Complexes and Homology Groups
2.4 Chains, cycles, boundaries
2.4.1 Algebraic structures
First, we recall briefly the definitions of some standard algebraic structures that are used in the book. For details we refer the reader to any standard book on algebra, e.g. [14].
Definition 2.18(Group; Homomorphism; Isomorphism). A setGtogether with a binary operation
‘+’ is a group if it satisfies the following properties: (i) for everya,b ∈ G, a+b ∈ G, (ii) for everya,b,c∈G, (a+b)+c=a+(b+c), (iii) there is anidentityelement denoted 0 inGso that
a+0 = 0+a = afor everya ∈G, and (iv) there is an inverse−a ∈Gfor everya ∈Gso that a+(−a) = 0. If the operation+commutes, that is,a+b = b+afor everya,b ∈G, thenGis calledabelian. A subsetH⊆Gis asubgroupof (G,+) if (H,+) is also a group.
Definition 2.19(Free abelian group; Basis; Rank; Generator). An abelian groupGis calledfree if there is a subsetB ⊆ G so that every element ofGcan be writtenuniquely as a finite sum of elements inBand their inverses disregarding trivial cancellationsa+b = a+c−c+b. Such a setBis called abasisofGand its cardinality is called itsrank. If the condition of uniqueness is dropped, thenBis called ageneratorofGand we also sayB generates G.
Definition 2.20(Coset; Quotient). For a subgroupH ⊆ Gand an elementa ∈ G, theleft coset isaH = {a+b | b ∈ H}and theright coset isHa= {b+a | b ∈ H}. For abelian groups, the left and right cosets are identical and hence are simply calledcosets. IfGis abelian, the quotient group ofGwith a subgroupH ⊆Gis given byG/H ={aH | a∈G}where the group operation is inherited fromGasaH+bH =(a+b)Hfor everya,b∈G.
Definition 2.21(Homomorphism; Isomorphism; Kernel; Image; Cokernel). A maph :G → H between two groups (G,+) and (H,∗) is called ahomomorphism ifh(a+b) = h(a)∗h(b) for every a,b ∈ G. If, in addition, h is bijective, it is called anisomorphism. Two groupsG and H with an isomorphism are calledisomorphicand denoted asG H. Thekernel, image, and cokernelof a homomorphismh :G → Hare defined as subgroups kerh = {a ∈G | h(a) = 0}, Imh={b∈H | ∃a∈Gwithh(a)=b}, and the quotient group cokerh=H/im hrespectively.
Definition 2.22(Ring). A setRequipped with two binary operations, addition ‘+’ and multiplica-tion ‘·’ is called a ring if (i)Ris an abelian group with the addition, (ii) the multiplication is asso-ciative, that is, (a·b)·c=a·(b·c) and is distributive with the addition, that is,a·(b+c)=a·b+a·c,
∀a,b,c∈R, and (iii) there is an identity for the multiplication.
The additive identity of a ringRis usually denoted as 0 whereas the multiplicative identity is denoted as 1. Observe that, by the definition of abelian group, the addition is commutative. How-ever, the multiplication need not be so. When the multiplication is also commutative,Ris called a commutativering. A commutative ring in which every nonzero element has a multiplicative inverse is called afield.
Definition 2.23 (Module). Given a commutative ring R with multiplicative identity 1, an R-module M is an abelian group with an operation R × M → M which satisfies the following properties∀r,r0∈Randx,y∈M:
• r·(x+y)=r·x+r·y
• (r+r0)x=r·x+r·x0
• 1·x= x
• (r·r0)·x=r·(r0·x)
Essentially, in anR-module, elements can be added and multiplied with coefficients in R.
However, ifRis taken as a fieldk, each non-zero element acquires a multiplicative inverse and we get a vector space.
Definition 2.24 (Vector space). AnR-moduleV is called avector spaceifRis a field. A set of elements{g1, . . . ,gk}is said togeneratethe vector spaceVif every elementa∈Vcan be written asa = α1g1+. . .+αkgk for someα1, . . . , αk ∈ R. The set{g1, . . . ,gk}is called abasisofV if everya∈ Vcan be written in the above wayuniquely. All bases ofV have the same cardinality which is called thedimensionofV. We say a set{g1, . . . ,gm} ⊆V isindependentif the equation α1g1+. . .+αmgm=0 can only be satisfied by settingαi =0 fori=1, . . . ,m.
Fact 2.7. A basis of a vector space is a generating set of minimal cardinality and an independent set of maximal cardinality.
