Geometric bordism and cobordism 1000-1M15BO
1. Review on smooth manifolds, tangent and normal bundles, transversality.
2. Vector bundles and basic constructions. Thom space. Universal bundle.
3. Geometric definition of bordism and cobordism as generalised homology and cohomology theories. Multiplicative structures. Information on bordism and cobordism of manifolds equipped with additional structure on the tangent bundle.
4. The Pontriagin-Thom construction. Homotopic interpretation of bordism and cobordism. Thom spectrum (info)
5. Axioms of the generalised multiplicative cohomology theory. Leray-Hirsch theorem. Bundle orientation. Characteristic classes. Canonical orientation of bundles in cohomology theory. Formal group of the cohomology theory.
6. Algebraic theory of formal groups. Universal formal group.
7. The Stiefel-Whitney characteristic classes in cobordism.
8. The Steenrod squares in cobordism. Algebra of cohomology operations in cobordism.
9. Computation of the ring of bordism of unoriented manifolds as a universal formal group over the field of two elements.
10. The Stiefel-Whitney characteristic numbers as invariants of unoriented bordism of manifolds.
Main fields of studies for MISMaP
mathematics
Type of course
Mode
Classroom
Prerequisites
Prerequisites (description)
Learning outcomes
1. Knows definitions of bordism and cobordism groups – both geometric and homotopical and can prove its equivalence.
2. Understands the notion of the canonical orientation of bundles in cohomology theory and the notion of the formal group.
3. Is familiar with basic algebraic theory of formal groups and can prove that unoriented bordism gives rise to universal formal group over the field of two elements.
4. Knows characteristic numbers and can prove that Stiefel Whitney characteristic numbers are invariants of unoriented cobordism.
Assessment criteria
Oral exam based on a set of published problems.
Bibliography
J.F. Adams Quillen's work on formal groups and complex cobordism.
Th. Broecker, T. tom Dieck Kobordismentheorie Lecture Notes in Math. 178, Springer 1970.
A. Bojanowska, S. Jackowski Geometric bordism and cobordism.
G.E.Bredon Topology and Geometry.
Th. Broecker, K. Jaenich Introduction to differential topology.
Dan Freed Bordism: Old and New. University of Texas
M. Hirsch Differential topology).
M.J. Hopkins Global methods in homotopy theory. W tomie Homotopy Theory London Math. Soc. Lect. Note Series 117 (1987), 73-96.
H. Miller Notes on cobordism. Lecture Notes MIT.
D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971).
R. Stong Notes on cobordism theory.
Additional information
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