Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group theory, nor methods used to prove geometrization. The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials. This result relies on normal surface theory, Mostow rigidity, and bounds on the computational complexity of solving algebraic equations.

[1]  M. Boileau,et al.  Non-zero degree maps and surface bundles over $S\sp 1$ , 1996 .

[2]  Klaus Johannson,et al.  Homotopy Equivalences of 3-Manifolds with Boundaries , 1979 .

[3]  A. Seidenberg A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA , 1954 .

[4]  J. W. Neuberger The Continuous Newton's Method, Inverse Functions, and Nash-Moser , 2007, Am. Math. Mon..

[5]  Udo Pachner,et al.  P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst , 1991, Eur. J. Comb..

[6]  W. Thurston,et al.  The Computational Complexity of Knot Genus and Spanning Area , 2002, math/0205057.

[7]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[8]  W. Thurston The geometry and topology of three-manifolds , 1979 .

[9]  W. Haken,et al.  Ein Verfahren zur Aufspaltung einer 3-Mannigfaltigkeit in irreduzible 3-Mannigfaltigkeiten , 1961 .

[10]  David Gabai,et al.  MOM TECHNOLOGY AND VOLUMES OF HYPERBOLIC 3-MANIFOLDS , 2006, math/0606072.

[11]  Matthias Aschenbrenner,et al.  Decision problems for 3-manifolds and their fundamental groups , 2014, 1405.6274.

[12]  Ivan Izmestiev Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional , 2014, Canadian Journal of Mathematics.

[13]  W. B. R. Lickorish Simplicial moves on complexes and manifolds , 1999 .

[14]  H. Short,et al.  The homeomorphism problem for closed 3–manifolds , 2012, 1211.0264.

[15]  John W. Morgan,et al.  Ricci Flow and Geometrization of 3-Manifolds , 2010 .

[16]  Z. Sela,et al.  The isomorphism problem for hyperbolic groups I , 1995 .

[17]  R. Edwards Suspensions of homology spheres , 2006, math/0610573.

[18]  Saul Schleimer,et al.  SPHERE RECOGNITION LIES IN NP , 2004, math/0407047.

[19]  B. Poonen UNDECIDABLE PROBLEMS: A SAMPLER , 2012, 1204.0299.

[20]  David Letscher,et al.  Algorithms for essential surfaces in 3-manifolds , 2002 .

[21]  Don Zagier,et al.  Volumes of hyperbolic three-manifolds , 1985 .

[22]  R. Kirby,et al.  On the triangulation of manifolds and the Hauptvermutung , 1969 .

[23]  J. Manning,et al.  Algorithmic detection and description of hyperbolic structures on closed 3{manifolds with solvable word problem , 2002 .

[24]  J. Rubinstein,et al.  An Algorithm to Recognize the 3-Sphere , 1995 .

[25]  William Jaco,et al.  Seifert fibered spaces in 3-manifolds , 1979 .

[26]  William Jaco,et al.  Algorithms for the complete decomposition of a closed $3$-manifold , 1995 .

[27]  Shmuel Weinberger,et al.  Algorithmic aspects of homeomorphism problems , 1997 .

[28]  H. Seifert,et al.  Topologie Dreidimensionaler Gefaserter Räume , 1933 .

[29]  Li Tao AN ALGORITHM TO FIND VERTICAL TORI IN SMALL SEIFERT FIBER SPACES , 2002 .