2.4.2 Chains
LetKbe a simplicialk-complex withmpnumber ofp-simplices,k ≤ p ≤0. Ap-chaincinKis a formal sum ofp-simplices added with some coefficients, that is,c=Pmp
i=1αiσiwhereσiare the
In general, coefficients can come from a ringRwith its associated additions making the chains constituting anR-module. For example, these additions can be integer additions where the coef-ficients are integers; e.g., from two 1-chains (edges) we get
(2e1+3e2+5e3)+(e1+7e2+6e4)=3e1+10e2+5e3+6e4.
Notice that while writing a chain, we only write the simplices that have non-zero coefficient in the chain. We follow this convention all along. In our case, we will focus on the cases where the coefficients come from a fieldk. In particular, we will mostly be interested ink=Z2. This means that the coefficients come from the fieldZ2whose elements can only be 0 or 1 with the modulo-2 additions 0+0=0, 0+1=1, and 1+1=0. This gives usZ2-additions of chains, for example, we have
(e1+e3+e4)+(e1+e2+e3)=e2+e4.
Observe thatp-chains withZ2-coefficients can be treated as sets: the chaine1+e3+e4is the set{e1,e3,e4}, andZ2-addition between two chains is simply the symmetric difference between the corresponding sets.
From now on, unless specified otherwise, we will consider all chain additions to be Z2 -additions. One should keep in mind that one can have parallel concepts for coefficients and additions coming from integers, reals, rationals, fields, and other rings. UnderZ2-additions, we have
Below, we show addition of chains shown in Figure 2.9:
0-chain: ({b}+{d})+({d}+{e}) = {b}+{e} (left) 1-chain: ({a,b}+{b,d})+({b,c}+{b,d}) = {a,b}+{b,c} (left) 2-chain: ({a,b,c}+{b,c,e})+({b,c,e}) = {a,b,c} (right)
a
b
c d
e
a
c d e b
Figure 2.9: Chains, boundaries, cycles.
The p-chains with theZ2-additions form a group where the identity is the chain 0 = Pmp
i=10σi, and the inverse of a chainciscitself sincec+c=0. This group, called the p-th chain group, is denotedCp(K). We also drop the complex Kand use the notationCp whenK is clear from the context. We do the same for other structures that we define afterward.
2.4.3 Boundaries and cycles
The chain groups at different dimensions are related by a boundary operator. Given a p-simplex σ={v0, . . . ,vp}(also denoted asv0v1· · ·vp), let
∂pσ=
p
X
i=0
{v0, . . . ,vˆi, . . . ,vp}
where ˆvi indicates that the vertexvi is omitted. Informally, we can view∂pas a map that sends a p-simplexσto the (p−1)-chain that has non-zero coefficients only onσ’s (p−1)-faces also referred asσ’s boundary. At this point, it is instructive to note that the boundary of a vertex is empty, that is,∂0σ=∅. Extending∂pto ap-chain, we obtain a homomorphism∂p :Cp→Cp−1 called theboundary operatorthat produces a (p−1)-chain when applied to ap-chain:
∂pc=
mp
X
i=1
αi(∂pσi) for ap-chainc=
mp
X
i=1
αiσi∈Cp.
Again, we note the special case of p=0 when we get∂0c=∅. The chain groupC−1has only one single element which is its identity 0. On the other end, we also assume that ifKis ak-complex, thenCpis 0 forp>k.
Consider the complex in Figure 2.9(right). For the 2-chainabc+bcdwe get
∂2(abc+bcd)=(ab+bc+ca)+(bc+cd+db)=ab+ca+cd+db.
It means that from the two triangles sharing the edgebc, the boundary operator returns the four boundary edges that are not shared. Similarly, one can check that the boundary of the 2-chains
consisting of all three triangles in Figure 2.9(right) contains all 7 edges. In particular, the edgebc does not get cancelled because of all three (odd) triangles adjoin it.
∂2(abc+bcd+bce)=ab+bc+ca+be+ce+bd+dc.
One important property of the boundary operator is that, applying it twice produces an empty chain.
Proposition 2.8. For p>0and any p-chain c,∂p−1◦∂p(c)=0.
Proof. Observe that∂0is a zero map by definition. Also, for ak-complex,∂poperates on a zero element forp>kby definition. Then, it is sufficient to show that, for 1≤ p≤k,∂p−1◦∂p(σ)=0 for a p-simplexσ. Observe that∂pσis the set of all (p−1)-faces ofσand every (p−2)-faces of σis contained in exactly two (p−1)-faces. Thus,∂p−1(∂pσ)=0.
