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Topology of four-manifolds with special homotopy groups

Published online by Cambridge University Press:  17 April 2009

Alberto Cavicchioli  and
Fulvia Spaggiari
Alberto Cavicchioli
Affiliation:
Dipartimento di Matematica, Università di Modena e di Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy, e-mail: cavicchioli.alberto@unimo.it, spaggiari.fulvia@unimo.it
Fulvia Spaggiari
Affiliation:
Dipartimento di Matematica, Università di Modena e di Reggio Emilia, Via Campi 213/B, 41100 Modena, Italy, e-mail: cavicchioli.alberto@unimo.it, spaggiari.fulvia@unimo.it
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We study the homotopy type and the s—cobordism class of a closed connected topological 4-manifold with vanishing second homotopy group. Our results are related to problem 4.53 of Kirby in Geometric Topology, Studies in Advanced Math. 2 (1997), and give a partial answer to a question stated by Hillman in Bull. London Math. Soc.27 (1995) 387–391.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 3 , December 2006 , pp. 321 - 335
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bauer, S., ‘The homotopy type of a 4-manifold with finite fundamental group’, in Lect. Notes in Math. 1361 (Springer-Verlag, Berlin, Heidelberg, New York, 1988), pp. 1–6.Google Scholar
[2]Baues, H.J., Obstruction theory, Lect. Notes in Math. 628 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
[3]Baues, H.J., Combinatorial Homotopy and 4-Dimensional Complexes (Walter de Gruyter, Berlin, New York, 1991).Google Scholar
[4]Baues, H.J., Homotopy type and homology, Oxford Science Publ. (Clarendon Press, Oxford, 1996).Google Scholar
[5]Bieri, R., Homological dimension of discrete groups (Queen Mary College Mathematics Notes, London, 1976).Google Scholar
[6]Cavicchioli, A. and Hegenbarth, F., ‘On 4-manifolds with free fundamental group’, Forum Math. 6 (1994), 415–429.CrossRefGoogle Scholar
[7]Cavicchioli, A. and Hegenbarth, F., ‘The homotopy classification of 4-manifolds having the fundamental group of an aspherical 4-manifold’, Osaka J. Math. 37 (2000), 859–871.Google Scholar
[8]Cavicchioli, A., Hegenbarth, F. and Repovš, D., ‘On the stable classification of certain 4-manifolds’, Bull. Austral. Math. Soc. 52 (1995), 385–398.Google Scholar
[9]Cavicchioli, A., Hegenbarth, F. and Repovš, D., ‘Four-manifolds with surface fundamental groups’, Trans. Amer. Math. Soc. 349 (1997), 4007–4019.Google Scholar
[10]Cavicchioli, A., Hegenbarth, F. and Spaggiari, F., ‘Embedding 4-manifolds with vanishing second homology’, Topology Appl. 123 (2002), 313–322.Google Scholar
[11]Freedman, M.H., ‘The topology of four-dimensional manifolds’, J. Differential Geom. 17 (1982), 357–453.CrossRefGoogle Scholar
[12]Freedman, M.H., ‘The disk theorem for four-dimensional manifolds’, in Proceed. Intern. Congress of Mathematicians, Warsaw 1983 (PWN, Warsaw, 1984), pp. 647–663.Google Scholar
[13]Freedman, M.H. and Quinn, F., Topology of 4-manifolds (Princeton Univ. Press, Princeton, N.J., 1990).Google Scholar
[14]Freedman, M.H. and Teichner, P., ‘4-Manifold topology I: Subexponential groups’, Invent. Math. 122 (1995), 509–529.Google Scholar
[15]Freedman, M. H. and Teichner, P., ‘4-Manifold topology II: Dwyer' s filtration and surgery kernels’, Invent. Math. 122 (1995), 531–557.CrossRefGoogle Scholar
[16]Hambleton, I. and Kreck, M., ‘On the classification of topological 4-manifolds with finite fundamental group’, Math. Ann. 280 (1988), 85–104.CrossRefGoogle Scholar
[17]Hegenbarth, F., Repovš, D. and Spaggiari, F., ‘Connected sums of 4-manifolds’, Topology Appl. 146 (2005), 209–225.Google Scholar
[18]Hillman, J.A., The Algebraic Characterization of Geometric 4-Manifolds, London Math. Soc. Lect. Note Ser. 198 (Cambridge Univ. Press, Cambridge, 1994).Google Scholar
[19]Hillman, J.A., ‘Free products and 4-dimensional connected sums’, Bull. London Math. Soc. 27 (1995), 387–391.CrossRefGoogle Scholar
[20]Hillman, J.A., ‘Complex surfaces which are fibre bundles’, Topology Appl. 100 (2000), 187–191.Google Scholar
[21]Hillman, J.A., Four-manifolds, geometries and knots 5 (Geometry and Topology Monographs, Coventry, 2002).Google Scholar
[22]Kirby, R., ‘Problems in low-dimensional topology’, in Geometric Topology, (Kazez, W. H., Editor), Studies in Advanced Math. 2, (Georgia Intern. Topology Conf., Athens, Georgia, August 2–3, 1993) (Amer. Math. Soc., Providence, R.I., 1997).Google Scholar
[23]Kreck, M., ‘Surgery and duality’, Ann. of Math. (2) 149 (1999), 707–754.Google Scholar
[24]Krushkal, V.S. and Lee, R., ‘Surgery on closed 4-manifolds with free fundamental group’, Math. Proc. Cambridge Philos. Soc. 133 (2002), 305–310.Google Scholar
[25]Krushkal, V.S. and Quinn, F., ‘Subexponential groups in 4-manifold topology’, Geom. Topol. 4 (2000), 407–430.CrossRefGoogle Scholar
[26]Luo, F., ‘The existence of K(π1, 1) 4-manifolds which are rational homology 4-spheres’, Proc. Amer. Math. Soc. 104 (1988), 1315–1321.Google Scholar
[27]Matumoto, T. and Katanaga, A., ‘On 4-dimensional closed manifolds with free fundamental groups’, Hiroshima Math. J. 25 (1995), 367–370.Google Scholar
[28]Mandelbaum, R., ‘Four-dimensional topology: an introduction’, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 1–159.CrossRefGoogle Scholar
[29]Robinson, D.J.S., A course in the theory of groups, Graduate Text in Math. 80 (Springer-Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar
[30]Ruberman, D., ‘Invariant knots of free involutions of 𝕊4’, Topology Appl. 18 (1984), 217–224.CrossRefGoogle Scholar
[31]Ruberman, D., ‘Seifert surfaces of knots in 𝕊4’, Pacific J. Math. 145 (1990), 97–116.CrossRefGoogle Scholar
[32]Spaggiari, F., ‘Four-manifolds with π1-free second homotopy’, Manuscripta Math. 111 (2003), 303–320.Google Scholar
[33]Spaggiari, F., ‘On the stable classification of spin four-manifolds’, Osaka J. Math. 40 (2003), 835–843.Google Scholar
[34]Specker, E., ‘Die erste cohomologie gruppe von überlagerungen und homotopie eigenschaften dreidimensionaler mannigfaltig keiten’, Comment. Math. Helvetici 23 (1949), 303–333.CrossRefGoogle Scholar
[35]Wall, C.T.C., Surgery on compact manifolds (Academic Press, London, New York, 1970).Google Scholar
[36]Wang, Z., ‘Classification of closed nonorientable 4-manifolds with infinite cyclic fundamental group’, Math. Res. Lett. 2 (1995), 339–344.Google Scholar
[37]Whitehead, G.W., Elements of homotopy theory (Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar