Hatcher Exercise 2.1.6
We compute the simplicial homology of the complex described in the text.
First, we fix notation to refer to all the faces of the complex: refers to the ‘th vertex of the ‘th -simplex. Next, we perform the indicated identifications in order to form equivalence classes of faces. We have two-simplices:
We have edges:
And finally, all the vertices end up identified to a single vertex:
Now, we compute homology groups. We have , , , the latter two isomorphic to .
For , we have and .
For , we have , for all (since there is only one vertex). As such, and .
For , we have:
So, can be described by:
Where the matrix on the right is the Smith Normal Form of .
and higher are the zero map.
We have .
To find , we first conclude that , so is a torsion group. The torsion coefficients are given by the non-trivial entries of . We conclude that .
It’s clear that is injective, so .