Find the homotopy type of $$R^5 \setminus \left\{ (x_1 = x_2 = 0) \ \cup \ (x_2 = x_3 = 0) \ \cup \ (x_3 = x_4 = 0) \ \cup \ (x_4 = x_5 = 0) \ \cup \ (x_5 = x_1 = 0) \right\}.$$
I guess it should be a wedge sum of spheres but can't prove it.
I tried to realise it as $S^4 \setminus S^2 \cup S^2\cup S^2\cup S^2\cup S^2$, but it seems a dead end to me.
I was told to somehow use the simplicial complexes, but I don't understand either how to do it or how it would make this problem easier.
So, what should I do? Any hints are welcome.