In this paper https://arxiv.org/pdf/math/0202260.pdf, the author proves that $BP$ is homotopic to $S^2$, using homology (In the first lemma of the paper).
Well, I tried to use the definition of geometric realization to see how $BP$ looks like. I can see that everything just collapses into five 1 simplices via the multiplication defined. But I can't seem to connect these together, to form $S^2$.
What am I missing here?
Thanks in advance.