cup product
The cup product is a way to multiply cohomology classes, turning a graded sequence of abelian groups into a ring. Cohomology in each degree is built from cochains, which are functions; the cup product simply multiplies two such functions together by feeding consecutive arguments to each factor, then checks that this respects the cocycle and coboundary structure. The payoff is that cohomology carries far more information as a ring than as a list of groups.
For a group G with modules A and B paired into a module C (via a G-equivariant bilinear map A ⊗ B -> C), the cup product is a map H^p(G, A) ⊗ H^q(G, B) -> H^{p+q}(G, C). On inhomogeneous cochains it is given by (f ∪ h)(g_1, ..., g_{p+q}) = f(g_1, ..., g_p) · (g_1 ... g_p)·h(g_{p+1}, ..., g_{p+q}). One checks d(f ∪ h) = df ∪ h ± f ∪ dh, so the product descends to cohomology and is well defined on classes.
The cup product is associative and graded-commutative: x ∪ y = (-1)^{pq} y ∪ x for classes of degrees p and q. With the trivial module Z, the total cohomology H^*(G, Z) = ⊕_n H^n(G, Z) becomes a graded-commutative ring, the cohomology ring of G, a powerful invariant studied intensively for finite groups. In Galois cohomology cup products assemble local and global pairings (such as the Hilbert symbol and Tate duality) that lie at the heart of class field theory.
For G = Z/2Z, the cohomology ring with F_2 coefficients is H^*(Z/2Z, F_2) = F_2[t], a polynomial ring on one generator t in degree 1; here t ∪ t = t^2 is nonzero because the characteristic is 2.
The mod-2 cohomology ring of Z/2Z is a polynomial ring.
Graded-commutativity means odd-degree classes square to elements killed by 2: if deg x is odd then x ∪ x = -x ∪ x, so 2(x ∪ x) = 0. Over a field of characteristic 2 this sign vanishes and odd classes can have nonzero squares.