Oriented and unitary equivariant bordism of surfaces

the unitary and oriented bordism groups of smooth G-equivariant compact surfaces, respectively, and we calculate them explicitly.Their ranks are determined by the possible representations around fixed points, while their torsion subgroups are isomorphic to the direct sum of the Bogomolov multipliers of the Weyl groups of representatives of conjugacy classes of all subgroups of G.We present an alternative proof of the fact that surfaces with free actions which induce non-trivial elements in the Bogomolov multiplier of the group cannot equivariantly bound.This result permits us to show that the 2-dimensional SK-groups (Schneiden und Kleben, or "cut and paste") of the classifying spaces of a finite group can be understood in terms of the bordism group of free equivariant surfaces modulo the ones that bound arbitrary actions. IEquivariant bordism groups have been a subject of ongoing research since the 1960s.Conner, Floyd, Landweber, Stong, Smith and tom Dieck, among others, laid the foundations for the extraordinary homology and cohomology theories obtained from equivariant bordism, and found many interesting properties of these groups.Given a finite group G, a particularly important problem is the explicit calculation of the oriented and complex G-equivariant bordism groups of a point, since they provide the coefficients for the theories.This turns out to be a complicated task.