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PUBLIC VIVA-VOCE 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. 29-06-21 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 non-orientable $3$--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 well-known theorem of V. Rohlin and C. T. C. Wall which states that all closed non-orientable $3$--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 4–manifold admits a simplified broken Lefschetz fibration. As an application of this theorem, we reprove the well-known 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 $\mathbb{C}P^3$ .
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