group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Euler class is a characteristic class of the special orthogonal group, hence of oriented real vector bundles.
The Euler class of the tangent bundle of a manifold is its Euler characteristic.
wikipedia Euler class
Robert F. Brown, On the Lefschetz number and the Euler class, Transactions of the AMS 118, (1965) (JSTOR)
Solomon Jekel, A simplicial formula and bound for the Euler class, Israel Journal of Mathematics 66, n. 1-3, 247-259 (1989)