Extending the boundary operator to the chains groups, we obtain the following sequence of homomorphisms satisfying Proposition 2.8 for a simplicialk-complex; such a sequence is also called achain complex:
0=Ck+1
∂k+1 //Ck
∂k //Ck−1 ∂k−1 //Ck−2 · · · C1 ∂1 //C0 ∂0 //C−1=0. (2.1) Fact 2.8.
1. For p≥ −1,Cpis a vector space because the coefficients are drawn from a fieldZ2–it has a basis so that every element can be expressed uniquely as a sum of the elements in the basis.
2. There is a basis forCpwhere every p-simplex form a basis element because any p-chain is a unique subset of the p-simplices. The dimension ofCpis therefore n, the number of p-simplices. When p=−1and p≥k+1, Cpis trivial with dimension0. In Figure 2.9(right) {abc,bcd,bce}is a basis forC2and so is{abc,(abc+bcd),bce}.
Cycle and boundary groups.
Definition 2.25(Cycle and cycle group). A p-chaincis a p-cycle if∂c = 0. In words, a chain that has empty boundary is a cycle. All p-cycles together form the p-th cycle group Zp under the addition that is used to define the chain groups. In terms of the boundary operator,Zpis the subgroup ofCpwhich is sent to the zero ofCp−1, that is, ker∂p=Zp.
For example, in Figure 2.9(right), the 1-chainab+bc+cais a 1-cycle since
∂1(ab+bc+ca)=(a+b)+(b+c)+(c+a)=0.
Also, observe that the above 1-chain is the boundary of the triangle abc. It’s not accident that the boundary of a simplex is a cycle. Thanks to Proposition 2.8, the boundary of a p-chain is a (p−1)-cycle. This is a fundamental fact in homology theory.
The set of (p−1)-chains that can be obtained by applying the boundary operator∂p on p-chains form a subgroup of (p−1)-chains, called the (p−1)-th boundary groupBp−1=∂p(Cp); or in other words, the image of the boundary homomorphism is the boundary group,Bp−1 =im ∂p. We have ∂p−1Bp−1 = 0 for p > 0 due to Proposition 2.8 and henceBp−1 ⊆ Zp−1. Figure 2.10 illustrates cycles and boundaries.
Figure 2.10: Each individual red, blue, green cycle is not a boundary because they do not bound any 2-chain. However, the sum of the two red cycles, and the sum of the two blue cycles each form a boundary cycle because they bound 2-chains consisting of redish and bluish triangles respectively.
Fact 2.9. For a simplicial k-complex, 1. C0 =Z0andBk =0.
2. For p≥0,Bp⊆Zp ⊆Cp.
3. LikeCp, bothBpandZpare vector spaces.
2.5 Homology
The homology groups classify the cycles in a cycle group by putting togther those cycles in the same class that differ by a boundary. From a group theoretic point of view, this is done by taking the quotient of the cycle groups with the boundary groups, which is allowed since the boundary group is a subgroup of the cycle group.
Definition 2.26 (Homology group). For p ≥ 0, the p-thhomology groupis the quotient group Hp = Zp/Bp. Since we use a field, namely Z2, for coefficients, Hp is a vector space and its dimension is called the p-th Betti number, denoted byβp:
βp :=dim Hp.
Every element ofHp is obtained by adding a p-cycle c ∈ Zp to the entire boundary group, c+Bp, which is a coset ofBpinZp. All cycles constructed by adding an element ofBptocform the class [c], referred to as thehomology classofc. Two cyclescandc0 in the same homology class are calledhomologous, which also means [c] = [c0]. By definition, [c] = [c0] if and only if c ∈ c0 + Bp, and under Z2 coefficients, this also means thatc+ c0 ∈ Bp. For example, in Figure 2.10, the outer cyclec5is homologous to the sumc2+c4because they together bound the 2-chain consisting of all triangles. Also, observe that the group operation for Hp is defined by [c]+[c0]=[c+c0].
(a) (b) (c) (d)
Figure 2.11: Complex K of a tetrahedron: (a) Vertices, (b) spanning tree of the 1-skeleton, (c) 1-skeleton, (d) 2-skeleton ofK.
Example. Consider the boundary complexKof a tetrahedron which consists of four triangles, six edges, and four vertices. Consider the 0-skeletonK0ofKwhich consists of four vertices only.
