CEU Electronic Theses and Dissertations, 2019
| Author | Matszangosz, Kyriakos Ákos |
|---|---|
| Title | Borel-Haefliger Type Theorems |
| Summary | The focus of this thesis are certain U(1)-manifolds, whose fixed point manifold is half-dimensional. We show that under certain additional conditions, the rational cohomology ring of the space is isomorphic to the cohomology ring of its fixed point set with the degrees halved. Such a phenomenon was first observed by Borel and Haefliger, for the Z_2-action of complex conjugation on complex algebraic varieties for mod 2 coefficient cohomology. The Borel-Haefliger theorem states that if X_\C is a smooth variety, which is the complexification of the real variety X_\R, and H^*(X_\R;\F_2) and H^*(X_\C;\F_2) are additively generated by real algebraic cycles and their complexifications respectively, then the complexification map is a multiplicative degree doubling isomorphism. More recently, Hausmann, Holm and Puppe introduced a class of Z_2-spaces called conjugation spaces with such a degree-halving ring isomorphism using equivariant cohomology. The relationship of conjugation spaces to the Borel-Haefliger theorem in terms of geometrically defined cycles was examined and clarified in a paper of van Hamel. The other main theorem in the paper of Borel and Haefliger relates equivariant fundamental classes of real and complex singularity loci, also known as Thom polynomials. The theorem states that the Thom polynomial of a complexified singularity locus \eta^\C expressed in terms of Chern classes is the same as the Thom polynomial of the real singularity locus \eta^\R expressed in terms of Stiefel-Whitney classes. In this thesis we obtain analogues of the Borel-Haefliger theorem and Hausmann, Holm and Puppe's theory of conjugation spaces for U(1)-actions and rational coefficient cohomology and we call the resulting spaces circle spaces. We also prove a theorem relating (rational) Thom polynomials of real singularities in Pontryagin classes to Thom polynomials of complex singularities in Chern classes. |
| Supervisor | László Fehér |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2019/matszangosz_kyriakos.pdf |
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