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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.
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.
|| Computations of complex equivariant bordism rings.
American Journal of Mathematics 123 (2001) 577-605.
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.
|| Bordism of semi-free S1-actions.
Mathematische Zeitschrift, Vol 249 No 2 (2005) 439-454.
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.
||The geometry of the local cohomology filtration in equivariant bordism.
Homotopy, Homology and Applications, vol 3(2), 2001, pp 385-406.
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.
||Real equivariant bordism and stable transversality obstructions
Proceedings of the AMS 130 (2002), No. 1, 271--281.