## Characters on homology groups of simplicial complexes

Hi all, nice to be here

I,ve been given this problem and have made some progress with it but am still slightly confused over the statement of it:

Essentially I need to find a simplicial complex (Simplicial complex - Wikipedia, the free encyclopedia) such that the alternating sum of the characters for the homology groups $\sum (-1)^i h_i$, where $h_i$ is a character for the homology group $H_i$ (Simplicial homology - Wikipedia, the free encyclopedia, better explained here) of the characters gives the character
$\chi (g)= |\{ (x,y) \in G | g=xyx^{-1}y^{-1}\}|$

To explain in more detail:

is there a natural looking way to write this character as a difference of two permutation
characters for some classes of groups? This would follow from the
simplicial complex statement by taking the permutation characters on
even/odd dimensional simplices and taking the difference of those
characters.

I think I have stated the problem correctly, but please ask if you think I am unclear as I don't quite understand it. My main issues are:

Surely you cannot make a simplicial complex with homology groups isomorphic to any group? If not how to the characters on these groups relate to other groups like $S_n$ or $SL(2,q)$ for example.

As regards to the statement on permutation characters, how can you have a permutation character on a homology group, when the group is not acting on anything? Is there a natural way to see this as acting on the simplicial complex?

Sorry for the long post, If you think you may be able to help with even just a tiny part of my question it would be a great help.

Many thanks,

Brothergomez