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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 ${\displaystyle (-)^{n}}$-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)*. Let ε = ±1; then εT is also an involution. Define the ε-symmetric forms as the invariants of εT, and the ε-quadratic forms are the coinvariants.

As an exact sequence,

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

As kernel (algebra) and cokernel,

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

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

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).

We obtain a homomorphism (1 + εT): Q_ε(M) → Q^ε(M) which is bijective if 2 is invertible in R. (The inverse is 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.

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, ${\displaystyle \lambda =\textstyle {\frac {1+i}{2}}}$ is such an element, though 1/2 ∉ R.

Intuition

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

• 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

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

which is a quotient map with kernel ${\displaystyle {\begin{pmatrix}0&b\\-b&0\end{pmatrix}}}$.

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 ${\displaystyle v^{2}=Q(v)}$. 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.

Examples

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

Manifolds

47 year-old Podiatrist Hyslop from Alert Bay, has lots of hobbies and interests that include fencing, property developers in condo new launch singapore and handball. Just had a family trip to Monasteries of Haghpat and Sanahin. 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 ${\displaystyle 4k+2,}$ this is skew-symmetric, while for doubly even dimension ${\displaystyle 4k,}$ 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 ${\displaystyle S^{2k}\times S^{2k}}$ and ${\displaystyle S^{2k+1}\times S^{2k+1}}$ 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 ${\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$ (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

Summary

restart; with(plots);

B := (a1, a2, a3) -> (GAMMA(1.0*a1) * GAMMA(1.0*a2) * GAMMA(1.0*a3)) / GAMMA(1.0*a1+1.0*a2+1.0*a3);

f := (x1, x2, a1, a2, a3) ->  (x1^(a1-1)) * (x2^(a2-1)) * ( (1-x1-x2)^(a3-1)) /B(a1,a2,a3) ;

animate ( plot3d,  [eval(log(f(x1, x2, a1, a2, a3)), {a1=a, a2=a, a3=a}), 
x1=0.00..1, x2=0.00..1, axes=BOXED, grid=[25,25],gridstyle=triangular,orientation=[-135, 60],
shading=zhue, contours=20, style=surfacecontour, view=-3..2 ], a=0.1..1.9, frames=100);

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