PUBLIC VIVAVOCE NOTIFICATION Tuesday, 29 June 2021, 4.15 pm Flexible surfaces in $4$manifolds and embeddings of low dimensional manifolds Ghanwat Abhijeet Atmaram Chennai Mathematical Institute. 290621 Abstract In this thesis, we study smooth embeddings of closed $3$manifolds and orientable $4$manifolds. First, we produce flexible embedding of every compact surface in various $4$manifolds(concretely, disk bundles over $\mathbb{S}^ 2$ ). In the case of closed orientable surfaces, we also establish that these embeddings are in a standard position. Then we use these flexible embeddings to show that every closed nonorientable $3$manifold admits an open book embedding into the $5$manifolds $\mathbb{S}^ 2 \times \mathbb{S}^ 3$ and $\matrhbb{S}^2 \widetilde{\times} \mathbb{S}^3$. The open book decompositions of $\mathbb{S}^ 2 \times \mathbb{S}^ 3$ and $\matrhbb{S}^2 \widetilde{\times} \mathbb{S}^3$ have disk bundles over $\mathbb{S}^ 2 $ as pages and the identity as monodromy. We use these open book embedding techniques and give new proof of the wellknown theorem of V. Rohlin and C. T. C. Wall which states that all closed nonorientable $3$manifolds embed in $\mathbb{S}^ 5$ . Finally, we show that every closed orientable $4$manifold embeds in a $6$manifold of the form $N \times \mathbb{C}P^1$ , where $N$ is $4$manifold which contains a $4$ball $\mathbb{D}^2 \times\mathbb{D}^2$ such that $\partial \mathbb{D}^2 \times \lbrace 0 \rbrace cup \lbrace 0 \rbrace times \partial \mathbb{D}^2$ bounds two disjoint discs in the complement of $(\mathbb{D}^2 \times\mathbb{D}^2 ) ^\circ$ . In the proof, we crucially use the flexible embeddings of orientable surfaces introduced earlier, together with the fact every orientable 4â€“manifold admits a simplified broken Lefschetz fibration. As an application of this theorem, we reprove the wellknown theorem of M. Hirsch which states that every $4$manifold embeds in $\mathbb{R}^7$ . Then we show that every $4$manifold admits an embedding into $\mathbb{C}P^3$ .
