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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Actions of the torus on $ 4$-manifolds. I


Authors: Peter Orlik and Frank Raymond
Journal: Trans. Amer. Math. Soc. 152 (1970), 531-559
MSC: Primary 57.47
DOI: https://doi.org/10.1090/S0002-9947-1970-0268911-3
MathSciNet review: 0268911
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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}\char93 - 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.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0268911-3
Keywords: Group action, $ 4$-manifold, torus, orbit space, cross-section, isotropy groups, equivariant map, equivariant connected sum
Article copyright: © Copyright 1970 American Mathematical Society