Definition
In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H∗(X) of a topological space X.
The construction starts with a product of cochains: if cp is a p-cochain and dq is a q-cochain, then
where σ is a (p + q) -singular simplex and is the canonical embedding of the simplex spanned by S into the -standard simplex.
Informally, is the p-th front face and is the q-th back face of σ, respectively.
The coboundary of the cup product of cocycles cp and dq is given by
The cup product of two cocycles is again a cocycle, and the product of a coboundary with a cocycle (in either order) is a coboundary. Thus, the cup product operation passes to cohomology, defining a bilinear operation
Read more about this topic: Cup Product
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)