I'm trying to understand how to find homotopy type of complement to a set of linear subspaces knowing their configuration (dimension, number and intersection).
Some low-dimensional examples are quite easy because we can directly find homotopy to a neat space:
- Complement to a set of subspaces $x_1=x_2=0,\ x_3=x_4=0$ in $\mathbb{R}^4$ ($x_i$ means real variable).
The complement to a two 2-dimeinsional subspaces in $\mathbb{R}^4$ that intersects by a point. We can remove the zero point and contract complement $\mathbb{R}^4\backslash 0$ to a sphere $S^3$; both planes now retracted to standard circles $S^1$ and those circles are linked in $S^3$. This means, this is homotopy equivalent to a torus $T^2\cong S^1\times S^1$.
- Complement to a set of subspaces $x_1=x_2=\ldots=x_k=0,\ x_{k+1}=x_{k+2}=\ldots =x_{2k}=0$ in $\mathbb{R}^{2k}$.
This is a direct generalization of a previous example and it gives product of spheres $S^{k-1}\times S^{k-1}$.
- Complement to a set of subspaces $x_1=x_2=\ldots = x_8=0,\ x_5=x_6=\ldots=x_{12}=0,\ x_9=x_{10}=\ldots=x_{12}=x_1=x_2=\ldots=x_4=0$.
This is a set of three four-dimensional subspaces in $\mathbb{R}^{12}$ and any two intersects by a point. We can use the same retraction of $\mathbb{R}^{12}\backslash 0$ to a sphere $S^{11}$. Now all three subspaces are retracted to spheres $S^3$. Now we can use that fact that any embedding of disjoint union of finite number of spheres $S^3$ in $S^{11}$ is isotopic to a regular, so this space is homeomorphic to a $S^{11}\backslash(S^3\sqcup S^3\sqcup S^3)\simeq S^7\vee S^7\vee S^7 \vee S^{10}\vee S^{10}$. I'm not confident about this reasoning though. Is this correct?
- Complement to a set of subspaces $z_1=z_2=0,\ z_2=z_3=0,\ z_3=z_4=0,\ z_4=z_5=0,\ z_5=z_1=0$ in $\mathbb{C}^5$ ($z_i$ means complex variable).
Now I have too few ideas. The first idea is to look at the intersection of all subspaces which is another set of subspaces, try to find its homotopy type and accurately try to see what happens with the original complement after this homotopy.
Another idea I have, but without deep analysys, is the same, but with cube - this should retracts to a subcomplex of cube $D^{10}\cong \mathbb{C}^5$. I have no idea how to do this strictly, so I would be a glad to see a some hint in this.
- Complement to a set of subspaces $q_1=q_2=0,\ q_2=q_3=0,\ q_3=q_4=0,\ q_4=q_1=0$ in $\mathbb{H}^4$.
I also doesn't have ideas here, so I would be glad to see any hint!
Thank you!