Definition
The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional Z/2Z-coefficient homology group
- q : H2m+1(M;Z/2Z) Z/2Z,
and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly, skew-quadratic form) is a quadratic refinement of the usual ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.
The quadratic form q can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections of immersions determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings (for ) and the mod 2 Hopf invariant of maps (for ).
Read more about this topic: Kervaire Invariant
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)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“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)