Syllabus:
Characteristic classes and cobordism
Characteristic classes are cohomological invariants of vector bundles on a topological space. In (...)
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Characteristic classes and cobordism
Characteristic classes are cohomological invariants of vector bundles on a topological space. In particular, if we use the tangent bundle of a manifold, they give invariants of differentiable manifolds. We will focus on the Chern and Pontryagin classes. An important theorem of Thom asserts that the knowledge of the Pontryagin classes of an oriented manifold is sufficient to determine the class of that manifold in the cobordism ring tensored with the rational numbers. The ring of cobordism is obtained by quotienting the set of all differentiable manifolds by the relation M=0 if M is the boundary of a manifold with boundary. This has a ring structure induced by disjoint union and cartesian product.
We will mostly follow the book :
Characteristic classes and cobordism
Characteristic classes are cohomological invariants of vector bundles on a topological space. In particular, if we use the tangent bundle of a manifold, they give invariants of differentiable manifolds. We will focus on the Chern and Pontryagin classes. An important theorem of Thom asserts that the knowledge of the Pontryagin classes of an oriented manifold is sufficient to determine the class of that manifold in the cobordism ring tensored with the rational numbers. The ring of cobordism is obtained by quotienting the set of all differentiable manifolds by the relation M=0 if M is the boundary of a manifold with boundary. This has a ring structure induced by disjoint union and cartesian product.
We will mostly follow the book :
John Milnor & James D. Stasheff, Characteristic classes. Princeton University Press.
John Milnor & James D. Stasheff, Characteristic classes. Princeton University Press.