Algebraic & Geometric Topology

The intersecting kernels of Heegaard splittings

Fengchun Lei and Jie Wu

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Abstract

Let VSW be a Heegaard splitting for a closed orientable 3–manifold M. The inclusion-induced homomorphisms π1(S)π1(V) and π1(S)π1(W) are both surjective. The paper is principally concerned with the kernels K= Ker(π1(S)π1(V)), L= Ker(π1(S)π1(W)), their intersection KL and the quotient (KL)[K,L]. The module (KL)[K,L] is of special interest because it is isomorphic to the second homotopy module π2(M). There are two main results.

(1)  We present an exact sequence of (π1(M))–modules of the form

( K L ) [ K , L ] R { x 1 , , x g } J T ϕ R { y 1 , , y g } θ R ϵ ,

where R=(π1(M)), J is a cyclic R–submodule of R{x1,,xg}, Tϕ and θ are explicitly described morphisms of R–modules and Tϕ involves Fox derivatives related to the gluing data of the Heegaard splitting M=VSW.

(2)  Let K be the intersection kernel for a Heegaard splitting of a connected sum, and K1, K2 the intersection kernels of the two summands. We show that there is a surjection KK1K2 onto the free product with kernel being normally generated by a single geometrically described element.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 887-908.

Dates
Received: 25 July 2010
Revised: 29 December 2010
Accepted: 12 January 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715212

Digital Object Identifier
doi:10.2140/agt.2011.11.887

Mathematical Reviews number (MathSciNet)
MR2782546

Zentralblatt MATH identifier
1215.57007

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57M99: None of the above, but in this section 20F38: Other groups related to topology or analysis
Secondary: 57M05: Fundamental group, presentations, free differential calculus 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

Keywords
Heegaard splitting intersecting kernel $3$–manifold mapping class Riemann surface

Citation

Lei, Fengchun; Wu, Jie. The intersecting kernels of Heegaard splittings. Algebr. Geom. Topol. 11 (2011), no. 2, 887--908. doi:10.2140/agt.2011.11.887. https://projecteuclid.org/euclid.agt/1513715212


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References

  • A J Berrick, F R Cohen, Y L Wong, J Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006) 265–326
  • J S Birman, On the equivalence of Heegaard splittings of closed, orientable $3$–manifolds, from: “Knots, groups, and $3$–manifolds (Papers dedicated to the memory of R H Fox)”, (L P Neuwirth, editor), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 137–164
  • J S Birman, The topology of $3$–manifolds, Heegaard distance and the mapping class group of a $2$–manifold, from: “Problems on mapping class groups and related topics”, (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 133–149
  • W A Bogley, J H C Whitehead's asphericity question, from: “Two-dimensional homotopy and combinatorial group theory”, (C Hog-Angeloni, W Metzler, A J Sieradski, editors), London Math. Soc. Lecture Note Ser. 197, Cambridge Univ. Press (1993) 309–334
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
  • R Brown, J-L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311–335 With an appendix by M Zisman
  • F R Cohen, J Wu, On braid groups, free groups, and the loop space of the $2$–sphere, from: “Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001)”, (G Arone, J Hubbuck, R Levi, M Weiss, editors), Progr. Math. 215, Birkhäuser, Basel (2004) 93–105
  • F R Cohen, J Wu, On braid groups and homotopy groups, from: “Groups, homotopy and configuration spaces”, (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13, Geom. Topol. Publ., Coventry (2008) 169–193
  • J Hempel, $3$-Manifolds, Ann. of Math. Studies 86, Princeton Univ. Press (1976)
  • W Jaco, Heegaard splittings and splitting homomorphisms, Trans. Amer. Math. Soc. 144 (1969) 365–379
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Ser. in Math. 43, Amer. Math. Soc. (1980)
  • F Lei, Haken spheres in the connected sum of two lens spaces, Math. Proc. Cambridge Philos. Soc. 138 (2005) 97–105
  • J Li, J Wu, Artin braid groups and homotopy groups, Proc. Lond. Math. Soc. $(3)$ 99 (2009) 521–556
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Math. und ihrer Grenzgebiete 89, Springer, Berlin (1977)
  • J Milnor, A unique decomposition theorem for $3$–manifolds, Amer. J. Math. 84 (1962) 1–7
  • J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, Amer. Math. Soc. (2007)
  • C D Papakyriakopoulos, A reduction of the Poincaré conjecture to group theoretic conjectures, Ann. of Math. $(2)$ 77 (1963) 250–305
  • M Scharlemann, Heegaard splittings of compact $3$–manifolds, from: “Handbook of geometric topology”, (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 921–953
  • J Stallings, How not to prove the Poincaré conjecture, Ann. of Math. Study 60 (1966) 83–88
  • J Wu, Combinatorial descriptions of homotopy groups of certain spaces, Math. Proc. Cambridge Philos. Soc. 130 (2001) 489–513