Contact surgery numbers of projective spaces

Abstract

We classify all contact projective spaces with contact surgery number one. In particular, this implies that there exist infinitely many non-isotopic contact structures on the real projective 3-space which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. Large parts of our proofs deal with a detailed analysis of Gompf's $\Gamma$-invariant of tangential 2-plane fields on 3-manifolds. From our main result we also deduce that the $\Gamma$-invariant of a tangential 2-plane field on the real projective 3-space only depends on its $d_3$-invariant.

arXiv comment23 pages, 11 figures, 2 tables. V2: Minor corrections. To appear in Mediterranean Journal of Mathematics

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