I would like to determine topological properties of $\mathbb R^8$ minus the set determined by the equation $ \mathrm{det}\begin{pmatrix} a-a' & b-b'\\ c-c' & d-d' \end{pmatrix}=0$ where $a,a',b,b',c,c',d,d'\in\mathbb R$.
How do I determine the homotopy type and how many connected components this space has? If this does not turn out to be a standard space, I would also like to determine (co)homology groups.