Ε-quadratic form explained

In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to

• -ring

s;, accordingly for symmetric or skew-symmetric. They are also called

( - ) ^{n} -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 be the "conjugate transpose" involution . Since multiplication by −1 is also an involution and commutes with linear maps, −T is also an involution. Thus we can write 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 ε-quadratic forms are the coinvariants.

As an exact sequence,

0 \to Q ^{\varepsilon(M)} \to B(M) \stackrel{ 1- \varepsilon T}{\longrightarrow} B(M) \to Q _{\varepsilon(M)} \to 0 As kernel and cokernel,Q ^{\varepsilon(M)} : = ker ( 1- \varepsilon T) Q _{\varepsilon(M)} : = coker ( 1- \varepsilon T)

The notation Q^ε(M), Q_ε(M) follows the standard notation M^G, M_G 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

Q ^{\varepsilon(M)} \to B(M) \to Q _{\varepsilon(M)} 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, 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,

( 1 + \varepsilon T)B(M) < Q ^{\varepsilon(M)} . 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 ε-quadratic form determines an ε-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 ε-quadratic form is called non-degenerate if the associated ε-symmetric form is non-degenerate.

Generalization from *

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 *, ε-symmetric forms are equivalent to ε-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 λ; for example, over the complex numbers any number with real part 1/2 is such a λ.

For instance, in the ring

R = Z \left[ style{ (1+i)/(2) }\right] (the integral lattice for the quadratic form), with complex conjugation, λ= style{ (1\pmi)/(2) } are two such elements, though .

Intuition

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

• matrices

\begin{pmatrix}a & b\\c & d\end{pmatrix} correspond to bilinear forms

• the subspace of symmetric matrices

\begin{pmatrix}a & b\\b & c\end{pmatrix} correspond to symmetric forms

• the subspace of (-1)-symmetric matrices

\begin{pmatrix} 0 & b\\ - b & 0 \end{pmatrix} correspond to symplectic forms

• the bilinear form

\begin{pmatrix}a & b\\c & d\end{pmatrix} yields the quadratic formax ^{2} + bxy + cyx + dy ^{2} = ax ^{2} + (b + c)xy + dy ^{2} ,

• the map 1 + T from quadratic forms to symmetric forms maps

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

If

\bar{ ⋅ } is complex conjugation, then\begin{pmatrix}a & z\ \bar z & c\end{pmatrix} \begin{pmatrix}bi & z\ - \bar z & di\end{pmatrix}

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: and

v ^{2=Q(v)} . 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

H _{\varepsilon(R)} \in Q _{\varepsilon(R} ⊕ R ^{*)} . (Here, denotes the dual of the R-module R.) It is given by the bilinear form ((v _{1,f} _{1),(v} _{2,f} _{2))} \mapsto f _{2(v} _{1)} . The standard hyperbolic ε-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 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.

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, 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 ε-symmetry. The simplest cases are for the product of spheres, where the product and respectively give the symmetric form

\left(\begin{smallmatrix} 0 & 1 \ 1 & 0 \end{smallmatrix}\right) and skew-symmetric form \left(\begin{smallmatrix} 0 & 1 \ -1 & 0 \end{smallmatrix}\right). 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.

\pi ^{s} _{1} .

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

\left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right) (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.)

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

1. math/0111315. Ranicki. Andrew. Foundations of algebraic surgery. 2001.