A new algorithm for 3-sphere recognition

2016/10/13 by Michael Heusener, Heusener, Michael, Raphael Zentner +1
Computer Science · Mathematics · #57M27 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR #math.GT #msc:57M27

paper · pdf · doi:10.48550/arxiv.1610.04092

8 pages, comments are welcome! v2: misspellings corrected

openalex publication_date 2016/10/13 · arxiv created 2016/10/14 · arxiv updated 2016/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of a new algorithm for 3-sphere recognition based on Groebner basis methods applied to the variety of \em SL(2,\C)-representation of the fundamental group. An essential input is a recent result of the second author, stating that any integer homology 3-sphere different from the 3-sphere admits an irreducible representation of its fundamental group in \em SL(2,\C). This result, and hence our algorithm, build on the geometrisation theorem of 3-manifolds.

Citations

• Recognizing geometric 3-manifold groups using the word problem

Related

• SL(2;R)-representations of a Brieskorn homology 3-sphere
• Fundamental classes of 3-manifold group representations in SL(4,R)
• Toroidal homology spheres and SU(2)-representations
• Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity
• Some numerical computations on Reidemeister torsion for homology 3-spheres obtained by Dehn surgeries along the figure-eight knot