The topological structure of $4$-manifolds with effective torus actions. I
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- by Peter Sie Pao PDF
- Trans. Amer. Math. Soc. 227 (1977), 279-317 Request permission
Abstract:
Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of ${S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3}$, and three families of elementary 4-manifolds, $R(m,n),T(m,n;m’,n’),L(n;p,q;m)$ has been obtained (where m, n, $m’,n’$, p, and q are integers). In addition, a stable homeomorphic relation for the manifolds $R(m,n)$ and $T(m,n;m’,n’)$ can also be found in [9]. But the topological classification of R’s, T’s, and especially L’s were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R’s and T’s, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds $L(n;p,q;m)$ form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of $L(n;p,q;m)$ is finite cyclic of order n. And it is proved that ${\pi _1}(L(n;p,q;m))$ and ${\pi _1}(L(n;p’,q’;m’))$ act identically on the second cohomology of their common universal covering space, $({S^2} \times {S^2})\# \cdots \# \;({S^2} \times {S^2})$ ($n - 1$ copies), even though $L(n;p,q;m)$ and $L(n;p’,q’;m’)$ are not homotopically equivalent for some $(n;p’,q’;m’)$’s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author’s doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.References
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- P. E. Conner and Frank Raymond, Actions of compact Lie groups on aspherical manifolds, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 227–264. MR 0271958
- André Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47–82 (French). MR 145538, DOI 10.1007/BF02566892 A. G. Kuroš, Theory of groups, Vol. 2, 2nd ed., GITTL, Moscow, 1953; Chelsea, New York, 1956. MR 15, 501; 18, 188.
- John Milnor, On simply connected $4$-manifolds, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 122–128. MR 0103472
- John Milnor, A procedure for killing homotopy groups of differentiable manifolds. , Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 39–55. MR 0130696
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
- Peter Orlik and Frank Raymond, Actions of the torus on $4$-manifolds. I, Trans. Amer. Math. Soc. 152 (1970), 531–559. MR 268911, DOI 10.1090/S0002-9947-1970-0268911-3
- Peter Orlik and Frank Raymond, Actions of the torus on $4$-manifolds. II, Topology 13 (1974), 89–112. MR 348779, DOI 10.1016/0040-9383(74)90001-9
- Frank Raymond, Classification of the actions of the circle on $3$-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78. MR 219086, DOI 10.1090/S0002-9947-1968-0219086-9
- H. Cišang, Torus bundles over surfaces, Mat. Zametki 5 (1969), 569–576 (Russian). MR 248835
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 227 (1977), 279-317
- MSC: Primary 57E10
- DOI: https://doi.org/10.1090/S0002-9947-1977-0431231-4
- MathSciNet review: 0431231