3 problems

Let $M$ be a $2$-connected $7$-manifold whose cohomology ring is that of an $\mathbf{S}^3$-bundle over $\mathbf{S}^4$. A non-negative curvature and Seifert fibration conjecture ass…

Let $N$ be a closed connected orientable $4$-manifold, let $f_0,f_1:N\to S^7$ be embeddings, and let $\varphi:\partial C_{f_0}\to\partial C_{f_1}$ be an orientation-preserving bund…

Let $N$ be a closed connected orientable $4$-manifold, let $f_0,f_1:N\to S^7$ be embeddings, and let $\varphi:\partial C_{f_0}\to\partial C_{f_1}$ be an orientation-preserving Spin…