arXiv subjects
Publications and source records attributed to Dishant Pancholi.
Following the overall strategy of the paper ``Convex hypersurfaces in contact topology" by Ko Honda and Yang Huang on contact convexity in high dimensions, we present a simplified proof of their main result.
We construct positive loops of Legendrian submanifolds in several instances. In particular, we partially recover G. Liu's result stating that any loose Legendrian admits a positive loop, under some mild topological assumptions on the Legendrian. Moreover, we show contractibility of the constructed loops under an extra topological assumption.
Given a group $G$ and a class of manifolds $\CC$ (e.g. symplectic, contact, Kähler etc), it is an old problem to find a manifold $M_G \in \CC$ whose fundamental group is $G$. This article refines it: for a group $G$ and a positive integer $r$ find $M_G \in \CC$ such that $π_1(M_G)=G$ and $π_i(M_G)=0$ for $1<i<r$. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest $r$ for which such an $M_G \in \CC$ can be found is called the homotopical height $ht_\CC(G)$. Homotopical height provides a dimensional obstruction to finding a $K(G,1)$ space within the given class $\CC$, leading to a hierarchy of these classes in terms of "softness" or "hardness" à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes $\SP$ and $\CA$ of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group $G$ can be realized as the fundamental group of a manifold in $\SP$ and a manifold in $\CA$. For these classes, $ht_\CC(G)$ provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.
We show that, for a closed non-orientable surface $F$, an automorphism of $H_1(F,\Z)$ is induced by a homeomorphism of $F$ if and only if it preserves the (mod 2) intersection pairing. We shall also prove the corresponding result for punctured surfaces.