Extending Proposition 4I.3, show that the \((2k+1)\)-skeleton of the suspension of a high-dimensional lens space with fundamental group of order \(p^n\) is homotopy equivalent to the wedge sum of the \((2k+1)\)-skeleta of the spaces \(X_i\), if these \(X_i\)’s are chosen to have the minimum number of cells in each dimension.
Solution
Let \[ K=K(\mathbb Z_{p^n},1) \] be the infinite-dimensional lens space, and let \[ \Sigma K\simeq X_1\vee\cdots\vee X_{p-1} \tag{1} \] be the splitting of Proposition 4I.3. Choose each \(X_i\) in the minimal-cell form described after the proof: one \(0\)-cell and one cell in each dimension congruent to \[ 2i\quad\text{or}\quad2i+1\pmod{2p-2}, \] and no other positive-dimensional cells.
A lens space of sufficiently high dimension with fundamental group \(\mathbb Z_{p^n}\) is a finite skeleton of the standard infinite lens space \(K\). Thus, provided its dimension is at least \(2k\), its suspension has \((2k+1)\)-skeleton equal to \[ (\Sigma K)^{(2k+1)}. \]
Recall the construction of (1). For a generator \(r\in\mathbb F_p^\times\), Proposition 4I.3 constructs maps \[ m_i:\Sigma K\to\Sigma K \] whose mapping telescopes are the \(X_i\)’s; on reduced homology, the \(i\)-th telescope retains exactly the dimensions congruent to \(2i\pmod{2p-2}\) and kills the others. The sum of the natural telescope maps gives a homology equivalence \[ F:\Sigma K\longrightarrow\bigvee_{i=1}^{p-1}X_i. \tag{2} \] By cellular approximation choose \(F\) cellular. Because the \(X_i\)’s have the minimal cell structures just described, (2) restricts to a cellular map \[ F^{(2k+1)}:(\Sigma K)^{(2k+1)}\longrightarrow \bigvee_{i=1}^{p-1}X_i^{(2k+1)}. \tag{3} \] The homology calculation in Proposition 4I.3 is degree-by-degree, so (3) induces an isomorphism on homology in every degree \(\le2k+1\). Above degree \(2k+1\), both sides have zero homology because they are \((2k+1)\)-dimensional CW complexes. Hence (3) is a homology isomorphism in all degrees.
Both sides of (3) are simply connected: they are skeleta of suspensions, with no \(1\)-cells in the minimal models. Therefore the homology Whitehead theorem implies that (3) is a homotopy equivalence. Identifying the source with the corresponding skeleton of the suspension of the chosen high-dimensional lens space gives \[ \boxed{ (\Sigma L)^{(2k+1)}\simeq \bigvee_{i=1}^{p-1}X_i^{(2k+1)}.} \] This is precisely the finite-skeletal extension of Proposition 4I.3.