Toroidal homology spheres and SU(2)-representations

We prove that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)representations. Our methods use instanton Floer homology, and in particular the surgery exact triangle, holonomy perturbations, and a non-vanishing result due to Kronheimer-Mrowka, as well as results about surgeries on cables due to Gordon.

[1]  Y. Eliashberg.  TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2 , 1990 .

[2]  P. Kronheimer,et al.  Witten's conjecture and Property P , 2003, math/0311489.

[3]  P. B. Kronheimer,et al.  Dehn surgery, the fundamental group and SU(2) , 2003 .

[4]  Kimihiko Motegi.  Haken manifolds and representations of their fundamental groups in SL(2, C) , 1988 .

[5]  Andrew J. Nicas,et al.  Varieties of group representations and Casson’s invariant for rational homology 3-spheres , 1990 .

[6]  Raphael Zentner,et al.  Integer homology 3-spheres admit irreducible representations in SL(2,C) , 2016, Duke Mathematical Journal.

[7]  A. Weil.  Remarks on the cohomology of groups , 2016 .

[8]  John Hempel,et al.  RESIDUAL FINITENESS FOR 3-MANIFOLDS , 1987 .

[9]  A. Floer.  Instanton homology and Dehn surgery , 1995 .

[10]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[11]  P. Kronheimer,et al.  Knot homology groups from instantons , 2008, 0806.1053.

[12]  P. Kronheimer,et al.  Knots, sutures, and excision , 2008, 0807.4891.

[13]  John A. Baldwin,et al.  Stein fillings and SU(2) representations , 2016, Geometry & Topology.

[14]  B. Tosun,et al.  MAZUR-TYPE MANIFOLDS WITH L –SPACE BOUNDARY , 2018 .

[15]  M. Boileau,et al.  Graph manifold ℤ‐homology 3‐spheres and taut foliations , 2013, 1303.5264.

[16]  R. Stern,et al.  Instanton Homology of Seifert Fibred Homology Three Spheres , 1990 .

[17]  H. Seifert.  ON THE HOMOLOGY INVARIANTS OF KNOTS , 1950 .

[18]  Yi Ni.  Nonseparating spheres and twisted Heegaard Floer homology , 2009, 0902.4034.

[19]  A. Lubotzky,et al.  Varieties of Representations of Finitely Generated Groups , 1985 .

[20]  A Menagerie of SU(2)-Cyclic 3-Manifolds , 2019, 1910.13270.

[21]  ifn.  DEHN SURGERY AND SATELLITE KNOTS , 2009 .

[22]  M. Culler.  Lifting Representations to Covering Groups , 1986 .

[23]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[24]  S. Donaldson,et al.  Floer’s work on instanton homology, knots and surgery , 1995 .

[25]  C. Livingston,et al.  Abelian invariants of satellite knots , 1985 .

[26]  A. Floer,et al.  An instanton-invariant for 3-manifolds , 1988 .

[27]  Eaman Eftekhary.  Bordered Floer homology and existence of incompressible tori in homology spheres , 2008, Compositio Mathematica.

[28]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[29]  Edited Rob Kirby,et al.  Problems in Low-Dimensional Topology , 1995 .

[30]  F. Bonahon,et al.  The characteristic toric splitting of irreducible compact 3-orbifolds , 1987 .

[31]  S. Yau,et al.  Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces , 1984 .

[32]  S. Donaldson,et al.  Floer Homology Groups in Yang-Mills Theory , 2002 .

[33]  Christopher R. Cornwell,et al.  Obstructions to Lagrangian concordance , 2014, 1411.1364.

[34]  Raphael Zentner.  A class of knots with simple SU(2)-representations , 2015, 1501.02504.