Hatcher Exercise 3.1.6


We compute the simplicial cohomology of some basic cell complexes directly from the definitions, using coefficients in both and .


(a) Let , with the cell complex structure given in Hatcher section 2.1. First work with coefficients in . The cochain complex is:

Specifically,

where we are viewing, for example , as the map assigning to the edge and to the other edges. Clearly then, the group is generated by , , and (similarly for the others).

We have for all edges , so . For , we have , and . Then, we can express as . Similarly, we find and . To summarize,

Determining the homology of the cochain complex is now a matter of algebra determined completely from the matrices (keeping in mind the homology groups are , instead of ), and we find:

Of course, this matches the homology of the torus exactly, because there is no torsion.

If we work with coefficients in , the maps change slightly. We still have , but . Then we have

and the homology is:


(b) Let , with the cell complex structure from Hatcher 2.1. First work with coefficients in . The cochain complex is:

We have , and likewise . Also . So . Similarly we find . For , we have and . Thus . Similarly we find and . Summarizing:

and we arrive at:

If we work with coefficients in , the maps change as follows:

and we arrive at:


(c) Finally, let with the cell structure from 2.1. Again, work with coefficients first. The cochain complex is:

We have , because there is only one vertex (reasoning as in part (a)). We have , and . So we find . For the others, we find and . So the maps are:

and we have:

If we work with coefficients in , changes as follows:

and we arrive at: