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The idea of using equivariant cobordism theory to study questions about the existence and structure of group actions on manifolds dates back to work of Conner and Floyd, soon after cobordism theory was invented. With the modern development of equivariant stable homotopy theory it is possible to carry this classical approach to group actions much further than in its first development. By recent work of Greenlees and May, equivariant cobordism also provides universal characteristic classes. Because transversality does not always hold in the equivariant world, geometric bordism theory differs from homotopical bordism theory and their interplay is a central theme in the subject. Work of Musin and (independently) Kosniowski on generators of geometric bordism and a result of Hanke, answering a question of mine, might lead to a computation of geometric complex S1 bordism.

PS PDF Computations of complex equivariant bordism rings.
American Journal of Mathematics 123 (2001) 577-605.
In this paper I give the first explicit complete computations of complex equivariant bordism ring structure, giving generators and relations when the group in question is S1. I make extensive use of localization by inverting Euler classes. Applications range from relating rigidity and strong multiplicativity in equivariant genera to showing that actions on stably-complex four-manifolds with three isolated fixed points are equivariantly cobordant to linear actions on CP2.

PS PDF Bordism of semi-free S1-actions.
Mathematische Zeitschrift, Vol 249 No 2 (2005) 439-454.
Semi-free actions are those in which points are either fixed by the whole group or are acted upon freely. I calculate semi-free S1 bordism in both homotopical and geometric settings. En route, I show that a semi-free S1 action with isolated fixed points is cobordant to a union of products of the standard action on CP1, complementing results of a similar flavor in symplectic geometry by Tolman and Weitsman. The geometric theory's ring structure is the first such computation, and I give a conjectural framework for further geometric work.

PS PDF The geometry of the local cohomology filtration in equivariant bordism.
Homotopy, Homology and Applications, vol 3(2), 2001, pp 385-406.
The local cohomology filtration in equivariant topology was initiated by Greenlees and has been used for example by Greenlees and Bruner to compute ku-homology of classifying spaces. I give a geometric construction of this filtration in the setting of bordism and show that the classes produced lend themselves to analysis through Atiyah-Segal-Wilson invariants.

PS PDF Real equivariant bordism and stable transversality obstructions for G=Z/2
Proceedings of the AMS 130 (2002), No. 1, 271--281.
Here I use the techniques developed in the complex setting to study real unoriented bordism for manifolds with an involution. Here it is possible to compute the quotient of the geometric and homotopical theories, algebraically encoding the lack of transversality in this setting.