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{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.

Contents

• 
  1 Definition
  
  
  
  • 
    1.1 Symmetrization
    
  
  
  • 
    1.2 Generalization from *
    
  
  
  
  


• 
  2 Intuition
  
  
  
  • 
    2.1 Refinements
    
  
  
  
  


• 
  3 Examples
  
  
  
  • 
    3.1 Manifolds
    
  
  
  
  


• 
  4 Applications
  


• 
  5 References
  

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→Qϵ(M)→B(M)⟶1−ϵTB(M)→Qϵ(M)→0{displaystyle 0to Q^{epsilon }(M)to B(M){stackrel {1-epsilon T}{longrightarrow }}B(M)to Q_{epsilon }(M)to 0}

As kernel and cokernel,

Qϵ(M):=ker(1−ϵT){displaystyle Q^{epsilon }(M):={mbox{ker}},(1-epsilon T)}

Qϵ(M):=coker(1−ϵT){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).

Composition of the inclusion and quotient maps (but not 1 − εT) as Qϵ(M)→B(M)→Qϵ(M){displaystyle 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+ϵT)B(M)<Qϵ(M){displaystyle (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=Z[1+i2]{displaystyle R=mathbf {Z} left[textstyle {frac {1+i}{2}}right]} (the integral lattice for the quadratic form 2x² − 2x+1), with complex conjugation, λ=1±i2{displaystyle lambda =textstyle {frac {1pm 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 (abcd){displaystyle {begin{pmatrix}a&b\c&dend{pmatrix}}} correspond to bilinear forms


• the subspace of symmetric matrices (abbc){displaystyle {begin{pmatrix}a&b\b&cend{pmatrix}}} correspond to symmetric forms


• the subspace of (−1)-symmetric matrices (0b−b0){displaystyle {begin{pmatrix}0&b\-b&0end{pmatrix}}} correspond to symplectic forms
  


• the bilinear form (abcd){displaystyle {begin{pmatrix}a&b\c&dend{pmatrix}}} yields the quadratic form

ax2+bxy+cyx+dy2=ax2+(b+c)xy+dy2{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 ex2+fxy+gy2{displaystyle ex^{2}+fxy+gy^{2}}
  

to (2eff2g){displaystyle {begin{pmatrix}2e&f\f&2gend{pmatrix}}}, for example by lifting to (ef0g){displaystyle {begin{pmatrix}e&f\0&gend{pmatrix}}} and then adding to transpose. Mapping back to quadratic forms yields double the original: 2ex2+2fxy+2gy2=2(ex2+fxy+gy2){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 (azz¯c){displaystyle {begin{pmatrix}a&z\{bar {z}}&cend{pmatrix}}}
  


• the subspace of skew-symmetric matrices are the skew-Hermitian matrices (biz−z¯di){displaystyle {begin{pmatrix}bi&z\-{bar {z}}&diend{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 v2=Q(v){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 Hϵ(R)∈Qϵ(R⊕R∗){displaystyle H_{epsilon }(R)in Q_{epsilon }(Roplus R^{*})}. (Here, R* := Hom_R(R,R) denotes the dual of the R-module R.) It is given by the bilinear form ((v1,f1),(v2,f2))↦f2(v1){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

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,{displaystyle 4k+2,} this is skew-symmetric, while for doubly even dimension 4k,{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 S2k×S2k{displaystyle S^{2k}times S^{2k}} and S2k+1×S2k+1{displaystyle S^{2k+1}times S^{2k+1}} respectively give the symmetric form (0110){displaystyle left({begin{smallmatrix}0&1\1&0end{smallmatrix}}right)} and skew-symmetric form (01−10).{displaystyle left({begin{smallmatrix}0&1\-1&0end{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 π1s{displaystyle pi _{1}^{s}}.

In the standard embedding of the torus, a (1,1) curve self-links, thus Q(1,1)=1{displaystyle Q(1,1)=1}.

For the standard embedded torus, the skew-symmetric form is given by (01−10){displaystyle {begin{pmatrix}0&1\-1&0end{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

1. 
   ^ Ranicki, Andrew (2001). "Foundations of algebraic surgery". arXiv:math/0111315..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
   

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