Equivariant homology for generating functions and orderability of lens spaces

Abstract

International audienceIn her PhD thesis [M08] Milin developed a Z k-equivariant version of the contact homology groups constructed in [EKP06] and used it to prove a Z k-equivariant contact non-squeezing theorem. In this article, we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in [S11]. As Milin showed, this result implies orderability of lens spaces

Quasimorphisms on contactomorphism groups and contact rigidity

“…Cosphere bundles of closed manifolds are known to be orderable [AF13, CN10, EKP06, EP00] and more generally Albers-Merry proved in [AM13b] that Liouville-fillable contact manifolds with nonvanishing Rabinowitz Floer homology are orderable. Using the connection between orderability and contact squeezing developed by Eliashberg-Kim-Polterovich [EKP06], Milin [Mil08] and Sandon [San11b] proved that lens spaces are orderable.…”

Givental’s Non-linear Maslov Index on Lens Spaces

“…Orderability of lens spaces was also proved with different methods by Milin [Mi08] and by the fourth author [Sa11b]. Regarding part (iii), this proves for the standard contact form of lens spaces the non-degenerate and cup-length variants of the following conjecture: if (V, ξ) is a compact contact manifold then any contactomorphism φ contact isotopic to the identity should have at least as many translated points (with respect to any contact form for ξ) as the minimal number of critical points of a smooth function on V .…”

Vanishing of Rabinowitz Floer homology on negative line bundles

“…Negative line bundles give rise to a rather special class of contact manifolds which nevertheless contains many interesting examples. They arise at many places in modern contact and symplectic geometry such as Givental's nonlinear Maslov index [Giv90] and more generally contact rigidity [EP00,San11,Bor13,BZ15] etc.…”

“…(e) Eliashberg and Polterovich [EP00] introduced the concept of orderability of contact manifolds and studied contact rigidity phenomena, see also [EKP06]. Examples of orderable contact manifolds include RP 2n+1 in [EP00], R 2n+1 in [Bhu01], lens spaces in [Mil08,San11,GKPS21], certain contact manifolds obtained as contact reduction of RP 2n+1 in [BZ15, Zap20], 1-jet bundles in [CN10a,CFP17], and cosphere bundles in [EKP06,CN10b,AF12,CFP17]. Some of these results rely on Givental's non-linear Maslov index on RP 2n+1 appeared in [Giv90] and constructed a quasi-morphism on Cont 0 in respective settings.…”