Hatcher Exercise 2.1.8

We compute the simplicial homology of the complex described in the text.

Fix notation to refer to the vertices of each tetrahedron in the complex: will refer to the ‘th vertex on the ‘th tetrahedron – throughout this problem, used as an index ranging from to will be understood to be taken mod . We perform the indicated identifications and examine equivalence classes of faces: We have 3-simplices (the tetrahedra):

For faces, we first identify the vertical faces with the next neighboring vertical face, in a cyclic fashion. This gives

Then we identify the bottom face of each tetrahedron with the top of the next tetrahedron around, giving:

So, we have faces:

Examining carefully the identifications this induces, we have edges:

And only vertices:

So we have , , , . We calculate the homology groups:

For , we have and .

For , we have:

For , we have:

For , we have:

It’s clear that . Let’s represent the rest of the boundary maps as matrices, to compute the homology:

Where the matrices on the right are the corresponding Smith normal forms. Then, we compute the Betti numbers:

So, besides , is the only homology group with non-trivial free rank, and has a non-trivial torsion coefficient coming from . We conclude: