Arf invariant

Arf and a formula for the Arf invariant appear on the reverse side of the 2009 Turkish 10 Lira note

In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2.

In the special case of the 2-element field F₂ the Arf invariant can be described as the element of F₂ that occurs most often among the values of the form. Two nonsingular quadratic forms over F₂ are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to Leonard Dickson (1901), even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. An assessment of Arf's results in the framework of the theory of quadratic forms can be found in.^[1]

The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of (4k + 2)-dimensional manifolds (singly even-dimensional manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4k-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.

Contents

• 
  1 Definitions
  


• 
  2 Arf's main results
  


• 
  3 Quadratic forms over F₂
  


• 
  4 The Arf invariant in topology
  
  
  
  • 
    4.1 Examples
    
  
  
  
  


• 
  5 See also
  


• 
  6 Notes
  


• 
  7 References
  


• 
  8 Further reading
  

Definitions

The Arf invariant is defined for a quadratic form q over a field K of characteristic 2 such that q is nonsingular, in the sense that the associated bilinear form b(u,v)=q(u+v)−q(u)−q(v){displaystyle b(u,v)=q(u+v)-q(u)-q(v)} is nondegenerate. The form b{displaystyle b} is alternating since K has characteristic 2; it follows that a nonsingular quadratic form in characteristic 2 must have even dimension. Any binary (2-dimensional) nonsingular quadratic form over K is equivalent to a form q(x,y)=ax2+xy+by2{displaystyle q(x,y)=ax^{2}+xy+by^{2}} with a,b{displaystyle a,b} in K. The Arf invariant is defined to be the product ab{displaystyle ab}. If the form q′(x,y)=a′x2+xy+b′y2{displaystyle q'(x,y)=a'x^{2}+xy+b'y^{2}} is equivalent to q(x,y){displaystyle q(x,y)}, then the products ab{displaystyle ab} and a′b′{displaystyle a'b'} differ by an element of the form u2+u{displaystyle u^{2}+u} with u{displaystyle u} in K. These elements form an additive subgroup U of K. Hence the coset of ab{displaystyle ab} modulo U is an invariant of q{displaystyle q}, which means that it is not changed when q{displaystyle q} is replaced by an equivalent form.

Every nonsingular quadratic form q{displaystyle q} over K is equivalent to a direct sum q=q1+…+qr{displaystyle q=q_{1}+ldots +q_{r}} of nonsingular binary forms. This was shown by Arf, but it had been earlier observed by Dickson in the case of finite fields of characteristic 2. The Arf invariant Arf(q{displaystyle q}) is defined to be the sum of the Arf invariants of the qi{displaystyle q_{i}}. By definition, this is a coset of K modulo U. Arf^[2] showed that indeed Arf(q{displaystyle q}) does not change if q{displaystyle q} is replaced by an equivalent quadratic form, which is to say that it is an invariant of q{displaystyle q}.

The Arf invariant is additive; in other words, the Arf invariant of an orthogonal sum of two quadratic forms is the sum of their Arf invariants.

For a field K of characteristic 2, Artin-Schreier theory identifies the quotient group of K by the subgroup U above with the Galois cohomology group H¹(K, F₂). In other words, the nonzero elements of K/U are in one-to-one correspondence with the separable quadratic extension fields of K. So the Arf invariant of a nonsingular quadratic form over K is either zero or it describes a separable quadratic extension field of K. This is analogous to the discriminant of a nonsingular quadratic form over a field F of characteristic not 2. In that case, the discriminant takes values in F^*/(F^*)², which can be identified with H¹(F, F₂) by Kummer theory.

Arf's main results

If the field K is perfect, then every nonsingular quadratic form over K is uniquely determined (up to equivalence) by its dimension and its Arf invariant. In particular, this holds over the field F₂. In this case, the subgroup U above is zero, and hence the Arf invariant is an element of the base field F₂; it is either 0 or 1.

If the field K of characteristic 2 is not perfect (that is, K is different from its subfield K² of squares), then the Clifford algebra is another important invariant of a quadratic form. A corrected version of Arf's original statement is that if the degree [K: K²] is at most 2, then every quadratic form over K is completely characterized by its dimension, its Arf invariant and its Clifford algebra.^[3] Examples of such fields are function fields (or power series fields) of one variable over perfect base fields.

Quadratic forms over F₂

Over F₂, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form xy{displaystyle xy}, and it is 1 if the form is a direct sum of x2+xy+y2{displaystyle x^{2}+xy+y^{2}} with a number of copies of xy{displaystyle xy}.

William Browder has called the Arf invariant the democratic invariant^[4] because it is the value which is assumed most often by the quadratic form.^[5] Another characterization: q has Arf invariant 0 if and only if the underlying 2k-dimensional vector space over the field F₂ has a k-dimensional subspace on which q is identically 0 – that is, a totally isotropic subspace of half the dimension. In other words, a nonsingular quadratic form of dimension 2k has Arf invariant 0 if and only if its isotropy index is k (this is the maximum dimension of a totally isotropic subspace of a nonsingular form).

The Arf invariant in topology

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Let M be a compact, connected 2k-dimensional manifold with a boundary ∂M{displaystyle partial M}
such that the induced morphisms in Z2{displaystyle mathbb {Z} _{2}}-coefficient homology

Hk(M,∂M;Z2)→Hk−1(∂M;Z2),Hk(∂M;Z2)→Hk(M;Z2){displaystyle H_{k}(M,partial M;mathbb {Z} _{2})to H_{k-1}(partial M;mathbb {Z} _{2}),quad H_{k}(partial M;mathbb {Z} _{2})to H_{k}(M;mathbb {Z} _{2})}

are both zero (e.g. if M{displaystyle M} is closed). The intersection form

λ:Hk(M;Z2)×Hk(M;Z2)→Z2{displaystyle lambda :H_{k}(M;mathbb {Z} _{2})times H_{k}(M;mathbb {Z} _{2})to mathbb {Z} _{2}}

is non-singular. (Topologists usually write F₂ as Z2{displaystyle mathbb {Z} _{2}}.) A quadratic refinement for λ{displaystyle lambda } is a function μ:Hk(M;Z2)→Z2{displaystyle mu :H_{k}(M;mathbb {Z} _{2})to mathbb {Z} _{2}} which satisfies

μ(x+y)+μ(x)+μ(y)≡λ(x,y)(mod2)∀x,y∈Hk(M;Z2){displaystyle mu (x+y)+mu (x)+mu (y)equiv lambda (x,y){pmod {2}};forall ,x,yin H_{k}(M;mathbb {Z} _{2})}

Let {x,y}{displaystyle {x,y}} be any 2-dimensional subspace of Hk(M;Z2){displaystyle H_{k}(M;mathbb {Z} _{2})}, such that λ(x,y)=1{displaystyle lambda (x,y)=1}. Then there are two possibilities. Either all of μ(x+y),μ(x),μ(y){displaystyle mu (x+y),mu (x),mu (y)} are 1, or else just one of them is 1, and the other two are 0. Call the first case H1,1{displaystyle H^{1,1}}, and the second case H0,0{displaystyle H^{0,0}}. Since every form is equivalent to a symplectic form, we can always find subspaces {x,y}{displaystyle {x,y}} with x and y being λ{displaystyle lambda }-dual. We can therefore split Hk(M;Z2){displaystyle H_{k}(M;mathbb {Z} _{2})} into a direct sum of subspaces isomorphic to either H0,0{displaystyle H^{0,0}} or H1,1{displaystyle H^{1,1}}. Furthermore, by a clever change of basis, H0,0⊕H0,0≅H1,1⊕H1,1.{displaystyle H^{0,0}oplus H^{0,0}cong H^{1,1}oplus H^{1,1}.} We therefore define the Arf invariant

Arf(Hk(M;Z2);μ){displaystyle Arf(H_{k}(M;mathbb {Z} _{2});mu )} = (number of copies of H1,1{displaystyle H^{1,1}} in a decomposition Mod 2) ∈Z2{displaystyle in mathbb {Z} _{2}}.

Examples

• Let M{displaystyle M} be a compact, connected, oriented 2-dimensional manifold, i.e. a surface, of genus g{displaystyle g} such that the boundary ∂M{displaystyle partial M} is either empty or is connected. Embed M{displaystyle M} in Sm{displaystyle S^{m}}, where m≥4{displaystyle mgeq 4}. Choose a framing of M, that is a trivialization of the normal (m-2)-plane vector bundle. (This is possible for m=3{displaystyle m=3}, so is certainly possible for m≥4{displaystyle mgeq 4}). Choose a symplectic basis x1,x2,…,x2g−1,x2g{displaystyle x_{1},x_{2},ldots ,x_{2g-1},x_{2g}} for H1(M)=Z2g{displaystyle H_{1}(M)=mathbb {Z} ^{2g}}. Each basis element is represented by an embedded circle xi:S1⊂M{displaystyle x_{i}:S^{1}subset M}. The normal (m-1)-plane vector bundle of S1⊂M⊂Sm{displaystyle S^{1}subset Msubset S^{m}} has two trivializations, one determined by a standard framing of a standard embedding S1⊂Sm{displaystyle S^{1}subset S^{m}} and one determined by the framing of M, which differ by a map S1→SO(m−1){displaystyle S^{1}to SO(m-1)} i.e. an element of π1(SO(m−1))≅Z2{displaystyle pi _{1}(SO(m-1))cong mathbb {Z} _{2}} for m≥4{displaystyle mgeq 4}. This can also be viewed as the framed cobordism class of S1{displaystyle S^{1}} with this framing in the 1-dimensional framed cobordism group Ω1framed≅πm(Sm−1)(m≥4)≅Z2{displaystyle Omega _{1}^{framed}cong pi _{m}(S^{m-1}),(mgeq 4)cong mathbb {Z} _{2}}, which is generated by the circle S1{displaystyle S^{1}} with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction. Define μ(x)∈Z2{displaystyle mu (x)in mathbb {Z} _{2}} to be this element. The Arf invariant of the framed surface is now defined

Φ(M)=Arf(H1(M,∂M;Z2);μ)∈Z2{displaystyle Phi (M)=Arf(H_{1}(M,partial M;mathbb {Z} _{2});mu )in mathbb {Z} _{2}}

Note that π1(SO(2))≅Z,{displaystyle pi _{1}(SO(2))cong mathbb {Z} ,} so we had to stabilise, taking m{displaystyle m} to be at least 4, in order to get an element of Z2{displaystyle mathbb {Z} _{2}}. The case m=3{displaystyle m=3} is also admissible as long as we take the residue modulo 2 of the framing.

• The Arf invariant Φ(M){displaystyle Phi (M)} of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing. This is because H1,1{displaystyle H^{1,1}} does not bound. H1,1{displaystyle H^{1,1}} represents a torus T2{displaystyle T^{2}} with a trivialisation on both generators of H1(T2;Z2){displaystyle H_{1}(T^{2};mathbb {Z} _{2})} which twists an odd number of times. The key fact is that up to homotopy there are two choices of trivialisation of a trivial 3-plane bundle over a circle, corresponding to the two elements of π1(SO(3)){displaystyle pi _{1}(SO(3))}. An odd number of twists, known as the Lie group framing, does not extend across a disc, whilst an even number of twists does. (Note that this corresponds to putting a spin structure on our surface.) Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed cobordism group Ω2framed≅πm(Sm−2)(m≥4)≅Z2{displaystyle Omega _{2}^{framed}cong pi _{m}(S^{m-2}),(mgeq 4)cong mathbb {Z} _{2}}, which is generated by the torus T2{displaystyle T^{2}} with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.

• Let (M2,∂M)⊂S3{displaystyle (M^{2},partial M)subset S^{3}} be a Seifert surface for a knot, ∂M=K:S1↪S3{displaystyle partial M=K:S^{1}hookrightarrow S^{3}}, which can be represented as a disc D2{displaystyle D^{2}} with bands attached. The bands will typically be twisted and knotted. Each band corresponds to a generator x∈H1(M;Z2){displaystyle xin H_{1}(M;mathbb {Z} _{2})}. x{displaystyle x} can be represented by a circle which traverses one of the bands. Define μ(x){displaystyle mu (x)} to be the number of full twists in the band modulo 2. Suppose we let S3{displaystyle S^{3}} bound D4{displaystyle D^{4}}, and push the Seifert surface M{displaystyle M} into D4{displaystyle D^{4}}, so that its boundary still resides in S3{displaystyle S^{3}}. Around any generator x∈H1(M,∂M){displaystyle xin H_{1}(M,partial M)}, we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding M↪D4{displaystyle Mhookrightarrow D^{4}} for 2 of the sections required. For the third, choose a section which remains normal to x{displaystyle x}, whilst always remaining tangent to M{displaystyle M}. This trivialisation again determines an element of π1(SO(3)){displaystyle pi _{1}(SO(3))}, which we take to be μ(x){displaystyle mu (x)}. Note that this coincides with the previous definition of μ{displaystyle mu }.

• The Arf invariant of a knot is defined via its Seifert surface. It is independent of the choice of Seifert surface (The basic surgery change of S-equivalence, adding/removing a tube, adds/deletes a H0,0{displaystyle H^{0,0}} direct summand), and so is a knot invariant. It is additive under connected sum, and vanishes on slice knots, so is a knot concordance invariant.

• The intersection form on the (2k + 1)-dimensional Z2{displaystyle mathbb {Z} _{2}}-coefficient homology H2k+1(M;Z2){displaystyle H_{2k+1}(M;mathbb {Z} _{2})} of a framed (4k + 2)-dimensional manifold M has a quadratic refinement μ{displaystyle mu }, which depends on the framing. For k≠0,1,3{displaystyle kneq 0,1,3} and x∈H2k+1(M;Z2){displaystyle xin H_{2k+1}(M;mathbb {Z} _{2})} represented by an embedding x:S2k+1⊂M{displaystyle x:S^{2k+1}subset M} the value μ(x)∈Z2{displaystyle mu (x)in mathbb {Z} _{2}} is 0 or 1, according as to the normal bundle of x{displaystyle x} is trivial or not. The Kervaire invariant of the framed (4k + 2)-dimensional manifold M is the Arf invariant of the quadratic refinement μ{displaystyle mu } on H2k+1(M;Z2){displaystyle H_{2k+1}(M;mathbb {Z} _{2})}. The Kervaire invariant is a homomorphism π4k+2S→Z2{displaystyle pi _{4k+2}^{S}to mathbb {Z} _{2}} on the (4k + 2)-dimensional stable homotopy group of spheres. The Kervaire invariant can also be defined for a (4k + 2)-dimensional manifold M which is framed except at a point.

