The tangential end fibration of an aspherical Poincaré complex (Q998017)

Suppose that \(X\) is a finite aspherical Poincaré complex with universal cover that is forward tame (meaning roughly that each co-compact subset is collared) and is simply-connected at infinity so that it only has one end. This paper gives an explicit construction of a spherical bundle over \(X\) that is shown to have the properties that one would require of a tangent bundle (inverse to the Spivak bundle and of the appropriate rank), in the case that either the formal dimension of \(X\) is even, or in case the formal dimension is odd that the diagonal \(X \rightarrow X \times X\) admits a Poincaré embedding. The proofs are homotopy theoretic and make use of explicit chain complexes.