oriented simplices. Definition 4.10 chain group The kth chain group of a simplicial complex K is C k K, +, the free Abelian group on the oriented k-simplices, where [σ] = −[τ] if σ = τ and σ and τ have different orientations. An element of C k K is a k-chain, ∑ q n q [σ q ], n q ∈ Z, σ q ∈ K. We often omit the complex in the notation. A simplicial complex has a chain group in every dimension. As stated earlier, homology examines the connec- tivity between two immediate dimensions. To do so, we define a structure- relating map between chain groups. Definition 4.11 boundary homomorphism Let K be a simplicial complex and σ ∈ K, σ = [v , v 1 , . . . , v k ]. The boundary homomorphism ∂ k : C k K → C k − 1 K is ∂ k σ = ∑ i −1 i [v , v 1 , . . . , ˆ v i , . . . , v n ], 4.3 where ˆ v i indicates that v i is deleted from the sequence. It is easy to check that ∂ k is well defined, that is, ∂ k is the same for every ordering in the same orientation. Example 4.4 boundaries Let us take the boundary of the oriented simplices in Figure 2.14. • ∂ 1 [a, b] = b − a. • ∂ 2 [a, b, c] = [b, c] − [a, c] + [a, b] = [b, c] + [c, a] + [a, b]. • ∂ 3 [a, b, c, d] = [b, c, d] − [a, c, d] + [a, b, d] − [a, b, c]. Note that the boundary operator orients the faces of an oriented simplex. In the case of the triangle, this orientation corresponds to walking around the triangle on the edges, according to the orientation of the triangle. If we take the boundary of the boundary of the triangle, we get: ∂ 1 ∂ 2 [a, b, c] = [c] − [b] − [c] + [a] + [b] − [a] = 0. 4.4 This is intuitively correct: The boundary of a triangle is a cycle, and a cycle does not have a boundary. In fact, this intuition generalizes to all dimensions. Theorem 4.3 ∂ k − 1 ∂ k = 0, for all k. Proof The proof is elementary: ∂ k − 1 ∂ k [v , v 1 , . . . , v k ] = ∂ k − 1 ∑ i −1 i [v , v 1 , . . . , ˆ v i , . . . , v k ] = ∑ j i −1 i −1 j [v , . . . , ˆ v j , . . . , ˆ v i , . . . , v k ] + ∑ j i −1 i −1 j − 1 [v , . . . , ˆ v i , . . . , ˆ v j , . . . , v k ] = 0, as switching i and j in the second sum negates the first sum. Using the boundary homomorphism, we have the following picture for an n- dimensional complex K: −→ C n ∂ n −→ C n − 1 ∂ n− 1 −→ . . . −→ C 1 ∂ 1 −→ C ∂ −→ 0, 4.5 with ∂ k ∂ k +1 = 0 for all k. Note that the sequence is augmented on the right by a 0, with ∂ = 0. On the left, C n +1 = 0, as there are no n + 1-simplices in K. Such a sequence is called a chain complex. Chain complexes are common in homology, but this is the only one we will see here. The images and kernels of these maps are subgroups of C k . Theorem 4.4 im ∂ k +1 and ker ∂ k are free Abelian normal subgroups of C k . im ∂ k +1 is a normal subgroup of ker ∂ k . Proof As in Section 3.2, both are subgroups by application of Theorem 3.3: A homomorphism preserves subgroups C k +1 and {0} ∈ C k , respectively. As C k is Abelian, both groups are normal. By Theorem 3.16, both groups are free Abelian. For the second statement, note that ∂ k ∂ k +1 = 0 implies im ∂ k +1 ⊆ ker ∂ k . We have already seen this subset is a group. Let ∂ k +1 σ, ∂ k +1 τ ∈ im ∂ k +1 . Then, ∂ k +1 σ + ∂ k +1 τ = ∂ k +1 σ + τ ∈ im ∂ k +1 , by the homomorphism property of ∂. Therefore, the set is closed and is a subgroup by definition. These subgroups are important enough to be named. Z 1 C 3 C 2 C 1 B 1 B C Z = Z 3 2 B 2 B Z 2 Fig. 4.8. A chain complex for a three-dimensional complex. z b z+b Fig. 4.9. A nonbounding oriented 1-cycle z ∈ Z k , z ∈ B k is added to an oriented 1- boundary b ∈ B k . The resulting cycle z + b is homotopic to z. The orientation on the cycles is induced by the arrows. Definition 4.12 cycle, boundary The kth cycle group is Z k = ker ∂ k . A chain that is an element of Z k is a k-cycle. The kth boundary group is B k = im ∂ k +1 . A chain that is an element of B k is a k-boundary. We also call boundaries bounding cycles and cycles not in B k nonbounding cycles . These names are self-explanatory: Bounding cycles bound higher dimensional cycles, as otherwise they would not be in the image of the boundary homomor- phism. We can think of them as “filled” cycles, as opposed to “empty” non- bounding cycles. Figure 4.8 shows a chain complex for a three-dimensional complex, along with the cycle and boundary subgroups.

