Actions of the torus on 4-manifolds. I

Orlik, Peter; Raymond, Frank

doi:10.1090/S0002-9947-1970-0268911-3

Actions of the torus on $4$-manifolds. I
HTML articles powered by AMS MathViewer

by Peter Orlik and Frank Raymond PDF

Trans. Amer. Math. Soc. 152 (1970), 531-559 Request permission

Abstract:

Smooth actions of the $2$-dimensional torus group $SO(2) \times SO(2)$ on smooth, closed, orientable $4$-manifolds are studied. A cross-sectioning theorem for actions without finite nontrivial isotropy groups and with either fixed points or orbits with isotropy group isomorphic to $SO(2)$ yields an equivariant classification for these cases. This classification is made numerically specific in terms of orbit invariants. A topological classification is obtained for actions on simply connected $4$-manifolds. It is shown that such a manifold is an equivariant connected sum of copies of complex projective space $C{P^2}, - C{P^2}$ (reversed orientation), ${S^2} \times {S^2}$ and the other oriented ${S^2}$ bundle over ${S^2}$. The latter is diffeomorphic (but not always equivariantly diffeomorphic) to $C{P^2}\# - C{P^2}$. The connected sum decomposition is not unique. Topological actions on topological manifolds are shown to reduce to the smooth case. In an appendix certain results are extended to torus actions on orientable $4$-dimensional cohomology manifolds.

References

Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341

Differentiable manifolds and quadratic forms

On simply connected $4$-manifolds

21

John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
Peter Orlik and Frank Raymond, Actions of $\textrm {SO}(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR 0263112
Frank Raymond, The end point compactification of manifolds, Pacific J. Math. 10 (1960), 947–963. MR 120637
Frank Raymond, The orbit spaces of totally disconnected groups of transformations on manifolds, Proc. Amer. Math. Soc. 12 (1961), 1–7. MR 141730, DOI 10.1090/S0002-9939-1961-0141730-4
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
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491