# 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. They are also called -quadratic forms, particularly in the context of surgery theory.

There is the related notion of **ε-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, ε-quadratic forms are equivalent to ε-symmetric forms: half the symmetrization map (below) gives an explicit isomorphism.

## Definition

ε-symmetric forms and ε-quadratic forms are defined as follows.^{[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)**. Let ε = ±1; then ε*T* is also an involution. Define the **ε-symmetric forms** as the invariants of ε*T*, and the **ε-quadratic forms** are the coinvariants.

As an exact sequence,

As kernel (algebra) and cokernel,

The notation *Q*^{ε}(*M*), *Q*_{ε}(*M*) follows the standard notation *M ^{G}*,

*M*for the invariants and coinvariants for a group action, here of the order 2 group (an involution).

_{G}We obtain a homomorphism (1 + ε*T*): *Q*_{ε}(*M*) → *Q*^{ε}(*M*) which is bijective if 2 is invertible in *R*. (The inverse is given by multiplication with 1/2.)

An ε-quadratic form ψ ∈ *Q*_{ε}(*M*) is called **non-degenerate** if the associated ε-symmetric form (1 + ε*T*)(ψ) is non-degenerate.

### Generalization from *

If the * is trivial, then ε = ±1, and "away from 2" means that 2 is invertible: 1/2 ∈ *R*.

More generally, one can take for ε ∈ *R* any element such that ε*ε =1. ε = ±1 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 *, ε-symmetric forms are equivalent to ε-quadratic forms if there is an element λ ∈ *R* such that λ* + λ = 1. If * is trivial, this is equivalent to 2λ = 1 or λ = 1/2.

For instance, in the ring (the integral lattice for the quadratic form 2*x*^{2}-2*x*+1), with complex conjugation, is such an element, though 1/2 ∉ *R*.

## Intuition

In terms of matrices, (we take *V* to be 2-dimensional):

- matrices correspond to bilinear forms
- the subspace of symmetric matrices correspond to symmetric forms
- the subspace of (−1)-symmetric matrices correspond to symplectic forms
- the bilinear form yields the quadratic form

- which is a quotient map with kernel .

### Refinements

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* = 2*B(v,w)* 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.

## Examples

An easy example for an ε-quadratic form is the **standard hyperbolic ε-quadratic form** . (Here, *R** := Hom_{R}(*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.

For the field of two elements *R* = **F**_{2} 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**_{2} is an **F**_{2}-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.

### Manifolds

Template:Rellink
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. In the case of singly even dimension this is skew-symmetric, while for doubly even dimension 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 ε-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 ε-symmetric form can be refined to an ε-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 Σ embedded in **R**^{3}, the middle homology group *H*_{1}(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 ⊂ **R**^{3}, 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. (This form has Arf invariant 0, and thus this embedded torus has Kervaire invariant 0.)

## Applications

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

## References

- ↑ Foundations of algebraic surgery, by Andrew Ranicki, p. 6