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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“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)