A1 Course. Differential Cohomology I

Cohomological Algebra

• Modules and algebras.
• Complexes, differential algebras, differential ideals.
• Differential modules and their morphisms.
• Cohomologies, homotopies.
• Basic techniques in homological algebra.
• Spectral sequences and their morphisms.
• Spectral sequence associated with a filtered complex.

de Rham Cohomology

• de Rham complex.
• Compact support and relative de Rham complexes.
• Differential homotopy.
• Suspension theorem.
• Cohomological surgery.
• de Rham cohomologies of "simple" manifolds.
• Cohomological Newton-Leibnitz formula.
• Cohomological theory of integration for orientable manifolds.
• Degree of a proper map of manifolds.
• CW-complexes and group-valued differential cohomology.
• Cohomology of compact surfaces.

Algebraic theory of Linear Connections

• Modules and vector bundles.
• Der-operators in a module.
• Linear connections in a module.
• Parallel transport.
• Curvature of a connection.
• Simple operations with connections.
• Bianchi identities.

Cohomological Bundles

• Fiber bundles and their morphisms.
• Vertical morphisms and vertical vector fields.
• Vertical forms and the vertical de Rham complex.
• Cohomological bundle associated with a fiber bundle.
• A natural flat connection in the cohomological bundle.

Differential Leray-Serre Spectral Sequence

• Filtered complex associated with a fiber bundle.
• Differential Leray-Serre spectral sequence (DLSS).
• The term E₀ of the DLSS.
• The term E₁ of the DLSS.
• The term E₂ of the DLSS.
• Leray-Serre theorem and Künneth formula

Applications of the Leray-Serre Theory

• Suspension theorem.
• Thom isomorphism.
• Fixed point theorem.
• Intersection index.
• Cohomological theory of integration for non-orientable manifolds.
• Computation of cohomologies via Leray-Serre theorem.