Show that if \(X\) is a complex of spaces in which all the maps are homeomorphisms, then the projection \(\Delta X \to \Gamma\) is a fiber bundle.
Solution
1. Structure of the total space \(\Delta X\):
1.1. A complex of spaces \(X\) over a graph \(\Gamma\) consists of a vertex space \(X_v\) for each \(v \in \Gamma\), an edge space \(X_e\) for each edge \(e \in \Gamma\), and attaching maps \(f_{e, 0}: X_e \to X_{d_0(e)}\) and \(f_{e, 1}: X_e \to X_{d_1(e)}\).The realization \(\Delta X\) is the quotient space:\[ \Delta X = \left( \bigsqcup_{v \in \Gamma^{(0)}} X_v \sqcup \bigsqcup_{e \in \Gamma^{(1)}} (X_e \times [0, 1]) \right) \Big/ \sim, \]where \((x, 0) \sim f_{e, 0}(x)\) and \((x, 1) \sim f_{e, 1}(x)\) for all \(x \in X_e\).
definition of the total space of a complex of spaces.
1.2. The projection \(p: \Delta X \to \Gamma\) maps \(X_v \mapsto v\) and \((x, t) \in X_e \times [0, 1] \mapsto t \in e\).Because all attaching maps \(f_{e, i}\) are homeomorphisms and \(\Gamma\) is connected, all vertex spaces \(X_v\) and edge spaces \(X_e\) are homeomorphic to a common fiber space \(F\).
composition of homeomorphisms along paths in \(\Gamma\).
2. Local triviality over open edges:
2.1. Let \(e\) be an edge of \(\Gamma\), and let \(U_e = \operatorname{int}(e) \cong (0, 1)\) be the open interior of \(e\).The preimage is \(p^{-1}(U_e) = X_e \times (0, 1)\).
interior of edge has no quotient identifications.
2.2. Using the homeomorphism \(\phi_e: X_e \xrightarrow{\sim} F\), we have a trivializing homeomorphism:\[ p^{-1}(U_e) = X_e \times U_e \xrightarrow{\phi_e \times \operatorname{id}} F \times U_e, \]which satisfies \(\operatorname{pr}_1 \circ (\phi_e \times \operatorname{id}) = p\).
direct Cartesian product.
3. Local triviality over vertex neighborhoods:
3.1. For each vertex \(v \in \Gamma^{(0)}\), let \(U_v \subset \Gamma\) be the open star of \(v\), consisting of \(v\) together with open half-edges \([0, 1/2)\) for all edges incident to \(v\).
open star basis for 1-complexes.
3.2. The preimage \(p^{-1}(U_v)\) is the union of \(X_v\) and \(X_e \times [0, 1/2)\) for each incident edge \(e\), glued via \((x, 0) \sim f_{e, v}(x)\).Since each \(f_{e, v}: X_e \xrightarrow{\sim} X_v\) is a homeomorphism, the map \((x, t) \mapsto (f_{e, v}(x), t)\) is a homeomorphism \(X_e \times [0, 1/2) \xrightarrow{\sim} X_v \times [0, 1/2)\).
product of homeomorphisms.
3.3. Under these maps, the gluing along \(t = 0\) corresponds to the identity map on \(X_v\).Thus we obtain a canonical homeomorphism:\[ \psi_v: p^{-1}(U_v) \xrightarrow{\sim} X_v \times U_v \cong F \times U_v, \]which commutes with the projection \(p\).
gluing local product charts along the identity at the vertex.
4. Conclusion:Every point in \(\Gamma\) has an open neighborhood \(U\) with \(p^{-1}(U) \cong F \times U\), so \(p: \Delta X \to \Gamma\) is a fiber bundle with fiber \(F\). Q.E.D.
2 and 3.