## Algebraic & Geometric Topology

### Reidemeister torsion of Seifert fibered homology lens spaces and Dehn surgery

#### Abstract

We provide necessary conditions on the Alexander polynomial of a knot $K$ in a homology sphere and on surgery coefficients $p∕q$ for the surgered manifold to be a Seifert fibered space over $S2$. As an application, we show that no $p∕q$–surgery with $p>3$ on a knot in a homology sphere with the same Alexander polynomial as the figure eight knot can produce a Seifert fibered space with base $S2$. The main tool is the abelian Reidemeister torsion.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 3 (2007), 1509-1529.

Dates
Revised: 20 September 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796751

Digital Object Identifier
doi:10.2140/agt.2007.7.1509

Mathematical Reviews number (MathSciNet)
MR2366168

Zentralblatt MATH identifier
1145.57004

#### Citation

Kadokami, Teruhisa. Reidemeister torsion of Seifert fibered homology lens spaces and Dehn surgery. Algebr. Geom. Topol. 7 (2007), no. 3, 1509--1529. doi:10.2140/agt.2007.7.1509. https://projecteuclid.org/euclid.agt/1513796751

#### References

• S Boyer, D Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990) 181–220
• M Brittenham, Y-Q Wu, The classification of exceptional Dehn surgeries on 2-bridge knots, Comm. Anal. Geom. 9 (2001) 97–113
• M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. $(2)$ 125 (1987) 237–300
• G de Rham, Complexes à automorphismes et homéomorphie différentiable, Ann. Inst. Fourier Grenoble 2 (1950) 51–67 (1951)
• R Fintushel, R J Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980) 33–51
• T Kadokami, Reidemeister torsion and lens surgeries on knots in homology spheres II, preprint
• T Kadokami, Reidemeister torsion and lens surgeries on knots in homology 3-spheres. I, Osaka J. Math. 43 (2006) 823–837
• T Kadokami, Y Yamada, Reidemeister torsion and lens surgeries on $(-2,m,n)$-pretzel knots, Kobe J. Math. 23 (2006) 65–78
• T Kadokami, Y Yamada, A deformation of the Alexander polynomials of knots yielding lens spaces, Bull. Austral. Math. Soc. 75 (2007) 75–89
• K Miyazaki, K Motegi, Seifert fibred manifolds and Dehn surgery, Topology 36 (1997) 579–603
• L Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 737–745
• A Némethi, L I Nicolaescu, Seiberg-Witten invariants and surface singularities, Geom. Topol. 6 (2002) 269–328
• L I Nicolaescu, The Reidemeister torsion of 3-manifolds, de Gruyter Studies in Mathematics 30, Walter de Gruyter & Co., Berlin (2003)
• P Orlik, Seifert manifolds, Lecture Notes in Mathematics 291, Springer, Berlin (1972)
• T Sakai, Reidemeister torsion of a homology lens space, Kobe J. Math. 1 (1984) 47–50
• N Saveliev, Invariants for homology $3$-spheres, Encyclopaedia of Mathematical Sciences 140, Springer, Berlin (2002)
• W P Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ. Math. Dept. Lecture Notes, Electronic Version 1.1 (2002) Available at \setbox0\makeatletter\@url www.msri.org/publications/books/gt3m/ \unhbox0
• V G Turaev, Reidemeister torsion in knot theory, Uspekhi Mat. Nauk 41 (1986) 97–147, 240
• V Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2001) Notes taken by Felix Schlenk