In mathematics, specifically the theory of quadratic forms, an õ-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; , accordingly for symmetric or skew-symmetric. They are also called -quadratic forms, particularly in the context of surgery theory.
There is the related notion of õ-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms. 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, õ-quadratic forms are equivalent to õ-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.
õ-symmetric forms and õ-quadratic forms are defined as follows.
Given a module M over a *-ring R, let B(M) be the space of bilinear forms on M, and let be the "conjugate transpose" involution . Since multiplication by âÂÂ1 is also an involution and commutes with linear maps, âÂÂT is also an involution. Thus we can write and õT is an involution, either T or âÂÂT (õ can be more general than ñ1; see below). Define the õ-symmetric forms as the invariants of õT, and the õ-quadratic forms are the coinvariants.
As an exact sequence,
The notation Q<sup>õ</sup>(M), Q<sub>õ</sub>(M) follows the standard notation M<sup>G</sup>, M<sub>G</sub> 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 ) as yields a map Q<sup>õ</sup>(M) â Q<sub>õ</sub>(M): every õ-symmetric form determines an õ-quadratic form.
Conversely, one can define a reverse homomorphism , called the symmetrization map (since it yields a symmetric form) by taking any lift of a quadratic form and multiplying it by . This is a symmetric form because , so it is in the kernel. More precisely, . The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of , but this vanishes after multiplying by . Thus every õ-quadratic form determines an õ-symmetric form.
Composing these two maps either way: or yields multiplication by 2, and thus these maps are bijective if 2 is invertible in R, with the inverse given by multiplication with 1/2.
An õ-quadratic form is called non-degenerate if the associated õ-symmetric form is non-degenerate.
If the * is trivial, then , and "away from 2" means that 2 is invertible: .
More generally, one can take for any element such that . always satisfy this, but so does any element of norm 1, such as complex numbers of unit norm.
Similarly, in the presence of a non-trivial *, õ-symmetric forms are equivalent to õ-quadratic forms if there is an element such that . If * is trivial, this is equivalent to or , while if * is non-trivial there can be multiple possible û; for example, over the complex numbers any number with real part 1/2 is such a û.
For instance, in the ring (the integral lattice for the quadratic form ), with complex conjugation, are two such elements, though .
In terms of matrices (we take V to be 2-dimensional), if * is trivial:
to , for example by lifting to and then adding to transpose. Mapping back to quadratic forms yields double the original: .
If is complex conjugation, then
An intuitive way to understand an õ-quadratic form is to think of it as a quadratic refinement of its associated õ-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: and . If 2 is invertible, this second relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary.
An easy example for an õ-quadratic form is the standard hyperbolic õ-quadratic form . (Here, denotes the dual of the R-module R.) It is given by the bilinear form . The standard hyperbolic õ-quadratic form is needed for the definition of L-theory.
For the field of two elements there is no difference between (+1)-quadratic and (−1)-quadratic forms, which are just called quadratic forms. The Arf invariant of a nonsingular quadratic form over F<sub>2</sub> is an F<sub>2</sub>-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 õ-symmetric form, via Poincaré duality, the intersection form. In the case of singly even dimension , this is skew-symmetric, while for doubly even dimension 4k, this is symmetric. 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 õ-symmetry. The simplest cases are for the product of spheres, where the product and respectively give the symmetric form and skew-symmetric form 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.
With additional structure, this õ-symmetric form can be refined to an õ-quadratic 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. For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant.
Given an oriented surface ã embedded in R<sup>3</sup>, the middle homology group H<sub>1</sub>(ã) 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 ã, whereas the skew-quadratic form is an invariant of the embedding , 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: : the basis curves don't self-link; and : a self-links, as in the Hopf fibration. (This form has Arf invariant 0, and thus this embedded torus has Kervaire invariant 0.)
A key application is in algebraic surgery theory, where even L-groups are defined as Witt groups of õ-quadratic forms, by C.T.C.Wall