All four vertices whose classes coincide with them are necessary to generateH0(K0). Therefore, these four vertices form a basis ofH0(K0). However, one can verify thatH0(K1) for the 1-skeleton K1 is generated by any one of the four vertices because all four vertices belong to the same class when we consider K1. This exemplifies the fact that rank ofH0(K) captures the number of connected components in a complexK.
The 1-skeletonK1 of the tetrahedron is a graph with four vertices and six edges. Consider a spanning tree with any vertex and the three edges adjoining it as in Figure 2.11(b). There is no 1-cycle in this configuration. However, each of the other three edges create a new 1-cycle which are not boundary because there is no triangle in K1. These three cyclesc1,c2, c3 as indicated in Figure 2.11(c) form their own classes in H1(K1). Observe that the 1-cycle at the base can be written as a combination of the other three and thus all classes in H1(K1) can be generated by only three classes [c1],[c2],[c3] and no fewer. Hence, these three classes form a basis ofH1(K1).
To develop more intuition, consider a simplicial surface Mwithout boundary embedded inR3. If the surface has genus g, that isgtunnels and handles in the complement space, thenH1(M) has dimension 2g(Exercise 4).
The 2-chain of the sum of four triangles inKmake a 2-cyclecbecause its boundary is 0. Since K does not have any 3-simplex (the tetrahedron is not part of the complex), this 2-cycle cannot be added to any 2-boundary other than 0 to form its class. Therefore, the homology class ofcis c itself, [c] = {c}. There is no other 2-cycle inK. Therefore, H2(K) is generated by [c] alone.
Its dimension is only one. If the tetrahedron is included in the complex,c becomes a boundary element, and hence [c]=[0]. In that case,H2(K)=0. Intuitively, one may thinkH2(K) capturing the voids in a complexKembedded inR3. (Convince yourself thatH1(K)=0 no matter whether the tetrahedron belongs toKor not.)
Fact 2.10. For p≥0,
1. Hpis a vector space (when defined overZ2),
2. Hp may not be a vector space when defined overZ, the integer coefficients. In this case, there could be torsion subgroups,
3. the Betti number,βp=dimHp, is given byβp=dimZp−dimBp,
4. there are exactly2βp homology classes inHpwhen defined withZ2coefficients.
2.5.1 Induced homology
Continuous functions from a topological space to another topological space takes cycles to cycles and boundaries to boundaries. Therefore, they induce a map in their homology groups as well.
Here we will restrict ourselves only to simplicial complexes and simplicial maps that are the counterpart of continuous maps between topological spaces. Simplicial maps between simplicial complexes take cycles to cycles and boundaries to boundaries with the following definition.
a
b
c d
e
g
K1 K2 h K3
Figure 2.12: Induced homology by simplicial map: Simplicial map f obtained by the vertex map a →e,b→e,c→g,d→ginduces a map at the homology level f∗:H1(K1) →H1(K2) which takes the only non-trivial class created by the empty triangleabcto zero thoughH1(K1)H1(K2).
Another simplicial mapK2 →K3destroys the single homology class born by the empty triangle eghinK2.
Definition 2.27 (Chain map). Let f : K1 → K2 be a simplicial map. The chain map f# : Cp(K1) → Cp(K2) corresponding to f is defined as follows. Ifc = Pαiσi is a p-chain, then
f#(c)=Pαiτiwhere
τi =
( f(σi), if f(σi) is ap-simplex inK2 0 otherwise.
For example, in Figure 2.12, the 1-cyclebc+cd+dbinK1is mapped to the 1-chaineg+eg=0 by the chain map f#.
Proposition 2.9. Let f : K1 → K2 a simplicial map. Let ∂Kp1 and ∂Kp2 denote the boundary homomorphisms in dimension p≥0. Then, the induced chain maps commute with the boundary homomorphisms, that is, f#◦∂Kp1 =∂Kp2 ◦ f#.
The statement in the above proposition can also be represented with the following diagram, which we saycommutessince starting from the top left corner, one reaches to the same chain at the lower right corner using both paths–first going right and then down, or first going down and then right (see Definition 3.15 in the next chapter).
Cp(K1)
∂Kp1
f# //Cp(K2)
∂Kp2
Cp−1(K1) f# //Cp−1(K2)
(2.2)
For example, in Figure 2.12, we have f#(c = ab +bd +da) = 0 and∂Kp1(c) = 0. Therefore,
∂Kp2(f#(c))=∂Kp2(0)=0= f#(0)= f#(∂Kp1(c)).
