aa r X i v : . [ m a t h . G T ] F e b COSMETIC SURGERIES AND THE POINCAR´E HOMOLOGY SPHERE
TYE LIDMAN
Abstract.
In this short note, we prove that if a knot in the Poincar´e homology sphere is homotopi-cally essential, then it does not admit any purely cosmetic surgeries.
Let K be a knot in a three-manifold Y . Recall that a knot K admits purely cosmetic surgeries if two different Dehn surgeries on K result in orientation-preserving homeomorphic manifolds. De-termining when knots have such surgeries is an interesting problem in low-dimensional topology, asit subsumes the knot complement problem and the nugatory crossing conjecture. Note that the un-knot in Y admits purely cosmetic surgeries. However, the Cosmetic Surgery Conjecture (see [Gor91,Conjecture 6.1]) asserts that this is the only such knot: Conjecture.
Let Y be a closed, oriented three-manifold and K a knot whose exterior is boundary-irreducible. If there exist two surgery slopes for K which produce orientation-preserving homeomor-phic manifolds, then there is a homeomorphism of the exterior sending one slope to the other. This problem is an effective testing ground for various Dehn surgery techniques. Consequently,many restrictions have been established recently using tools from hyperbolic geometry, curve com-plexes, character varieties, quantum invariants, and Heegaard Floer homology. For example, buildingon work of Ozsv´ath-Szab´o [OS11] and J. Wang [Wan06], Z. Wu showed that if p/q and p ′ /q ′ arepurely cosmetic non-trivial surgeries on a non-trivial knot in S , then p/q and p ′ /q ′ have oppositesign [Wu11]. This was extended by Ni and Wu [NW15] to show that p/q = − p ′ /q ′ , and gave severalother constraints on the surgery coefficients and the knot. We will not use this strengthening, butwill use the fact that Wu’s original results apply more generally: they hold for non-trivial knots inany integer homology sphere L-space. One example of such a 3-manifold is the Poincar´e homologysphere (and conjecturally all other examples are connected sums thereof). Using this result, we willdeduce that “most” knots in the Poincar´e homology sphere cannot admit purely cosmetic surgeries. Theorem.
Let K be a knot in the Poincar´e homology sphere which is not nullhomotopic. Then K does not admit any cosmetic surgeries. The rough strategy is as follows. We use the cosmetic surgeries to build a positive definite cobor-dism from the Poincar´e homology sphere to itself. By choosing K to be homotopically essential, theresulting cobordism has no non-trivial SU (2) representations. Such a four-manifold cannot exist byTaubes’s periodic ends theorem [Tau87]. We now give the proof. Proof.
Without loss of generality, we orient the Poincar´e homology sphere as the boundary of thepositive-definite E Y . First, suppose that −∞
The above arguments can be applied to give some more general results about knots in arbi-trary homology spheres. For example, let Z be a homology sphere that bounds a simply-connected,non-standard, definite four-manifold. If K normally generates π ( Z ), then a positive surgery on K cannot be homeomorphic to negative surgery on any knot in Z . Additionally, K is determined byits complement. References [Auc97] David Auckly. Surgery numbers of 3-manifolds: a hyperbolic example. In
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Department of Mathematics, North Carolina State University, Raleigh, NC 27607
E-mail address : [email protected]