ε-quadratic form

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 ε-quadratic forms are the coinvariants.

As an exact sequence,

${\displaystyle 0\to Q^{\varepsilon }(M)\to B(M){\stackrel {1-\varepsilon T}{\longrightarrow }}B(M)\to Q_{\varepsilon }(M)\to 0}$ 

As kernel and cokernel,

${\displaystyle Q^{\varepsilon }(M):={\mbox{ker}}\,(1-\varepsilon T)}$ 

${\displaystyle Q_{\varepsilon }(M):={\mbox{coker}}\,(1-\varepsilon T)}$ 

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 1 − εT) as ${\displaystyle Q^{\varepsilon }(M)\to B(M)\to Q_{\varepsilon }(M)}$  yields a map Q^ε(M) → Q_ε(M): every ε-symmetric form determines an ε-quadratic form.

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 − T² = 0, so it is in the kernel. More precisely, ${\displaystyle (1+\varepsilon T)B(M)<Q^{\varepsilon }(M)}$ . The map is well-defined by the same equation: choosing a different lift corresponds to adding a multiple of (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.

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, 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 ${\displaystyle R=\mathbf {Z} \left[\textstyle {\frac {1+i}{2}}\right]}$  (the integral lattice for the quadratic form 2x² − 2x + 1), with complex conjugation, ${\displaystyle \lambda =\textstyle {\frac {1\pm i}{2}}}$  are two such elements, though 1/2 ∉ R.

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

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

${\displaystyle ax^{2}+bxy+cyx+dy^{2}=ax^{2}+(b+c)xy+dy^{2}\,}$ ,

• the map 1 + T from quadratic forms to symmetric forms maps ${\displaystyle ex^{2}+fxy+gy^{2}}$ 

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

If ${\displaystyle {\bar {\cdot }}}$  is complex conjugation, then

• the subspace of symmetric matrices are the Hermitian matrices ${\displaystyle {\begin{pmatrix}a&z\\{\bar {z}}&c\end{pmatrix}}}$ 
• the subspace of skew-symmetric matrices are the skew-Hermitian matrices ${\displaystyle {\begin{pmatrix}bi&z\\-{\bar {z}}&di\end{pmatrix}}}$ 

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 ${\displaystyle v^{2}=Q(v)}$ . 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.

An easy example for an ε-quadratic form is the standard hyperbolic ε-quadratic form ${\displaystyle H_{\varepsilon }(R)\in Q_{\varepsilon }(R\oplus R^{*})}$ . (Here, R* := Hom_R(R, R) denotes the dual of the R-module R.) It is given by the bilinear form ${\displaystyle ((v_{1},f_{1}),(v_{2},f_{2}))\mapsto f_{2}(v_{1})}$ . The standard hyperbolic ε-quadratic form is needed for the definition of L-theory.

For the field of two elements R = F₂ 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.

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 4k + 2, 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 S^2k × S^2k and S^2k+1 × S^2k+1 respectively give the symmetric form ${\displaystyle \left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}$  and skew-symmetric form ${\displaystyle \left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right).}$  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.

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 ${\displaystyle \pi _{1}^{s}}$ .

For the standard embedded torus, the skew-symmetric form is given by ${\displaystyle \left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right)}$  (with respect to the standard symplectic basis), and the skew-quadratic refinement is given by xy with respect to this basis: 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.)

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