# Homology and Burnside ring

If $G$ is a finite groupe, we denote $\mathcal{S}(G)$ the category of finite $G$-sets and $\mbox{I}(G)$ the set of isomorphism classes of it's objects. The Burnside ring of $G$, denoted by $\Omega(G)$, is the grothendieck ring of the semi-ring $\mbox{Is}(G)$ for addition $[X]+[Y]=[X\sqcup Y]$ and multiplication $[X].[Y]=[X\times Y]$.

Let $G$ be a finite group. For $n\in\mathbb{N}$, we denote $G_n=G^{n+1}$. If $X$ is an object of $\mathcal{S}(G_{n+1})$, we define an object $\alpha_{i}X$ of $\mathcal{S}(G_{n})$ (for $0\leqslant i\leqslant n$) as follows : $\alpha_{i}X$ is the set $X$ with the action of the groupe $G^n$ defined by, $(g_0,...,g_n).x=(g_{0},...,g_{i-1},e,g_{i+1},...,g_n).x$, for all $x\in X$ and $(g_0,...,g_n)\in G_n$.

We define then a chain complex $(A_n,d_n)$ as follows :

1) for all $n\in\mathbb{N}$, $A_n=\Omega(G_n)$.

2) $d_n:A_{n+1}\longrightarrow A_n$ is defined by, $d_n([X])=\sum_{k=0}^{n}(-1)^k[\alpha_k X]$.

This defines a chain complex, we denote then $H^{bur}_n(G,\mathbb{Z})$ the nth homology group and we call it the nth Burnside homology group of $G$.

My question is : the homology $H^{bur}(G,\mathbb{Z})$ can be interesting and nontrivial ?

• Is the definition of $(A_n,d_n)$ new? – Dima Pasechnik Apr 11 '15 at 19:28
• Yes, I think so, the definition is mine but i'm not sure that it give's something new. – Anis Rajhi Apr 11 '15 at 20:17
• well, try this for $G$ being something small and see... – Dima Pasechnik Apr 11 '15 at 20:22
• Ok I will try it. I don't know if there is such definition or construction ? – Anis Rajhi Apr 11 '15 at 20:49