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I'm reading Ghrist's paper "Configuration spaces and braid groups on graphs in robotics". In this discussion, counterexamples are shown for both Theorem 2.3 and the implication "Theorem 2.3 $\Rightarrow$ Corollary 2.4". Are there other proofs, in literature, that configuration spaces of trees are Eilenberg-MacLane spaces? Or, is there a way to "fix" Ghrist's proof?

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If the tree is a single edge, its configuration spaces (for at least two points) are disconnected and thus not a $K(\pi,1)$. (I know you only said "Eilenberg MacLane space" without saying "1", but the linked discussion says Ghrist claimed it was a $K(\pi,1)$.) – Omar Antolín-Camarena Jun 6 '14 at 13:24
Yes, I actually meant "aspherical". – Giove Jun 6 '14 at 15:40
up vote 4 down vote accepted

The paper

Patty, C. W., Homotopy groups of certain deleted product spaces, Proc. Amer. Math. Soc. 12 (1961) 369–373.

contains a proof that for any connected, finite graph having essential vertices (of degree $\ge3$) the "deleted product" $$C_2(\Gamma) = \Gamma\times\Gamma \setminus \Delta\Gamma$$ is aspherical. The proof uses the following Theorem of J.H.C. Whitehead:

Suppose $X=A\cup B$, where $X$, $A$ and $B$ are connected polyhedra. Suppose that $A\cap B$ has finitely many path components $C_i$. If:

  1. each of $A$, $B$ and $C_i$ are aspherical, and
  2. each map $\pi_1(C_i)\to \pi_1(A)$ and $\pi_1(C_i)\to \pi_1(B)$ induced by inclusion is injective,

then $X$ is aspherical.

I don't know if this proof can be adapted to give asphericity of $C_N(\Gamma)$ for $N>2$ when $\Gamma$ is a tree, but perhaps this is what Ghrist had in mind.

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