Ε-quadratic form

ε-symmetric forms and ε-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 −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) := \mbox{ker}\,(1-\varepsilon T) :Q_\varepsilon(M) := \mbox{coker}\,(1-\varepsilon T) The notation Qε(M), Qε(M) follows the standard notation MG, MG 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 Qε(M) → Qε(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) . 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=\mathbf{Z}\left[\textstyle{\frac{1+i}{2}}\right] (the integral lattice for the quadratic form ), with complex conjugation, \lambda=\textstyle{\frac{1\pm i}{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 form ::ax^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}2e & f\\f & 2g\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: 2ex^2 + 2fxy + 2gy^2 = 2(ex^2 + fxy + gy^2). If \bar{\cdot } is complex conjugation, then • the subspace of symmetric matrices are the Hermitian matrices \begin{pmatrix}a & z\\ \bar z & c\end{pmatrix} • the subspace of skew-symmetric matrices are the skew-Hermitian matrices \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 \oplus 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. 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 , 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 \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==