Dual decompositions of 4manifolds
Author:
Frank Quinn
Journal:
Trans. Amer. Math. Soc. 354 (2002), 13731392
MSC (2000):
Primary 57R65, 57M20
Published electronically:
November 8, 2001
MathSciNet review:
1873010
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper concerns decompositions of smooth 4manifolds as the union of two handlebodies, each with handles of index . In dimensions results of Smale (trivial ) and Wall (general ) 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 2deformation of their spines. In higher dimensions spine 2deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2complexes are (up to 2deformation) spines of a dual decomposition of the 4sphere if and only if they satisfy the conclusions of AlexanderLefshetz duality ( and ). (3.3) If is 1connected then there is a ``pseudo'' handle decomposition without 1handles, in the sense that there is a pseudo collar (a relative 2handlebody with spine that 2deforms to ) and is obtained from this by attaching handles of index .
 [AM]
S.
Akbulut and R.
Matveyev, A convex decomposition theorem for 4manifolds,
Internat. Math. Res. Notices 7 (1998), 371–381. MR 1623402
(99f:57030), http://dx.doi.org/10.1155/S1073792898000245
 [AC]
J.
J. Andrews and M.
L. Curtis, Free groups and handlebodies,
Proc. Amer. Math. Soc. 16 (1965), 192–195. MR 0173241
(30 #3454), http://dx.doi.org/10.1090/S00029939196501732418
 [DK]
S.
K. Donaldson and P.
B. Kronheimer, The geometry of fourmanifolds, Oxford
Mathematical Monographs, The Clarendon Press, Oxford University Press, New
York, 1990. Oxford Science Publications. MR 1079726
(92a:57036)
 [FKT]
Maria
Rita Casali, The average edge order of 3manifold coloured
triangulations, Canad. Math. Bull. 37 (1994),
no. 2, 154–161. MR 1275697
(95c:57003), http://dx.doi.org/10.4153/CMB1994022x
 [FQ]
M. H. Freedman and F. Quinn, Topology of 4manifolds, Princeton University Press, 1990.
 [GS]
Robert
E. Gompf and András
I. Stipsicz, 4manifolds and Kirby calculus, Graduate Studies
in Mathematics, vol. 20, American Mathematical Society, Providence,
RI, 1999. MR
1707327 (2000h:57038)
 [H]
Günther
Huck, Embeddings of acyclic 2complexes in 𝑆⁴ with
contractible complement, Topology and combinatorial group theory
(Hanover, NH, 1986/1987; Enfield, NH, 1988) Lecture Notes in Math.,
vol. 1440, Springer, Berlin, 1990, pp. 122–129. MR 1082987
(92a:57002), http://dx.doi.org/10.1007/BFb0084457
 [K1]
Vyacheslav
S. Krushkal, Embedding obstructions and
4dimensional thickenings of 2complexes, Proc.
Amer. Math. Soc. 128 (2000), no. 12, 3683–3691. MR 1690995
(2001b:57006), http://dx.doi.org/10.1090/S0002993900054587
 [KT]
Vyacheslav
S. Krushkal and Peter
Teichner, Alexander duality, gropes and link homotopy, Geom.
Topol. 1 (1997), 51–69 (electronic). MR 1475554
(98i:57013), http://dx.doi.org/10.2140/gt.1997.1.51
 [Q]
Frank
Quinn, Handlebodies and 2complexes, Geometry and topology
(College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167,
Springer, Berlin, 1985, pp. 245–259. MR 827274
(88a:57008), http://dx.doi.org/10.1007/BFb0075228
 [W1]
C.
T. C. Wall, Geometrical connectivity. I, J. London Math. Soc.
(2) 3 (1971), 597–604. MR 0290387
(44 #7569a)
 [W2]
C.
T. C. Wall, Formal deformations, Proc. London Math. Soc. (3)
16 (1966), 342–352. MR 0193635
(33 #1851)
 [W3]
C.
T. C. Wall, Finiteness conditions for
𝐶𝑊complexes, Ann. of Math. (2) 81
(1965), 56–69. MR 0171284
(30 #1515)
 [W4]
C.
T. C. Wall, Finiteness conditions for 𝐶𝑊 complexes.
II, Proc. Roy. Soc. Ser. A 295 (1966), 129–139.
MR
0211402 (35 #2283)
 [AM]
 S. Akbulut and R. Matveyev, A convex decomposition theorem for manifolds, Internat. Math. Res. Notices 7 (1998), 371381. MR 99f:57030
 [AC]
 J. J. Andrews and M. L. Curtis, Free groups and handlebodies, Proc. Amer. Math. Soc. 16 (1965), 192195. MR 30:3454
 [DK]
 S. K. Donaldson and P. B. Kronheimer, The geometry of fourmanifolds, 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 2complexes in , Math. Res. Lett. 1 (1994), 167176. MR 95c:57003
 [FQ]
 M. H. Freedman and F. Quinn, Topology of 4manifolds, Princeton University Press, 1990.
 [GS]
 R. E. Gompf and A. I. Stipsicz, 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 complexes in with contractible complement, Springer Lecture Notes in Math. 1440 (1990), 122129. MR 92a:57002
 [K1]
 V. S. Krushkal, Embedding obstructions and 4dimensional thickenings of 2complexes, Proc. Amer. Math Soc. 128 (2000), 36833691. MR 2001b:57006
 [KT]
 V. S. Krushkal and P. Teichner, Alexander duality, gropes and link homotopy, Geometry & Topology 1 (1997), 5169. MR 98i:57013
 [Q]
 F. Quinn, Handlebodies and 2complexes, Springer Lecture Notes in Math. 1167 (1985), 245259. MR 88a:57008
 [W1]
 C. T. C. Wall, Geometrical Connectivity, J. London Math. Soc 3 (1971), 597608. MR 44:7569a
 [W2]
 , Formal deformations, Proc. London Math. Soc 16 (1966), 342352. MR 33:1851
 [W3]
 , Finiteness conditions for CWcomplexes, Ann. Math. 81 (1965), 5669. MR 30:1515
 [W4]
 , Finiteness conditions for CWcomplexes II, Proc. Roy. Soc. 295 (1966), 129139. 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 240610123
Email:
quinn@math.vt.edu
DOI:
http://dx.doi.org/10.1090/S0002994701029403
PII:
S 00029947(01)029403
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
