at.algebraic topology - homotopy type of complement of subspace arrangement

I am studying the homotopy type of a space,and i hope it would be a $K(pi,1)$ space.
now i have find its covering,once we can say the covering is $K(pi,1)$,so is the space
itself.and the covering is

$mathbb{R}^4-M$ where $M=M_1cup M_2cup M_3cup M_4$,

$M_1={(x,y,z,w)|x,y in mathbb R,z,w inmathbb Z}$

$M_2={(x,y,z,w)|y,z in mathbb R,x,w inmathbb Z}$

$M_3={(x,y,z,w)|x,w in mathbb R,y,z inmathbb Z}$

$M_4={(x,y,z,w)|z,w in mathbb R,x,y inmathbb Z}$

I guess $mathbb{R}^4-M$ is $K(pi,1)$ space,can someone help prove this?