used to distinguish manifolds from wedges of spaces with identical cohomology groups. The space has the same cohomology groups as the torus T, but with a different cup product. In the case of X the multiplication of the cochains associated to the copies of is degenerate, whereas in T multiplication in the first cohomology group can be used to decompose the torus as a 2-cell diagram, thus having product equal to Z (more generally M where this is the base module). Other definitions Cup product and differential forms In de Rham cohomology, the cup product of differential forms is induced by the wedge product. In other words, the wedge product of two closed differential forms belongs to the de Rham class of the cup product of the two original de Rham classes. Cup product and geometric intersections For oriented manifolds, there is a geometric heuristic that "the cup product is dual to intersections." Indeed, let be an oriented smooth manifold of dimension . If two submanifolds of codimension and intersect transversely, then their intersection is again a submanifold of codimension . By taking the images of the fundamental homology classes of these manifolds under inclusion, one can obtain a bilinear product on homology. This product is Poincaré dual to the cup product, in the sense that taking the Poincaré pairings then there is the following equality : . Similarly, the linking number can be defined in terms of intersections, shifting dimensions by 1, or alternatively in terms of a non-vanishing cup product on the complement of a link. Massey products The cup product is a binary (2-ary) operation; one can define a ternary (3-ary) and higher order operation called the Massey product, which generalizes the cup product. This is a higher order cohomology operation, which is only partly defined (only defined for some triples). See also Singular homology Homology theory Cap product Massey product Torelli group References James R. Munkres, "Elements of Algebraic Topology", Perseus Publishing, Cambridge Massachusetts (1984) (hardcover) (paperback) Glen E. Bredon, "Topology and Geometry", Springer-Verlag, New York (1993) Allen Hatcher, "Algebraic Topology", Cambridge Publishing Company (2002) Homology theory Algebraic topology Binary operations