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Article overview
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Contact surgery numbers | | John Etnyre
; Marc Kegel
; Sinem Onaran
; | | Date: |
1 Jan 2022 | | Abstract: | It is known that any contact 3-manifold can be obtained by rationally contact
Dehn surgery along a Legendrian link $L$ in the standard tight contact
3-sphere. We define and study various versions of contact surgery numbers, the
minimal number of components of a surgery link L describing a given contact
3-manifold under consideration.
In the first part of the paper, we relate contact surgery numbers to other
invariants in terms of various inequalities. In particular, we show that the
contact surgery number of a contact manifold is bounded from above by the
topological surgery number of the underlying topological manifold plus three.
In the second part, we compute contact surgery numbers of all contact
structures on the 3-sphere. Moreover, we completely classify the contact
structures with contact surgery number one on $S^1 imes S^2$, the Poincar’e
homology sphere and the Brieskorn sphere $Sigma(2,3,7)$. We conclude that
there exist infinitely many non-isotopic contact structures on each of the
above manifolds which cannot be obtained by a single rational contact surgery
from the standard tight contact 3-sphere. We further obtain results for the
3-torus and lens spaces.
As one ingredient of the proofs of the above results we generalize
computations of the homotopical invariants of contact structures to contact
surgeries with more general surgery coefficients which might be of independent
interest. | | Source: | arXiv, 2201.00157 | | Services: | Forum | Review | PDF | Favorites | |
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