group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The integrability of G-structures exists to first order, precisely if a certain torsion obstruction vanishes. This is the first in an infinite tower of tensor invariants in Spencer cohomology associated with a $G$-structure that obstruct its integrability (local flatness) (Guillemin 65).
The torsion of a $G$-structure is defined to be the space in which the invariant part of the torsion of a Cartan connection takes values, for any Cartan connection compatible with the $G$-structure (see at Cartan connection – Examples – G-Structure) (Sternberg 64, from p. 317 on, Guillemin 65, section 4), for review see also (Lott 90, p.10, Joyce 00, section 2.6).
The order $k$-torsion of a $G$-structure (counting may differ by 1) is an element in a certain Spencer cohomology group (Guillemin 65, prop. 4.2) and is the obstruction to lifting an order-$k$-integrable G-structure to order $k+1$ (Guillemin 65, theorem 4.1).
The concept goes back to the work of Eli Cartan (Cartan geometry).
Textbook accounts include
Discussion including the higher order obstructions in Spencer cohomology to integrability of G-structures is in
Formalization in homotopy type theory is in
Discussion with an eye towards torsion constraints in supergravity is in
Discussion with an eye towards special holonomy is in
Further mentioning of the higher order torsion invariants includes
See also