I am trying to calculate the homology of the chain complex defined as follows
or 0 otherwise.
We defined the boundary operator
First question - I presume the notation
means the free abelian group with generators
Second - I don't really know where to go with this question.
I might as well start by assuming k odd and
I am not really sure how to calculate the image/kernel of these maps? Any hints appreciated
First of all, yes, it is the free abelian group on two generators. The point of the presentation you're given is to calculate the kernal using linear algebra.
Notice that the matrix for the even ones is:
and for the odds is:
so clearly the nullspace (kernel) has dimension 1 for each of them, and in the odd case it is obviously spanned by
and the even case by
, since those are sent to zero clearly from the description. As the matrix has rank 1, that's the entire kernel. By the rank nullity theorem you should be able to deduce that the image has dimension 1 as well, and go from there.