Our goal now is to define the simplicial homology groups of a∆ complex X . Let
∆n(X) be the free abelian group with basis the open n simplices enα ofX . Elements
of ∆n(X) , called n chains, can be written as finite formal sums P
αnαenα with co-efficients nα ∈ Z. Equivalently, we could write P
αnασα where σα:∆n
→
X is thecharacteristic map of eαn, with image the closure of enα as described above. Such a sumP
αnασα can be thought of as a finite collection, or ‘chain,’ of n simplices in X with integer multiplicities, the coefficientsnα.
As one can see in the next figure, the boundary of then simplex [v0,··· , vn] con-sists of the various(n−1) dimensional simplices [v0,··· , bvi,··· , vn] , where the ‘hat’
symbol b over vi indicates that this vertex is deleted from the sequencev0,··· , vn. In terms of chains, we might then wish to say that the boundary of[v0,··· , vn] is the (n− 1) chain formed by the sum of the faces [v0,··· , bvi,··· , vn] . However, it turns out to be better to insert certain signs and instead let the boundary of[v0,··· , vn] be P
i(−1)i[v0,··· , bvi,··· , vn] . Heuristically, the signs are inserted to take orientations into account, so that all the faces of a simplex are coherently oriented, as indicated in the figure.
v0
v1 v2 v3
v
v0 1
v2
v0
-
+v1 ∂[v0, v1]= [v1]− [v0]
∂[v0, v1, v2]= [v1, v2]− [v0, v2]+ [v0, v1]
∂[v0, v1, v2, v3]= [v1, v2, v3]− [v0, v2, v3] + [v0, v1, v3]− [v0, v1, v2]
In the last case, the orientations of the two hidden faces are also counterclockwise when viewed from outside the 3 simplex.
With this geometry in mind we define for a general ∆ complex X a boundary homomorphism ∂n:∆n(X)
→
∆n−1(X) by specifying its values on basis elements:∂n(σα)=X
i
(−1)iσα||[v0,··· , bvi,··· , vn]
Note that the right side of this equation does indeed lie in∆n−1(X) since each restric-tionσα||[v0,··· , bvi,··· , vn] is the characteristic map of an (n− 1) simplex of X .
L
emma 2.1. The composition ∆n(X)---→
∂n ∆n−1(X)---→
∂n−1 ∆n−2(X) is zero.P
roof: We have ∂n(σ )=Pi(−1)iσ ||[v0,··· , bvi,··· , vn] , and hence
∂n−1∂n(σ )=X
j<i
(−1)i(−1)jσ ||[v0,··· , bvj,··· , bvi,··· , vn]
+X
j>i
(−1)i(−1)j−1σ ||[v0,··· , bvi,··· , bvj,··· , vn]
The latter two summations cancel since after switchingi and j in the second sum, it
becomes the negative of the first. tu
The algebraic situation we have now is a sequence of homomorphisms of abelian groups
···
-→
Cn+1---→
∂n+1 Cn---→
∂n Cn−1-→
···-→
C1---→
∂1 C0---→
∂0 0with∂n∂n+1= 0 for each n. Such a sequence is called a chain complex. Note that we have extended the sequence by a 0 at the right end, with ∂0 = 0. From ∂n∂n+1 = 0 it follows that Im∂n+1⊂ Ker ∂n, where Im and Ker denote image and kernel. So we can define the nth homology group of the chain complex to be the quotient group Hn= Ker ∂n/ Im ∂n+1. Elements of Ker∂n are calledcycles and elements of Im ∂n+1 areboundaries. Elements of Hnare cosets of Im∂n+1, calledhomology classes. Two cycles representing the same homology class are said to behomologous. This means their difference is a boundary.
Returning to the case thatCn= ∆n(X) , the homology group Ker ∂n/ Im ∂n+1 will be denotedHn∆(X) and called the nthsimplicial homology group of X .
E
xample 2.2. X = S1, with one vertex v and one edge e . Then ∆0(S1)v e and∆1(S1) are bothZ and the boundary map ∂1 is zero since∂e= v −v . The groups ∆n(S1) are 0 for n≥ 2 since there are no simplices in these dimensions. Hence
Hn∆(S1)≈
Z forn= 0, 1 0 forn≥ 2
This is an illustration of the general fact that if the boundary maps in a chain complex are all zero, then the homology groups of the complex are isomorphic to the chain groups themselves.
E
xample 2.3. X= T , the torus with the ∆ complex structure pictured earlier, having one vertex, three edgesa , b , and c , and two 2 simplices U and L . As in the previous example, ∂1= 0 so H0∆(T )≈ Z. Since ∂2U = a + b − c = ∂2L and {a, b, a + b − c} is a basis for ∆1(T ) , it follows that H1∆(T )≈ Z⊕Z with basis the homology classes [a]and [b] . Since there are no 3 simplices, H2∆(T ) is equal to Ker ∂2, which is infinite cyclic generated byU− L since ∂(pU + qL) = (p + q)(a + b − c) = 0 only if p = −q.
Thus
Hn∆(T )≈
Z⊕Z forn= 1 Z forn= 0, 2
0 forn≥ 3
E
xample 2.4. X = RP2, as pictured earlier, with two vertices v and w , three edges a , b , and c , and two 2 simplices U and L . Then Im ∂1 is generated by w− v , so H0∆(X)≈ Z with either vertex as a generator. Since ∂2U= −a+b+c and ∂2L= a−b+c , we see that∂2 is injective, soH2∆(X)= 0. Further, Ker ∂1≈ Z⊕Z with basis a−b and c , and Im ∂2 is an index-two subgroup of Ker∂1 since we can choose c and a− b + cas a basis for Ker∂1 anda− b + c and 2c = (a − b + c) + (−a + b + c) as a basis for Im∂2. Thus H1∆(X)≈ Z2.
E
xample 2.5. We can obtain a∆ complex structure on Sn by taking two copies of∆n and identifying their boundaries via the identity map. Labeling these twon simplices U and L , then it is obvious that Ker ∂n is infinite cyclic generated by U − L. Thus Hn∆(Sn) ≈ Z for this ∆ complex structure on Sn. Computing the other homology groups would be more difficult.Many similar examples could be worked out without much trouble, such as the other closed orientable and nonorientable surfaces. However, the calculations do tend to increase in complexity before long, particularly for higher-dimensional complexes.
Some obvious general questions arise: Are the groups Hn∆(X) independent of the choice of ∆ complex structure on X ? In other words, if two ∆ complexes are homeomorphic, do they have isomorphic homology groups? More generally, do they have isomorphic homology groups if they are merely homotopy equivalent? To answer such questions and to develop a general theory it is best to leave the rather rigid simplicial realm and introduce the singular homology groups. These have the added advantage that they are defined for all spaces, not just ∆ complexes. At the end of this section, after some theory has been developed, we will show that simplicial and singular homology groups coincide for ∆ complexes.
Traditionally, simplicial homology is defined forsimplicial complexes, which are the ∆ complexes whose simplices are uniquely determined by their vertices. This amounts to saying that eachn simplex has n+ 1 distinct vertices, and that no other n simplex has this same set of vertices. Thus a simplicial complex can be described combinatorially as a set X0 of vertices together with sets Xn of n simplices, which are(n+1) element subsets of X0. The only requirement is that each(k+1) element subset of the vertices of ann simplex in Xn is ak simplex, in Xk. From this combi-natorial data a ∆ complex X can be constructed, once we choose a partial ordering of the vertices X0 that restricts to a linear ordering on the vertices of each simplex in Xn. For example, we could just choose a linear ordering of all the vertices. This might perhaps involve invoking the Axiom of Choice for large vertex sets.
An exercise at the end of this section is to show that every ∆ complex can be subdivided to be a simplicial complex. In particular, every∆ complex is then homeo-morphic to a simplicial complex.
Compared with simplicial complexes,∆ complexes have the advantage of simpler computations since fewer simplices are required. For example, to put a simplicial complex structure on the torus one needs at least 14 triangles, 21 edges, and 7 vertices, and forRP2 one needs at least 10 triangles, 15 edges, and 6 vertices. This would slow down calculations considerably!