MSC in Mathematics, Pure Mathematics, Elective Course


 Differential Forms


 Dr. Balázs Csikós (associate professor)

 Dr. Gyula Lakos (assistant)

 Dr. Gábor Moussong (assistant professor)

 Dr. László Verhóczki (associate professor)




 3rd semester


The notion of ordinary de Rham cohomology and de Rham cohomology with compact supports. Poincaré-lemma. Algebraic homotopies of complexes. Mayer-Vietoris sequence. Poncaré pairing. Poincaré duality. Künneth theorem. Leray-Hirsch theorem. Fiber bundles, cohomology with vertically compact supports. Thom isomorphism theorem. Thom forms. Poincaré dual of submanifolds. Multiplicative structure of cohomology. Intersections. Pfaff forms, Mathai-Quillen form. Euler form, Poincaré-Hopf theorem.

Hodge operator. Hodge Laplacian. Harmonic forms. The Hodge decomposition theorem on compact manifolds. Čech cohomology. The Čech-de Rham complex. Isomorphism of de Rham cohomology and Čech cohomology with real coefficients.        


R. Bott – L. W. Tu: Differential forms in algebraic geometry. Springer Verlag, New York, Berlin, 1982.

F. W. Warner: Foundations of differentiable manifolds and Lie groups. Springer Verlag, New York, Berlin, 1983.

form of tuition

Two hours of lecture per week.

mode of assessment

 oral exam