Realization of equivariant chain complexes. (Q1430400)

For an \(n\)-dimensional Poincaré duality space \(X\) over \(\mathbb{F}_p\), where \(p\) is an odd prime, \(\mathbb{Z}/p\) acts nicely if \(H^*(X,\mathbb{F}_p)\) as an \(\mathbb{F}_p[\mathbb{Z}/p]\) module is a direct sum of a free and a trivial \(\mathbb{F}_p[\mathbb{Z}/p]\) module. \textit{A.~Sikora} [Topology 43, No. 3, 725--748 (2004; Zbl 1060.55008)], asked whether for such a nice action on \(X\), the differential \(d_r:E_r^{i,j}\to E_r^{i+r,j-r+1}\) in the Leray Serre spectral sequence vanishes for odd \(r>1\) and \(i\geq n\), when the action has non-empty fixed point set. In the present paper, the author constructs a counterexample to this. First, an equivariant chain complex with the required properties is constructed. Then, this is realized \(p\)-locally as the cellular chain complex of a 10-dimensional \(\mathbb{Z}/p\)-CW-complex which is constructed inductively. This is then fattened up via an equivariant smooth thickening, and one takes the oriented double of that to get an oriented closed smooth \(\mathbb{Z}/p\)-manifold \(X\). For this manifold, the differential \(d_3:E_3^{i,*}\to E_3^{i+3,*-2}\) is nontrivial whenever \(i\) is even and \(i>2\). Moreover, the author proves that the answer to Sikoras question is positive when one adds the assumption that \(H^*(X,\mathbb{Z}_{(p)})\) does not contain \(\mathbb{Z}/p\) as a direct summand.