Cup Product - Definition

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:

    Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?
    Richard Eberhart (b. 1904)

    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)

    Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.
    Walter Pater (1839–1894)