In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can replace $\Delta^{n}$ by the orthonormal group $O(n)$. There are $n+1$ (both topological and group theoretical) obvious embedding $\epsilon_{i}:O(n)\to O(n+1),\;\;i=1,2,\ldots,n+1$. These obvious embeddings are as follows:
$$\epsilon_{1}(A)=1 \oplus A,\;\;\\ \epsilon_{i}(A)=\lambda_{i}\epsilon_{1}(A)\lambda_{i}^{-1}$$ where $\lambda_{i}$ is the elementary matrix obtaining from the identity matrix by replacing first row by $i_{th}$ row.
For example, for $n=2$, the matrix $\epsilon_{i} \left (\begin{pmatrix} a&b\\c&d \end{pmatrix} \right)$ is $\begin{pmatrix}1&0&0\\0&a&b\\0&c&d \end{pmatrix}$, $\begin{pmatrix}a&0&b\\0&1&0\\c&0&d \end{pmatrix}$ and $\begin{pmatrix} a&b&0\\c&d&0\\0&0&1\end{pmatrix}$, for $i=1,2,3,\;$ respectively.
For a topological space $X$, one can define $\overline{C_{n}(X)}$ as the free abelian group generated by all continuous maps from $O(n)$ to $X$.
The boundary maps $\delta: \overline{C_{n}(X)} \to \overline{C_{n-1}(X)}$ is defined by $\delta(\phi)=\sum (-1)^i \phi \circ \epsilon_i$. Then we have $\delta \circ \delta =0$.
So we obtain a kind of "homology" as a functor on the category of topological spaces. Of course homeomorphic spaces have isomorphic homology.
(However this functor is not necessarily a homotopoic invariant functor but it impose an equivalent relation on the space of all continuous maps $f,g:X\to Y:\; f\simeq g$ iff $f_{*}=g_{*}$.
Is this equivalent relation, stronger than the homotopy equivalent?
On the other hand, for a group $G$, one can define the free abelian group generated by all group homomorphism from $O(n)$ to $G$. The same processes as above, gives us a homology of groups.
Are these type of homologies studied already?