4.2.2 Simplicial Homology

Chains and cycles are simplicial analogs of the maps called paths and loops in the continuous domain. Following the construction of the fundamental group, we now need a simplicial version of a homotopy to form equivalent classes of cycles. Consider the sum of the nonbounding 1-cycle and a bounding 1-cycle in Figure 4.9. The two cycles z , b have a shared boundary. The edges in the shared boundary appear twice in the sum z + b with opposite signs, so they are eliminated. The resulting cycle z + b is homotopic to z: We may slide the shared portion of the cycles smoothly across the triangles that b bounds. But Table 4.2. Homology of basic 2-manifolds. 2-manifold H H 1 H 2 sphere Z {0} Z torus Z Z × Z Z projective plane Z Z 2 {0} Klein bottle Z Z × Z 2 {0} such homotopies exist for any boundary b ∈ B 1 . Generalizing this argument to all dimensions, we look for equivalent classes of z + B k for a k-cycle. But these are precisely the cosets of B k in Z k by Definition 3.8. As B k is normal in Z k , the cosets form a group under coset addition. Definition 4.13 homology group The kth homology group is H k = Z k B k = ker ∂ k im ∂ k +1 . 4.6 If z 1 = z 2 + B k , z 1 , z 2 ∈ Z k , we say z 1 and z 2 are homologous and denote it with z 1 ∼ z 2 . By Corollary 3.3, homology groups are finitely generated Abelian, as they are factor groups of two free Abelian groups. Therefore, the fundamental theo- rem of finitely generated Abelian groups Theorem 3.10 applies. Homology groups describe spaces through their Betti numbers and the torsion subgroups. Definition 4.14 kth Betti number The kth Betti number β k of a simplicial complex K is β H k , the rank of the free part of H k . By Corollary 3.3, β k = rank H k = rank Z k − rank B k . The description given by homology is finite, as an n-dimensional simplicial space has at most n + 1 nontrivial homology groups.

4.2.3 Understanding Homology

The description provided by homology groups may not be transparent at first. In this section, we look at a few examples to gain an intuitive understanding of what homology groups capture. Table 4.2 lists the homology groups of our basic 2-manifolds shown in Figure 4.1. Because they are 2-manifolds, the highest nontrivial homology group for any of them is H 2 . Torsion-free spaces have homology that does not have a torsion subgroup, that is, terms that are v v v v v a a b b v b b a a v w w v v v v b b a a Fig. 4.10. Diagrams for our basic 2-manifolds from Figure 4.1. finite cyclic groups Z m . Most of the spaces we are interested are torsion-free. In fact, any space that is a subcomplex of S 3 , the three-dimensional sphere, is torsion-free. We deal with S 3 as it is compact and does not create special boundary cases that need to be resolved in algorithms. To avoid these difficul- ties, we add a point at infinity and compactify R 3 to get S 3 . This construction mirrors that of the two-dimensional sphere in Figure 4.1. Algorithmically, the one point compactification of R 3 is easy, as we have a simplicial representation of space. So what does homology capture? For torsion-free spaces in three dimen- sions, the Betti numbers the number of Z terms in the description have in- tuitive meaning as a consequence of the Alexander duality. β measures the number of components of the complex. β 1 is the rank of a basis for the tunnels. As H 1 is free, it is a vector-space and β 1 is its rank. β 2 counts the number of voids in the complex. Tunnels and voids exist in the complement of the com- plex in S 3 . The distinction might seem tenuous, but this is merely because of our familiarity with the terms. For example, the complex encloses a void, and the void is the empty space enclosed by the complex. Using this understanding, we may now examine Table 4.2. All four spaces have a single component, so H = Z and β = 1. The sphere and the torus enclose a void, so H 2 = Z and β 2 = 1. The nonorientable spaces, on the other hand, are one-sided and cannot enclose any voids, so they have trivial homol- ogy in dimension 2. To see what H 1 captures, we look again at the diagrams for the 2-manifolds, as shown in Figure 4.1 for convenience. We may, of course, triangulate these diagrams to obtain abstract simplicial complexes for comput- ing simplicial homology. For now, though, we assume that whatever curve we draw on these manifolds could be “snapped” to some triangulation of the dia- grams. To understand 1-cycles and torsion, we need to pay close attention to the boundaries in the diagrams. Recall that a boundary is simply a cycle that bounds . In each diagram, we have a boundary, simply, the boundary of the