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 $(-)^{n}$ -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 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,

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

As kernel and cokernel,

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

Q_{\epsilon }(M):={\mbox{coker}}\,(1-\epsilon 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 $Q^{\epsilon }(M)\to B(M)\to Q_{\epsilon }(M)$ 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 "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, $(1+\epsilon T)B(M)<Q^{\epsilon }(M)$ . 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.

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, 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 2x² − 2x+1), with complex conjugation, $\lambda =\textstyle {{\frac {1\pm i}{2}}}$ are two such elements, though 1/2 ∉ R.

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 * 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 ${\begin{pmatrix}bi&z\\-z&di\end{pmatrix}}$

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: vw + wv = 2B(v,w) 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 $H_{\epsilon }(R)\in Q_{\epsilon }(R\oplus R^{*})$ . (Here, R* := Hom_R(R,R) 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 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.

Manifolds

Further information: Intersection product

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 $S^{{2k}}\times S^{{2k}}$ and $S^{{2k+1}}\times S^{{2k+1}}$ 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 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 $\pi _{1}^{s}$ .

In the standard embedding of the torus, a (1,1) curve self-links, thus

Q(1,1)=1

For the standard embedded torus, the skew-symmetric form is given by ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ (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

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