Restriction-lift conjecture for 13-manifolds with H*(CP^3 × S^7; Z)

Determine whether every simply connected, closed, smooth 13-dimensional manifold M whose cohomology ring is isomorphic to H*(CP^3 × S^7; Z) admits a restriction lift; specifically, ascertain whether there exists a map f: M → CP^n for some n such that the composition of f with the natural inclusion CP^n → CP^∞ is homotopic to the map x: M → CP^∞ corresponding to a chosen generator of H^2(M).

Background

The paper studies simply connected, closed, smooth 13-manifolds M whose integral cohomology ring is isomorphic to H*(CP^3 × S^7; Z). A central technical notion is a “restriction lift,” defined by the existence of a map f: M → CP^n whose composition with the inclusion CP^n → CP^∞ is homotopic to the map x: M → CP^∞ representing a generator of H^2(M).

The main classification results (Theorem 1.2) for nonspin manifolds rely on the existence of such restriction lifts. Many natural examples, including total spaces of S^7-bundles over CP^3, admit restriction lifts, and the author has not found a counterexample. Proving the conjecture that all such M admit restriction lifts would complete the nonspin classification via the main theorem.