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On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds (2406.01148v1)
Published 3 Jun 2024 in math.SG, math.DG, and math.GT
Abstract: We study the global topology of the space $\mathcal L$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M{2n+1}, \alpha)$. We filter $\mathcal L$ by a quantitative measure of the ``positivity'' of the loops and describe the topology of $\mathcal L$ in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of $\mathcal L$ are subgroups of the homotopy groups of the subspace of positive loops $\mathcal L+$. We obtain analogous results for the space of loops of Legendrian submanifolds in $(M{2n+1}, \alpha)$.