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Let $X$ be a fibrant connected simplicial set. There is a simplicial map $h_n\colon X^{2n}\to \mathbb{Z}[X]$, defined on points by $(x_1, \ldots x_{2n})\mapsto \sum\limits_{i=1}^{2n}(-1)^ix_i$. Here $\mathbb{Z}[X]$ is the simplicial abelian group associated to $X$ (the object that appears in the statement of Dold-Thom theorem). There is a natural map $f\colon F_{h_n} \to h_n^{-1}(0)$ between the fiber and the homotopy fiber of $h_n$.

Question: What can be said about the connectivity of this map $f$? Can one expect that it is $c(n)$-connected for some sequence $c(n)$ that tends to infinity as $n\to \infty$?

What I think can be shown in this situation is the following:

  • For $n=\infty$ $f$ is a weak equivalence. Here $h_\infty$ means the following thing: one chooses a basepoint $x_0\in X$ and considers the colimit of $h_n$ with respect to "stabilization" maps $X^{2n} \to X^{2(n+1)}$, $x \mapsto (x, x_0, x_0)$.
  • For $n\geq 3$ and $X=BG$ (the classifying space of a discrete group $G$) I can show via some lengthy computation that $f$ is $1$-connected, i.e. $\pi_1(F_{h_n})\cong \pi_1(h_n^{-1}(0))$.

I am also interested whether there is any general method which could produce a simple proof of the latter statement (or any reference if this is already known).

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