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Involutions with 1- or 2-dimensional fixed point sets on orientable torus bundles over a 1-sphere and on unions of orientable twisted I-bundles over a Klein bottle

University of British Columbia

We obtain a complete equivariant torus theorem for involutions on 3-manifolds M. M is not required to be orientable nor is H,(M) restricted to be infinite. The proof proceeds by a surgery argument. Similar theorems are given for annuli and for discs. These are used to classify involutions on various spaces such as orientable twisted I-bundles over a Klein bottle. Next we restrict our attention to orientable torus bundles over S¹ or unions of orientable twisted I-bundles over a Klein bottle. The equivariant torus theorem is applied to the problem of determining which of these spaces have involutions with 1-dimensional fixed point sets. It is shown that the fixed point set must be one, two, three, or four 1-spheres. Matrix conditions that determine which of these spaces have involutions with a given number of V-spheres as the fixed point sets are obtained. The involutions with 2-dimensional fixed point sets on orientable torus bundles over S¹ and on unions of orientable twisted I-bundles over a Klein bottle are classified. Only the orientable flat 3-space forms M₁, M₂ and M₆ have involutions with 2-dimensional fixed sets. Up to conjugacy, M₁ has two involutions, M₂ has four involutions, and M₆ has a unique involution.

Text

2010-06-01

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http://hdl.handle.net/2429/25303

Mathematics

University of British Columbia

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