• In surgery theory, for any 4k+2{displaystyle 4k+2}-dimensional normal map (f,b):M→X{displaystyle (f,b):Mto X} there is defined a nonsingular quadratic form (K2k+1(M;Z2),μ){displaystyle (K_{2k+1}(M;mathbb {Z} _{2}),mu )} on the Z2{displaystyle mathbb {Z} _{2}}-coefficient homology kernel

K2k+1(M;Z2)=ker(f∗:H2k+1(M;Z2)→H2k+1(X;Z2)){displaystyle K_{2k+1}(M;mathbb {Z} _{2})=ker(f_{*}:H_{2k+1}(M;mathbb {Z} _{2})to H_{2k+1}(X;mathbb {Z} _{2}))}

refining the homological intersection form λ{displaystyle lambda }. The Arf invariant of this form is the Kervaire invariant of (f,b). In the special case X=S4k+2{displaystyle X=S^{4k+2}} this is the Kervaire invariant of M. The Kervaire invariant features in the classification of exotic spheres by Kervaire and Milnor, and more generally in the classification of manifolds by surgery theory. Browder defined μ{displaystyle mu } using functional Steenrod squares, and Wall defined μ{displaystyle mu } using framed immersions. The quadratic enhancement μ(x){displaystyle mu (x)} crucially provides more information than λ(x,x){displaystyle lambda (x,x)} : it is possible to kill x by surgery if and only if μ(x)=0{displaystyle mu (x)=0}. The corresponding Kervaire invariant detects the surgery obstruction of (f,b){displaystyle (f,b)} in the L-group L4k+2(Z)=Z2{displaystyle L_{4k+2}(mathbb {Z} )=mathbb {Z} _{2}}.

See also

• 
  de Rham invariant, a mod 2 invariant of (4k+1){displaystyle (4k+1)}-dimensional manifolds

Notes

1. 
   ^ Falko Lorenz and Peter Roquette. Cahit Arf and his invariant.
   


2. 
   ^ Arf (1941)
   


3. 
   ^ Falko Lorenz and Peter Roquette. Cahit Arf and his invariant. Section 9.
   


4. 
   ^ Martino and Priddy, p.61
   


5. 
   ^ Browder, Proposition III.1.8
   

References

• See Lickorish (1997) for the relation between the Arf invariant and the Jones polynomial.


• See Chapter 3 of Carter's book for another equivalent definition of the Arf invariant in terms of self-intersections of discs in 4-dimensional space.


• 
  Arf, Cahit (1941), "Untersuchungen über quadratische Formen in Körpern der Charakteristik 2, I", J. Reine Angew. Math., 183: 148–167.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}
  


• 
  Glen Bredon: Topology and Geometry, 1993,
  ISBN 0-387-97926-3.


• 
  Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
  


• J. Scott Carter: How Surfaces Intersect in Space, Series on Knots and Everything, 1993,
  ISBN 981-02-1050-7.


• 
  A.V. Chernavskii (2001) [1994], "Arf invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  Dickson, Leonard Eugene (1901), Linear groups: With an exposition of the Galois field theory, New York: Dover Publications, MR 0104735
  


• 
  Kirby, Robion (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag, doi:10.1007/BFb0089031, ISBN 0-387-51148-2, MR 1001966
  


• 
  W. B. Raymond Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Springer, 1997,
  ISBN 0-387-98254-X
  


• 
  Martino, J.; Priddy, S. (2003), "Group Extensions And Automorphism Group Rings", Homology, Homotopy and Applications, 5 (1): 53–70, arXiv:0711.1536, doi:10.4310/hha.2003.v5.n1.a3
  


• 
  Lev Pontryagin, Smooth manifolds and their applications in homotopy theory American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959)

Further reading

• 
  Lorenz, Falko; Roquette, Peter (2013), "Cahit Arf and his invariant", Contributions to the history of number theory in the 20th century (PDF), Heritage of European Mathematics, Zürich: European Mathematical Society, pp. 189–222, ISBN 978-3-03719-113-2, MR 2934052, Zbl 1276.11001
  


• 
  Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Berlin: Springer-Verlag, pp. 211–222, doi:10.1007/978-3-642-75401-2, ISBN 3-540-52117-8, MR 1096299, Zbl 0756.11008
  

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