*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 [math]\displaystyle{ (-)^n }[/math]-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*)*. Since multiplication by −1 is also an involution and commutes with linear maps, −*T* is also an involution. Thus we can write *ε* = ±1 and *εT* is an involution, either *T* or −*T* (ε can be more general than ±1; see below). Define the ** ε-symmetric forms** as the invariants of

*εT*, and the

**are the coinvariants.**

*ε*-quadratic formsAs an exact sequence,

- [math]\displaystyle{ 0 \to Q^\varepsilon(M) \to B(M) \stackrel{1-\varepsilon T}{\longrightarrow} B(M) \to Q_\varepsilon(M) \to 0 }[/math]

- [math]\displaystyle{ Q^\varepsilon(M) := \mbox{ker}\,(1-\varepsilon T) }[/math]
- [math]\displaystyle{ Q_\varepsilon(M) := \mbox{coker}\,(1-\varepsilon T) }[/math]

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}Composition of the inclusion and quotient maps (but not 1 − *εT*) as [math]\displaystyle{ Q^\varepsilon(M) \to B(M) \to Q_\varepsilon(M) }[/math] yields a map *Q*^{ε}(*M*) → *Q*_{ε}(*M*): every *ε*-symmetric form determines an *ε*-quadratic form.

### Symmetrization

Conversely, one can define a reverse homomorphism "1 + *εT*": *Q*_{ε}(*M*) → *Q*^{ε}(*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*. This is a symmetric form because (1 − *εT*)(1 + *εT*) = 1 − *T*^{2} = 0, so it is in the kernel. More precisely, [math]\displaystyle{ (1 + \varepsilon T)B(M) \lt Q^\varepsilon(M) }[/math]. 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*. Thus every *ε*-quadratic form determines an *ε*-symmetric form.

Composing these two maps either way: *Q*^{ε}(*M*) → *Q*_{ε}(*M*) → *Q*^{ε}(*M*) or *Q*_{ε}(*M*) → *Q*^{ε}(*M*) → *Q*_{ε}(*M*) 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 *ε*-quadratic form *ψ* ∈ *Q*_{ε}(*M*) is called **non-degenerate** if the associated *ε*-symmetric form (1 + *εT*)(*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 2*λ* = 1 or *λ* = 1/2, while if * is non-trivial there can be multiple possible *λ*; for example, over the complex numbers any number with real part 1/2 is such a *λ*.

For instance, in the ring [math]\displaystyle{ R=\mathbf{Z}\left[\textstyle{\frac{1+i}{2}}\right] }[/math] (the integral lattice for the quadratic form 2*x*^{2} − 2*x* + 1), with complex conjugation, [math]\displaystyle{ \lambda=\textstyle{\frac{1\pm i}{2}} }[/math] are two such elements, though 1/2 ∉ *R*.

## Intuition

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

- matrices [math]\displaystyle{ \begin{pmatrix}a & b\\c & d\end{pmatrix} }[/math] correspond to bilinear forms
- the subspace of symmetric matrices [math]\displaystyle{ \begin{pmatrix}a & b\\b & c\end{pmatrix} }[/math] correspond to symmetric forms
- the subspace of (−1)-symmetric matrices [math]\displaystyle{ \begin{pmatrix}0 & b\\-b & 0\end{pmatrix} }[/math] correspond to symplectic forms
- the bilinear form [math]\displaystyle{ \begin{pmatrix}a & b\\c & d\end{pmatrix} }[/math] yields the quadratic form

- [math]\displaystyle{ ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2\, }[/math],

- the map 1 + T from quadratic forms to symmetric forms maps [math]\displaystyle{ ex^2 + fxy + gy^2 }[/math]

to [math]\displaystyle{ \begin{pmatrix}2e & f\\f & 2g\end{pmatrix} }[/math], for example by lifting to [math]\displaystyle{ \begin{pmatrix}e & f\\0 & g\end{pmatrix} }[/math] and then adding to transpose. Mapping back to quadratic forms yields double the original: [math]\displaystyle{ 2ex^2 + 2fxy + 2gy^2 = 2(ex^2 + fxy + gy^2) }[/math].

If [math]\displaystyle{ \bar{\cdot } }[/math] is complex conjugation, then

- the subspace of symmetric matrices are the Hermitian matrices [math]\displaystyle{ \begin{pmatrix}a & z\\ \bar z & c\end{pmatrix} }[/math]
- the subspace of skew-symmetric matrices are the skew-Hermitian matrices [math]\displaystyle{ \begin{pmatrix}bi & z\\ -\bar z & di\end{pmatrix} }[/math]

### 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 [math]\displaystyle{ v^2=Q(v) }[/math]. 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** [math]\displaystyle{ H_\varepsilon(R) \in Q_\varepsilon(R \oplus R^*) }[/math]. (Here,

*R** := Hom

_{R}(

*R*,

*R*) denotes the dual of the

*R*-module

*R*.) It is given by the bilinear form [math]\displaystyle{ ((v_1,f_1),(v_2,f_2)) \mapsto f_2(v_1) }[/math]. 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

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 4*k* + 2, this is skew-symmetric, while for doubly even dimension 4*k*, 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 *S*^{2k} × *S*^{2k} and *S*^{2k+1} × *S*^{2k+1} respectively give the symmetric form [math]\displaystyle{ \left(\begin{smallmatrix} 0 & 1\\ 1 & 0\end{smallmatrix}\right) }[/math] and skew-symmetric form [math]\displaystyle{ \left(\begin{smallmatrix} 0 & 1\\ -1 & 0\end{smallmatrix}\right). }[/math] 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 [math]\displaystyle{ \pi^s_1 }[/math].

For the standard embedded torus, the skew-symmetric form is given by [math]\displaystyle{ \left(\begin{smallmatrix}0 & 1\\-1 & 0\end{smallmatrix}\right) }[/math] (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

- ↑ Ranicki, Andrew (2001). "Foundations of algebraic surgery". arXiv:math/0111315.

Original source: https://en.wikipedia.org/wiki/ Ε-quadratic form.
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