Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T06:45:07.811Z Has data issue: false hasContentIssue false

Fundamental groups of 4-manifolds with circle actions

Published online by Cambridge University Press:  24 October 2008

Sławomir Kwasik
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, U.S.A.
Reinhard Schultz
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907., U.S.A.

Abstract

Topological circle actions on 4-manifolds are studied using modifications of known techniques for smooth actions. This yields topological versions of some previously known restrictions on the fundamental groups of 4-manifolds admitting smooth circle actions.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ba]Bauer, S.. The homotopy type of a 4-manifold with finite fundamental group. In Algebraic Topology and Transformation Groups (Proceedings, Göttingen, 1987), Lecture Notes in Math. vol. 1361 (Springer, 1988), 16.Google Scholar
[Bi1]Bing, R. H.. The cartesian product of a certain nonmanifold and a line is E 4. Ann. of Math. 70 (1959), 399412.CrossRefGoogle Scholar
[Bi2]Bing, R. H.. Inequivalent families of periodic homeomorphisms of E 3. Ann. of Math. 80 (1964), 7893.CrossRefGoogle Scholar
[Bo]Borel, A. (ed.). Seminar on transformation groups (Princeton University Press, 1960).Google Scholar
[Br1]Bredon, G.. Orientation in generalized manifolds and applications to the theory of transformation groups. Mich. Math. J. 7 (1960), 3564.CrossRefGoogle Scholar
[Br2]Bredon, G.. Generalized manifolds revisited. In Topology of manifolds (Proc. Inst., Univ. of Georgia, 1969) (Markham, 1970), pp. 461469.Google Scholar
[Br3]Bredon, G.. Introduction to compact transformation groups (Academic Press, 1972).Google Scholar
[CV]Christensen, C. O. and Voxman, W. L.. Aspects of topology (Marcel Dekker, 1977).Google Scholar
[DnS]Donnelly, H. and Schultz, R.. Compact group actions and maps into aspherical manifolds. Topology 21 (1982), 443455.CrossRefGoogle Scholar
[E1]Edmonds, A.. Taming free circle actions. Proc. Amer. Math. Soc. 62 (1977), 337343.CrossRefGoogle Scholar
[E2]Edmonds, A.. Transformation groups and low-dimensional manifolds. In Group actions on manifolds (Conference Proceedings, University of Colorado, 1983), Contemp. Math. 36 (1985), 341368.Google Scholar
[Fi1]Fintushel, R.. Circle actions on simply connected 4-manifolds. Trans. Amer. Math. Soc. 230 (1977), 147171.Google Scholar
[Fi2]Fintushel, R.. Classification of circle actions on 4-manifolds. Trans. Amer. Math. Soc. 242 (1978), 377390.CrossRefGoogle Scholar
[FP]Fintushel, R. and Pao, P.. Circle actions on spheres with codimension four fixed point set. Pac. J. Math. 109 (1983), 349362.CrossRefGoogle Scholar
[FL]Floyd, E. E.. Orbits of torus groups operating on manifolds. Ann. of Math. 65 (1957), 505512.CrossRefGoogle Scholar
[HY]Hocking, J. G. and Young, G. S.. Topology (Addison-Wesley, 1961).Google Scholar
[Hs]Hsiang, W.-Y.. Cohomology theory of topological transformation groups (Springer, 1975).CrossRefGoogle Scholar
[KwS0]Kwasik, S. and Schultz, R.. Desuspensions of group actions and the ribbon theorem. Topology 27 (1988), 443457.CrossRefGoogle Scholar
[KwS1]Kwasik, S. and Schultz, R.. Horaological properties of periodic homeomorphisms of 4-manifolds. Duke Math. J. 58 (1989), 241250.CrossRefGoogle Scholar
[KwS2]Kwasik, S. and Schultz, R.. Finite restrictions of pseudofree circle actions on 4-manifolds. Quart. J. Math. Oxford (2), 45 (1994), 227241.CrossRefGoogle Scholar
[Ml1]Milnor, J.. Groups which act on Sn without fixed points. Amer. J. Math. 79 (1957), 623630.CrossRefGoogle Scholar
[MZ]Montgomery, D. and Zippin, L.. Examples of transformation groups. Proc. Amer. Math. Soc. 5 (1954), 460465.CrossRefGoogle Scholar
[Mos]Mostert, P. S.. On a compact Lie group acting on a manifold. Ann. of Math. 65 (1957), 447455.CrossRefGoogle Scholar
[OrRa]Orlik, P. and Raymond, F.. Actions of SO(2) on 3-manifolds. In Proceedings of the Conference on Transformation Groups (New Orleans, 1967) (Springer, 1968), pp. 297318.Google Scholar
[OVZ]Orlik, P., Vogt, E. and Zieschang, H.. Zur Topologie dreidimensionaler gefaserter Mannigfaltigkeiten. Topology 6 (1967), 4975.CrossRefGoogle Scholar
[Pao]Pao, P. S.. Nonlinear circle actions on the 4-sphere and twist spun knots. Topology 17 (1978), 291296.CrossRefGoogle Scholar
[Par]Parker, J.. Four-dimensional G-manifolds with 3-dimensional orbits. Pac. J. Math. 125 (1986), 187204.CrossRefGoogle Scholar
[Pl]Plotnik, S.. Circle actions and fundamental groups for homology 4-spheres. Trans. Amer. Math. Soc. 273 (1982), 393404.CrossRefGoogle Scholar
[Q]Quinn, F.. Resolution of homology manifolds and the topological characterization of manifolds. Invent. Math. 72 (1983), 267284; corrigendum, 85 (1986), 653.CrossRefGoogle Scholar
[Ra1]Raymond, F.. Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 (1968), 5178.CrossRefGoogle Scholar
[Ra2]Raymond, F.. Separation and union theorems for generalized manifolds with boundary. Mich. Math. J. 7 (1960), 721.CrossRefGoogle Scholar
[Re]Repovš, D.. The recognition problem for topological manifolds. In Geometric and Algebraic Topology (Proceedings, Warsaw, 1984) (PWN-Polish Scientific Publishers, 1986), pp. 77108.Google Scholar
[Ru]Rushing, T. B.. Topological embeddings (Academic Press, 1973).Google Scholar
[St]Stein, E. V.. On the orbit types in a circle action. Proc. Amer. Math. Soc. 66 (1977), 143147.CrossRefGoogle Scholar
[T]Turaev, V. G.. Three-dimensional Poincaré complexes: homotopy classification and splitting [Russian], Mat. Sbornik 180 (1989), 809830Google Scholar
Turaev, V. G.. English translation Math. U.S.S.R. Sbornik 6 (1990), 261282.CrossRefGoogle Scholar
[vK]van Kampen, E. R.. On some characterizations of 2-dimensional manifolds. Duke Math. J. 1 (1935), 7493.CrossRefGoogle Scholar
[Wal]Wall, C. T. C.. Poincaré complexes. I. Ann. of Math. 86 (1967), 213245.CrossRefGoogle Scholar
[Wil]Wilder, R. L.. Topology of manifolds (American Mathematical Society, 1949).CrossRefGoogle Scholar
[Y]Yoshida, T.. Simply connected smooth 4-manifolds which admit nontrivial S 1 actions. J. Math. Okayama Univ. 20 (1978), 2540.Google Scholar