Simplicial Homology Groups

Examples of families of sets that satisfy the conditions of Lemma 3.1 are the kernel and the image of every boundary map since Z 2) = 

An important property of this boundary operator is that when applied twice it gets reduced to the trivial map. 

Lemma 3.4. Let A be a finite abstract simplicial complex. Then for all n 0, 

For any E Cn+ A), the simplices in dn dn+i E)) are obtained by taking some simplex in E and deleting from it two different vertices. This can be done in two different ways, depending on the order in which the two vertices are removed. In the total symmetric difference these two simplices give the combined contribution 0, and hence the total value of 5 (5 +i S)) is equal to 0 as well.  

We shall now upgrade our discussion from Section 3.1 in two ways first we replace abstract simplicial complexes with arbitrary trisps second we now phrase our invariants algebraically as groups. 

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Theorem 3.26. For an arbitrary trisp A, its simplicial and singular homology groups are isomorphic. 

We conclude that a simplicial map between abstract simplicial complexes induces a chain map between associated simplicial chain complexes, and hence also the homomorphisms between the corresponding homology groups. ... 

The main reason why CW complexes are so handy for concrete computations of homology groups is that it turns out that one can substitute the simplicial chain complex with another, usually much smaller, chain complex, which we now proceed to define. 

For example, an action on an abstract simplicial complex induces an action on the underlying topological space this is a composition with the geometric realization functor. An action on a poset P induces an action on the order complex A P), which is defined in Chapter 9 this is a composition with the order complex functor. Furthermore, it induces a G-action on any given homology group Hi A P)] TZ), which, in case 7 . is a field, is the same as a linear representation of G over TZ. 

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