# Postnikov system for a tree

The Postnikov system of a path-connected space X is a tower of spaces $...\longrightarrow X_{n+1}\longrightarrow X_{n}\longrightarrow ...\longrightarrow X_{1}\longrightarrow X_{0}$ with the following properties:

1) each map $X_{n+1}\longrightarrow X_{n}$ is a fibration.

2) $\pi_{k}(X_{n})=\pi_{k}(X)$ for $k \leqslant n$.

3) $\pi_{k}(X_{n})=0$ for $k > n$.

Every CW-complexes has such a Postnikov system, and it is unique up to homotopy. The space X can be reconstructed from the Postnikov system as its inverse limit.

If $X$ is the Bruhat-Tits tree of $PGL(2,F)$, where $F$ a locally compact non-archimidian fields. As a topological space $X$ is CW-complexe and then it posseses a postnikov system. We can construct a tower of graphs over $X$

$...\longrightarrow X_{n+1}\longrightarrow X_{n}\longrightarrow ...\longrightarrow X_{1}\longrightarrow X_{0}$

such that their invere limit is disjoint union of $X$ indexeces by the points of the board of $X$.

The set of vertex of $X_{n}$ is the set of geodesic path of length $n$ in $X$ and the set of edges is the set of geodesic path of length $n+$ in $X$.

My question is : Is this tower of graphs is Postinikov System for $X$. If it is not clear that this tower is Postnikov System or not, how we can construct a Postnikov system for $X$

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What known properties of this tower of graphs make you suspect that it may be a Postnikov tower? Concerning your second question, there's a general procedure to construct Postnikov towers for arbitrary (nice enough) spaces, but usually it doesn't look very nice even if you apply it to a space you're familiar with. – Fernando Muro Jun 28 '13 at 17:34
What does the phrase "points of the board of $X$" mean? – Lee Mosher Jun 28 '13 at 18:40
I'm sorry Mr Lee for my bad english. It means the ends of the tree X, that is the set of semi-geodesic with basepoint s for a fixed vertex s of X. There is another definition of the ends of a tree or more genaraly of a graph. – Rajkarov Jun 29 '13 at 19:22
Thank you Mr Fernando for your responses. Actually I don't know I if this tower of graphs is postnikov system or not. But I understend the combinatoral prprieties of this graphs. for exemple every graph $X_{k}$ contain a loop and then it's not contractible. But we can taking a subgraphs $X_{k}^{x}$, where x is a point of the ends of X and $X_{k}^{x}$ is the subgraph of $X_{k}$ consists of the edges a such that there is a sequence $(s_{i})$ of vertex of X such that $(s_{0},...,s_{k+1})=a$ and $(s_{i})$ converge to x. I think that this tower of subgraphs is Postnikov System for $X$ . – Rajkarov Jun 29 '13 at 19:42