SinceBp(K1)⊆Zp(K1), we have that f#(Bp(K1))⊆ f#(Zp(K1)). Thus, the induced map in the quotient space, namely,
f∗(Zp(K1)/Bp(K1)) := f#(Zp(K1))/f#(Bp(K1))
is well defined. Furthermore, by the commutativity of the Diagram (2.2), f#(Zp(K1)) ⊆ Zp(K2) and f#(Bp(K1))⊆Bp(K2), which gives an induced homomorphism in the homology groups:
f∗:Zp(K1)/Bp(K1)→Zp(K2)/Bp(K2) or equivalently f∗:Hp(K1)→Hp(K2)
A homology class [c] = c+BpinK1 is mapped to the homology class f#(c)+ f#(Bp) inK2 by f∗. In Figure 2.12, we have B1 = {0,ab+bd +da}. Then, for c = bd+dc+cb, we have
f∗([c])={f#(c),f#(c)+ f#(ab+bd+da)}={0,0}=[0].
Now we can state a result relating contiguous maps (Definition 2.7) and homology groups that we promised in Section 2.1.
Fact 2.11. For two contiguous maps f1 : K1 → K2 and f2 : K1 → K2, the induced maps f1∗:Hp(K1)→Hp(K2)and f2∗:Hp(K1)→Hp(K2)are equal.
2.5.2 Relative homology
As the name suggests, we can define a homology group of a complex relative to a subcomplex.
Let K0 be a subcomplex of K. By definition, the chain groupCp(K0) is a subgroup ofCp(K).
Therefore, the quotient groupCp(K)/Cp(K0) is well defined which is called arelative chain group and is denotedCp(K,K0). It is an abelian group whose elements are the cosets [cp]=cp+Cp(K0) for every chaincp∈Cp(K).
The boundary operator ∂p : Cp(K) → Cp−1(K) extends to the relative chain groups in a natural way:
∂K,Kp 0 :Cp(K,K0)→Cp−1(K,K0), [cp]7→[∂pcp].
One may verify that∂K,Kp−10 ◦∂K,Kp 0 =0 as before. Therefore, we can define Zp(K,K0)=ker∂K,Kp 0, thep-th relative cycle group Bp(K,K0)=Im∂K,Kp+10, thep-th relative boundary group Hp(K,K0)=Zp(K,K0)/Bp(K,K0), thep-th relative homology group.
The relative homologyHp(K,K0) is related to a coned complexK∗. A coned complex K∗ of a simplicial complex K w.r.t. to the pair (K,K0) is a simplicial complex which has all simplices from Kand every coned simplexσ∪ {x}from an additional vertex x to every simplexσ ∈ K0. Figure 2.13 shows the coned complexes on right in each case. The following fact is useful to build an intuition about relative homology groups.
Fact 2.12. Hp(K,K0)Hp(K∗)for all p>0andβ0(H0(K,K0))=β0(H0(K∗))−1.
For example, considerKto be an edge{a,b,ab}withK0 ={a,b}as in Figure 2.13(left). The 1-chainab is a relative 1-cycle because∂1(ab) = a+b ∈ C0(K0) and hence ∂K,K1 0([ab]) is 0 in C0(K,K0). This is indicated by the presence of the loop in the coned space.
a
Figure 2.13: Illustration for relative homology: the subcomplex K0 consists of (left) verticesa andb, (right) verticesa,b,c, and the edgeab; the coned complexK∗are indicated with a coning from a dummy vertexx.
Now, consider K to be a triangle {a,b,c,ab,ac,bc,abc} with K0 = {a,b,c,ab} as in Fig-ure 2.13(right). The 1-chainsbcandacboth are relative 1-cycles because∂1(bc)=b+c∈C0(K0) and hence∂1K,K0([bc]) is 0 inC0(K,K0); similarly,∂K,K1 0([ac]) = 0. The 1-chainab is of course a relative 1-cycle because it is already 0 as a relative chain. Therefore, the relative 1-cycle group Z1(K,K0) has a basis {[bc],[ac]}. The relative 1-boundary group B1(K,K0) is given by
∂K,K2 0(abc) = [ab]+[bc]+[ac] = [bc]+[ac]. The relative homology groupH1(K,K0) has one non-trivial class, namely the class of either [bc] or [ac] but not both because [bc]+[ac] is a relative boundary.
