restriction map

The restriction map is the natural way to view the cohomology of a group from the vantage point of a subgroup. If a cocycle is a rule defined for all elements of the big group, you can simply forget about the elements outside the subgroup and keep only the rule on the subgroup. That forgetful operation respects cocycles and coboundaries, so it descends to a homomorphism between cohomology groups, going from the whole group down to the subgroup.

Given a group G, a subgroup H, and a G-module A (which is also an H-module by restricting the action), restriction is the homomorphism res : H^n(G, A) -> H^n(H, A) induced on cochains by restricting a function on G^n to a function on H^n. It is functorial and commutes with connecting homomorphisms, so it gives a map of long exact sequences. Restriction is one half of the inflation–restriction machinery; the other half, inflation, goes the opposite way for a quotient.

A central structural fact is the corestriction (transfer) map cor : H^n(H, A) -> H^n(G, A) in the opposite direction when [G : H] is finite, satisfying cor ∘ res = multiplication by [G : H]. This forces H^n(G, A) to be annihilated by the group order for finite G (take H = 1), and it underlies the standard reduction of cohomology to Sylow subgroups: the p-primary part of H^n(G, A) injects into H^n(P, A) for a Sylow p-subgroup P.