Poincaré duality embeddings and fiberwise homotopy theory (Q1292693)

A map \(f:K\to X\) from a finite complex \(K\) into an \(n\)-dimensional Poincaré duality space \(X\) is said to Poincaré embed if \(f\) extends to a homotopy equivalence \(K\cup_A C\cong X\) such that each piece of the decomposition satisfies Poincaré duality. This means that \((K,A)\) and \((C,A)\) are Poincaré pairs having the same dimension. Moreover, the fundamental class in each case is included from the fundamental class of \(X\). Thus, to specify the Poincaré embedding we have to find the complement \(C\) and the way in which it is glued to \(K\) to give \(X\). For such a decomposition of \(X\) to exist it is necessary that the homology of \(K\) vanishes in degrees greater than \(n\). We write \(\text{hodim} K\leq k\) if \(K\) is homotopy equivalent to a CW complex of dimension \(\leq k\). The author works with the codimension greater than 3 and \(k \leq n-3\). The author poses a question: Given a map \(f:\kappa\to X\), when does it Poincaré embed? He gives a partial answer by proving that if \(f:\kappa\to X\) is \(r\)-connected, then \(f\) Poincaré embeds provided that \(k\leq n\) and \(r\geq 2k-n+2\).