Homomorphisms on Homology Induced by Trisp Maps

This convention will simplify our arguments considerably, since by using it we can avoid dealing with different cases, and can perform a unified computation. 

Proof of Proposition 3.33. By Proposition 3.30 we need to show that a sim-plicial map f A2 will induce a chain map between the corresponding 

By definition of chain maps, to see that the collection of maps / defines a chain map, we need to see that the following diagram commutes  

By linearity, it is enough to verify this fact for oriented simphces, which is straightforward, as the following diagram shows  

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.  

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