ε-quadratic form

In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric.

-quadratic forms, particularly in the context of surgery theory.

More briefly, one may refer to quadratic, skew-quadratic, symmetric, and skew-symmetric forms, where "skew" means (−) and the * (involution) is implied.

The theory is 2-local: away from 2, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.

[1] Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let T : B(M) → B(M) be the "conjugate transpose" involution B(u, v) ↦ B(v, u)*.

As an exact sequence, As kernel and cokernel, The notation Qε(M), Qε(M) follows the standard notation MG, MG for the invariants and coinvariants for a group action, here of the order 2 group (an involution).

Composition of the inclusion and quotient maps (but not 1 − εT) as

Conversely, one can define a reverse homomorphism "1 + εT": Qε(M) → Qε(M), called the symmetrization map (since it yields a symmetric form) by taking any lift of a quadratic form and multiplying it by 1 + εT.

The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of (1 − εT), but this vanishes after multiplying by 1 + εT.

An ε-quadratic form ψ ∈ Qε(M) is called non-degenerate if the associated ε-symmetric form (1 + εT)(ψ) is non-degenerate.

(the integral lattice for the quadratic form 2x2 − 2x + 1), with complex conjugation,

are two such elements, though 1/2 ∉ R. In terms of matrices (we take V to be 2-dimensional), if * is trivial: to

Mapping back to quadratic forms yields double the original:

is complex conjugation, then An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form.

For instance, in defining a Clifford algebra over a general field or ring, one quotients the tensor algebra by relations coming from the symmetric form and the quadratic form: vw + wv = 2B(v, w) and

(Here, R* := HomR(R, R) denotes the dual of the R-module R.) It is given by the bilinear form

The standard hyperbolic ε-quadratic form is needed for the definition of L-theory.

The Arf invariant of a nonsingular quadratic form over F2 is an F2-valued invariant with important applications in both algebra and topology, and plays a role similar to that played by the discriminant of a quadratic form in characteristic not equal to two.

The free part of the middle homology group (with integer coefficients) of an oriented even-dimensional manifold has an ε-symmetric form, via Poincaré duality, the intersection form.

Geometrically this corresponds to intersection, where two n/2-dimensional submanifolds in an n-dimensional manifold generically intersect in a 0-dimensional submanifold (a set of points), by adding codimension.

For singly even dimension the order switches sign, while for doubly even dimension order does not change sign, hence the ε-symmetry.

In dimension two, this yields a torus, and taking the connected sum of g tori yields the surface of genus g, whose middle homology has the standard hyperbolic form.

For doubly even dimension this is integer valued, while for singly even dimension this is only defined up to parity, and takes values in Z/2.

For example, given a framed manifold, one can produce such a refinement.

Given an oriented surface Σ embedded in R3, the middle homology group H1(Σ) carries not only a skew-symmetric form (via intersection), but also a skew-quadratic form, which can be seen as a quadratic refinement, via self-linking.

The skew-symmetric form is an invariant of the surface Σ, whereas the skew-quadratic form is an invariant of the embedding Σ ⊂ R3, e.g. for the Seifert surface of a knot.

The Arf invariant of the skew-quadratic form is a framed cobordism invariant generating the first stable homotopy group

For the standard embedded torus, the skew-symmetric form is given by

(with respect to the standard symplectic basis), and the skew-quadratic refinement is given by xy with respect to this basis: Q(1, 0) = Q(0, 1) = 0: the basis curves don't self-link; and Q(1, 1) = 1: a (1, 1) self-links, as in the Hopf fibration.

A key application is in algebraic surgery theory, where even L-groups are defined as Witt groups of ε-quadratic forms, by C.T.C.Wall