Monopoles and lens space surgeries

By P. Kronheimer, T. Mrowka, Peter S. Ozsv'ath, Zolt'an Szab'o. 2003

301 citations recorded by the source corpus.

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot. In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff966036d82c731ee02898ccfefed48183e5ef08

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Grid Homology for Knots and Links (2015, background): “It can also be used to give obstructions on knots admitting lens space surgeries [80, 177, 192]; see also [114].”
Knot homology groups from instantons (2008, methodology): “It seems that a proof of such a long exact sequence can be given using the same ideas that were used in [18] to prove a surgery exact sequence for Seiberg-Witten Floer homology, and the authors hope to return to this and other related issues in a future paper.”
Lens space surgeries and a conjecture of Goda and Teragaito (2004, background): “AMS Classification numbers Primary: 57M25 Secondary: 57R58 Keywords: Lens space surgery, Seifert genus, Heegaard Floer homology Proposed: Peter Ozsvath Received: 13 May 2004 Seconded: Tomasz Mrowka, Peter Kronheimer Accepted: 11 July 2004 c© Geometry & Topology Publications”
Instantons and odd Khovanov homology (2014, methodology): “The use of metric stretching maps in this context was applied in [22] by Kronheimer, Mrowka, and the previous two authors to prove an exact triangle in monopole Floer homology.”
Knot Floer homology and rational surgeries (2005, methodology): “The above result, which was conjectured by Gordon in [8], was first established using Seiberg–Witten monopoles by Kronheimer, Mrowka and the authors [10].”
Involutive Heegaard Floer homology (2015, methodology): “The maps α, β, γ are analogous to the Frøyshov-type correction terms arising from monopole or Heegaard Floer homology [6, 13, 35], but have the additional property that their reduction mod 2 is equal to the Rokhlin invariant.”
Knots with unknotting number one and Heegaard Floer homology (2004, background): “his establishes the second inequality. The ﬁrst is proved along the same lines. The following result about L-spaces will be useful to us, as well. A more general result is established in Section 7 of [12] in the context of the Seiberg-Witten monopole Floer homology (though the proof adapts readily t”
Lectures on open book decompositions and contact structures (2004, background): “Szabó in [34] gave an alternate proof of a characterization of the unknot via surgery which was originally established in [29].”
L-space surgeries, genus bounds, and the cabling conjecture (2010, methodology): “We begin with the method introduced and carried out in [21, 25], which uses a version of Theorem 2.”
Link Floer homology and the Thurston norm (2006, result): “Monopole Floer homology [12] detects the Thurston norm of a closed three-manifold, according to a fundamental result of Kronheimer and Mrowka [13], see also [14], building on results of Gabai [9] and Eliashberg and Thurston [5].”
Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture (2013, methodology): “Correction terms were defined first by Frøyshov in instanton (Yang-Mills) Floer homology [19], then by Ozsváth-Szabó in Heegaard Floer homology [44], and by Frøyshov and Kronheimer-Mrowka in monopole (Seiberg-Witten) Floer homology [20,32,33].”
Bordered Heegaard Floer homology (2008, background): “(The result was rst proved in [32], but it has its origins in Floer's theory of instanton homology, see [9], compare also [19] for a Seiberg-Witten analogue.”
Lens space surgeries and L-space homology spheres (2007, methodology): “s proved by Hedden in [17], using a diﬀerent method. Also, the second half of the theorem was originally proved (in the context of monopole Floer homology) by Kronheimer, Mrowka, Ozsv´ath, and Szabo´ [18]. If K ⊂ S3 is a knot with a lens space surgery L(p,q), there is a dual knot Ke ⊂ L(p,q) which a”