Let \(X\) be a complex of spaces over a simplicial complex \(\Gamma\), as defined in §4.G. Show that the natural projection \(\Delta X \to \Gamma\) is a quasifibration if all the maps in \(X\) associated to edges of \(\Gamma\) are weak homotopy equivalences.
Solution
Let \[ p:\Delta X\longrightarrow\Gamma \] be the canonical projection. We prove that it is a quasifibration.
First restrict over a simplex \[ \sigma=[v_0,\ldots,v_q]\subset\Gamma. \] By definition, \(p^{-1}(\sigma)\) is the iterated mapping cylinder of the sequence of edge maps \[ X_{v_0}\longrightarrow X_{v_1}\longrightarrow\cdots\longrightarrow X_{v_q}. \] Collapse \(\sigma\) linearly to its terminal vertex \(v_q\). The corresponding deformation of the iterated mapping cylinder collapses it to \(X_{v_q}\). For a point \(b\in\sigma\), the induced map from the fiber \(p^{-1}(b)\) to the terminal fiber \(X_{v_q}\) is a composite of edge maps. By hypothesis every edge map is a weak homotopy equivalence, hence so is every such composite. Lemma 4K.3(c) therefore implies \[ p:p^{-1}(\sigma)\to\sigma \] is a quasifibration. The same argument applies to every face of \(\sigma\).
Now pass from a single simplex to the whole complex. Over each finite subcomplex \(L\subset\Gamma\), glue the simplexwise quasifibrations one simplex at a time. Near the attaching boundary of a simplex use a collar, so Lemma 4K.3(a) applies; the intersections are precisely restrictions over faces, already known to be quasifibrations. Hence \[ p:p^{-1}(L)\to L \] is a quasifibration for every finite subcomplex \(L\).
Finally exhaust \(\Gamma\) by its finite subcomplexes. Every compact subset of a CW complex is contained in a finite subcomplex, so the corrected form of Lemma 4K.3(b) applies. Therefore \[ \boxed{\Delta X\longrightarrow\Gamma\text{ is a quasifibration}.} \] The fiber over a vertex \(v\) is the corresponding vertex space \(X_v\), and fibers over arbitrary points are weakly equivalent to these via the edge maps.