Homotopy classification of twisted complex projective spaces of dimension 4 (Q2483795)

A twisted \(\mathbb{C}\text{P}^n\) is a simply-connected finite CW complex \(M\) which is a Poincaré space and whose homology satisfies \(H_*(M;\mathbb{Z}) \cong H_*(\mathbb{C}\text{P}^n;\mathbb{Z})\). Such a space necessarily has the homotopy type of a topological manifold, and is homotopic to a CW complex with cells in even dimensions. It is known that there are no nontrivial examples for \(n=1,2\), and the case \(n=3\) is well studied. The paper under review examines the case \(n=4\). The study of twisted \(\mathbb{C}\text{P}^4\)'s \(M\) begins by noting that (with suitable generators), the cohomology ring \(H^*(M;\mathbb{Z})\) is determined by a unique integer \(m\geq 0\) such that \[ x_2 x_2 = mx_4, \;x_2 x_4 = m x_6, \;\text{and} \;x_2 x_6 = x_4 x_4 = x_8. \] Such a complex \(M\) is called an \(m\)-twisted \(\mathbb{C}\text{P}^4\), and the set of all homotopy tyes of such complexes is denoted by \(\mathcal{M}_m\). The second step is to examine the \(6\)-skeleta of these \(m\)-twisted \(\mathbb{C}\text{P}^4\)'s; it is shown that there are three possiblities: \(M^{(6)}\) is one of the spaces denoted here by \(X_m\), \(Y_m\) and \(Z_m\). The final ingredient concerns the attaching map of the \(8\)-cell, and it is quantified by studying the Steenrod square \[ \text{Sq}^2: H^6(M;\mathbb{Z}/2) \to H^8(M;\mathbb{Z}/2). \] If this operation is trivial, then \(M\) has type \((X,0)\), \((Y,0)\) or \((Z,0)\), depending on the \(6\)-skeleton, and otherwise \(M\) has type \((X,1)\), \((Y,1)\) or \((Z,1)\). With this terminology, the main results state that (1) If \(m\) is odd, then there are \(m\)-twisted \(\mathbb{C}\text{P}^4\)'s, and they all have type \((X,1)\), and (2) If \(m\) is even but not divisible by \(8\), then there are no \(m\)-twisted \(\mathbb{C}\text{P}^m\)'s. The case \(m \equiv 0\) mod \(8\) is also worked out, but it is more complicated. It turns out that there are no twisted \(\mathbb{C}\text{P}^4\)s of type \((Z,0)\) or \((Z,1)\). Estimates are made of the cardinality of \(\mathcal{M}_m\).