Classification for smooth manifolds looking like $\mathbb{CP}^3\times S^7$
Wen Shen
TL;DR
The paper achieves a comprehensive classification of simply connected, closed smooth $13$-manifolds with cohomology ring $H^*(\mathbb{CP}^3\times S^7)$, showing that the homeomorphism type is determined by the first Pontrjagin class $p_1$ and that, under various arithmetic conditions on $p_1$, one can refine to diffeomorphism or homotopy equivalence classes — with the potential involvement of an exotic $13$-sphere in the diffeomorphism type. The authors develop a normal $BO\langle 8\rangle$-structure framework, place the manifolds into bordism classes $\Omega_{13}^{O\langle 8\rangle}(\xi)$, and apply the Atiyah–Hirzebruch spectral sequence to extract a filtration where the main invariant sits at $F^{4,9}$. They extend the analysis to the PL category and use surgery arguments to obtain precise equivalence results, culminating in a classification of diagonal $S^1$-actions on $S^7\times S^7$ via explicit invariants such as $p_1$ and $w_2$, with explicit formulas for the action quotients and associated total spaces. Additionally, curvature consequences are established: if $p_1(M)\ge 4$, then under certain conditions either $M$ or $M\#\Sigma^{13}$ admits a metric with nonnegative sectional curvature, tying topology to geometric realizability. Overall, the work provides a rigorous bridge between high-dimensional manifold topology, bordism theory, and geometric applications, including a concrete classification of certain $S^1$-quotients and their smooth structures.
Abstract
In this paper, we classify simply connected closed smooth $13$-dimensional manifolds whose cohomology ring is isomorphic to that of $\mb{CP}^3\times S^7$, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a manifold satisfies certain conditions, either itself or its connected sum with an exotic $13$-sphere $Σ^{13}$ admits a Riemannian metric of non-negative sectional curvature. As an additional application of our classification, we classify the diagonal $S^1$-actions on $S^7\times S^7$.