Classification of homotopy Dold manifolds (Q1419676)

For \(r,s > 1\), the Dold manifold \(P(r,s)\) is a quotient of \(S^r \times \mathbb{C}P^s\) obtained by identifying \((x,y)\sim (x',y')\) iff \(x'=-x\) and \(y'=\bar{y}\). Thus, \(P(r,s)\) fibers over \(\mathbb{R}P^r\) with fiber \(\mathbb{C}P^s\). In this paper all manifolds homotopy equivalent to \(P(r,s)\) are classified in the piecewise linear and topological categories from the point of view of surgery theory. That is, the groups and maps in the surgery exact sequence are calculated. Moreover, the same is done for products of \(P(r,s)\) with disks of all dimensions. Partial results are obtained in the smooth category.