Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dual decompositions of 4-manifolds

Author: Frank Quinn
Journal: Trans. Amer. Math. Soc. 354 (2002), 1373-1392
MSC (2000): Primary 57R65, 57M20
Published electronically: November 8, 2001
MathSciNet review: 1873010
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index $\leq 2$. In dimensions $\geq 5$results of Smale (trivial $\pi _{1}$) and Wall (general $\pi _{1}$) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( $H_{1}K\simeq H^{2}L$ and $H_{2}K\simeq H^{1}L$). (3.3) If $(N,\partial N)$ is 1-connected then there is a ``pseudo'' handle decomposition without 1-handles, in the sense that there is a pseudo collar $(M,\partial N)$ (a relative 2-handlebody with spine that 2-deforms to $\partial N$) and $N$ is obtained from this by attaching handles of index $\geq 2$.

References [Enhancements On Off] (What's this?)

  • [AM] S. Akbulut and R. Matveyev, A convex decomposition theorem for $4$-manifolds, Internat. Math. Res. Notices 7 (1998), 371-381. MR 99f:57030
  • [AC] J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192-195. MR 30:3454
  • [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Science Publications. Oxford University Press, New York, 1990. MR 92a:57036
  • [FKT] M. H. Freedman, V. S. Krushkal and P. Teichner, Van Kampen's embedding obstruction is incomplete for 2-complexes in $R^{4}$, Math. Res. Lett. 1 (1994), 167-176. MR 95c:57003
  • [FQ] M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton University Press, 1990.
  • [GS] R. E. Gompf and A. I. Stipsicz, $4$-manifolds and Kirby calculus., Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 2000h:57038
  • [H] G. Huck, Embeddings of acyclic $2$-complexes in $S^{4}$with contractible complement, Springer Lecture Notes in Math. 1440 (1990), 122-129. MR 92a:57002
  • [K1] V. S. Krushkal, Embedding obstructions and 4-dimensional thickenings of 2-complexes, Proc. Amer. Math Soc. 128 (2000), 3683-3691. MR 2001b:57006
  • [KT] V. S. Krushkal and P. Teichner, Alexander duality, gropes and link homotopy, Geometry & Topology 1 (1997), 51-69. MR 98i:57013
  • [Q] F. Quinn, Handlebodies and 2-complexes, Springer Lecture Notes in Math. 1167 (1985), 245-259. MR 88a:57008
  • [W1] C. T. C. Wall, Geometrical Connectivity, J. London Math. Soc 3 (1971), 597-608. MR 44:7569a
  • [W2] -, Formal deformations, Proc. London Math. Soc 16 (1966), 342-352. MR 33:1851
  • [W3] -, Finiteness conditions for CW-complexes, Ann. Math. 81 (1965), 56-69. MR 30:1515
  • [W4] -, Finiteness conditions for CW-complexes II, Proc. Roy. Soc. 295 (1966), 129-139. MR 35:2283

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R65, 57M20

Retrieve articles in all journals with MSC (2000): 57R65, 57M20

Additional Information

Frank Quinn
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123

Received by editor(s): October 2, 2000
Received by editor(s) in revised form: August 4, 2001
Published electronically: November 8, 2001
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society