2.5.3 Singular Homology
So far we have considered only simplicial homology which is defined on a simplicial complex without any assumption of a particular topology. Now, we extend this definition to topological spaces. Let X be a topological space. We bring the notion of simplices in the context of X by considering maps from the standardd-simplices toX. A standardp-simplex∆pis defined by the convex hull of p+1 points
(x1, . . . ,xi, . . . ,xp+1)|xi=1and xj=0 f or j,ii=1,...,p+1inRp+1. Definition 2.28(Singular simplex). A singular p-simplex for a topological spaceXis defined as a mapσ:∆p→X.
Notice that the mapσneed not be injective and thus∆pmay be ‘squashed’ arbitrarily in its image. Nevertheless, we can still have a notion of the chains, boundaries, and cycles which are the main ingredients for defining a homology group called thesingular homologyofX.
The boundary of ap-simplexσis given by∂σ=τ0+τ2+. . .+τpwhereτi : (∂∆p)i →Xis the restriction of the mapσon theith facet (∂∆p)iof∆p.
A p-chain is a sum of singular p-simplices with coefficients from integers, reals, or some appropriate rings. As before, under our assumption ofZ2coefficients, a singular p-chain is given byP
iαiσiwhereαi =0 or 1. The boundary of a singularp-chain is defined the same way as we did for simplicial chains, only difference being that we have to accommodate for infinite chains.
∂(cp =σ1+σ2+. . .)=∂σ1+∂σ2+. . . We get the usual chain complex with∂p◦∂p−1=0 for allp>0
· · ·∂→p+1 Cp
∂p
→Cp−1∂→ · · ·p−1
and can define the cycle and boundary groups asZp = ker∂p andBp = im∂p+1. We have the singular homology defined as the quotient groupHp=Zp/Bp.
A useful fact is that singular and simplicial homology coincide when both are well defined.
Theorem 2.10. Let X be a topological space with a triangulation K, that is, the underlying space
|K|is homeomorphic to X. ThenHp(K)Hp(X)for any p≥0.
Note that the above theorem also implies that different triangulations of the same topological space give rise to isomorphic simplicial homology.
2.5.4 Cohomology
There is a dual concept to homology called cohomology. Although cohomology can be defined with coefficients in rings as in the case of homology groups, we will mainly focus on defining it over a field thus becoming a vector space.
A vector spaceV defined with a field kadmits a dual vector spaceV∗ whose elements are linear functionsφ:V →k. These linear functions themselves can be added and multiplied over kforming the dual vector spaceV∗. The homology groupHp(K) as we defined in Definition 2.26 over the fieldZ2is a vector space and hence admits a dual vector space which is usually denoted asHom(Hp(K),Z2). The p-th cohomology group denotedHp(K) is not equal to this dual space, though over the coefficient field Z2, one has thatHp(K) is isomorphic to Hom(Hp(K),Z2) and Hp(K) is also defined with spaces of linear maps.
Cochains, cobounadries, and cocycles. Ap-cochain is a homomorphismφ :Cp → Z2from the chain group to the coefficient ring over whichCpis defined which isZ2 here. In this case, a p-cochainφis given by its evaluationφ(σ) (0 or 1) on everyp-simplexσinK, or more precisely, a p-chainc=Pmp
i=1αiσigets a value
φ(c)=α1φ(σ1)+α2φ(σ2)+· · ·+αmpφ(σmp).
Also, verify thatφ(c+c0) = φ(c)+φ(c0) satisfying the property of group homomorphism. For a chainc, the particular cochain that assigns 1 to a simplex if and only if it has a non-zero coefficient inc, is called its dual cochainc∗. Thep-cochains form a cochain groupCpdual toCpwhere the addition is defined by (φ+φ0)(c) =φ(c)+φ0(c) by takingZ2-addition on the right. We can also define a scalar multiplication (αφ)(c) = αφ(c) by using theZ2-multiplication. This makesCp a vector space.
Similar to boundaries of chains, we have the notion of coboundaries of cochainsδp : Cp → Cp+1. Specifically, for a p-cochain φ, its (p+1)-coboundaryis given by the homomorphism δφ : Cp+1 → Z2 defined asδφ(c) = φ(∂c) for any (p+1)-chainc. Therefore, the coboundary operatorδtakes a p-cochain and produces a (p+1)-cochain giving the sequence for a simplicial
Similar to boundaries of chains, we have the notion of coboundaries of cochainsδp : Cp → Cp+1. Specifically, for a p-cochain φ, its (p+1)-coboundaryis given by the homomorphism δφ : Cp+1 → Z2 defined asδφ(c) = φ(∂c) for any (p+1)-chainc. Therefore, the coboundary operatorδtakes a p-cochain and produces a (p+1)-cochain giving the sequence for a simplicial