Arf invariant
Encyclopedia
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Arf invariant of a nonsingular quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
over a field of characteristic 2 was defined by
when he started the systematic study of quadratic forms over arbitrary fields of characteristic
2. The Arf invariant is the substitute, in
characteristic 2, of 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
GF(2)
GF is the Galois field of two elements. It is the smallest finite field.- Definition :The two elements are nearly always called 0 and 1, being the additive and multiplicative identities, respectively...
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 , even for any finite field of characteristic 2, and it follows from Arf's results for an arbitrary perfect field. An assessment of Arf's results in the framework of the theory of quadratic forms can be found in ,
The Arf invariant is particularly applied in geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
, where it is primarily used to define an invariant of (4k+2)-dimensional manifolds (singly even-dimension 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
L-theory
Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...
. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.
Definitions
The Arf invariant belongs to a quadratic formQuadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
over a field K of characteristic 2.
Any binary non-singular
quadratic form over K is equivalent
to a form
with 
in K.The Arf invariant is defined to be the product
.If the form
isequivalent to
, then the products 
and 
differ
by an element of the form
with 
in K. Theseelements form an additive subgroup U of K. Hence the
coset of
modulo U is an invariant of 
, whichmeans that it is not changed when
is replaced byan equivalent form.
Every nonsingular quadratic form
over K is equivalentto a direct sum
of nonsingularbinary forms. This has been shown by Arf but it had
been earlier observed by Dickson in the case of finite
fields of characteristic 2. The Arf invariant Arf(
) isdefined to be the sum of the Arf invariants of the
. By definition, this is a cosetof K modulo U. Arf has shown that indeed Arf(
)does not change if
is replaced by an equivalentquadratic form, which is to say that it is an invariant of
.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.
Arf's main results
If the field K is perfect thenevery 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 U=0 and hence
the Arf invariant is an element of the base field
F2; it is either 0 or 1.
If the field is not perfect then the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
is another important invariant of a quadratic form.
For various fields Arf has shown that every quadratic
form is
completely characterized by its dimension, its Arf
invariant
and its Clifford algebra. Examples of such fields are
function fields (or power series fields) of one
variable over perfect base fields.
Quadratic forms over F2
Over F2the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form
, and it is 1 if the form is a direct sum of 
with a number of copies of 
.William Browder
William Browder (mathematician)
William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
has called the Arf invariant the democratic invariant because it is the value which is assumed most often by the quadratic form. 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; its isotropy index is k (this is the maximum dimension of a totally isotropic subspace of a nonsingular form).
The Arf invariant in topology
Let M be a compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
2k-dimensional manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
with a boundary
such that the induced morphisms in
-coefficient homology
, 
are both zero (e.g. if
is closed). The intersection formIntersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
is non-singular. (Topologists usually write F2 as
.) A quadratic refinement for 
is a function 
which satisfies
Let
be any 2-dimensional subspace of 
, such that 
. Then there are two possibilities. Either all of 
are 1, or else just one of them is 1, and the other two are 0. Call the first case 
, and the second case 
.Since every form is equivalent to a symplectic form, we can always find subspaces
with x and y being 
-dual. We can therefore split 
into a direct sum of subspaces isomorphic to either 
or 
.Furthermore, by a clever change of basis,
.We therefore define the Arf invariant
= (number of copies of 
in a decomposition Mod 2) 
.Examples
- Let
be a compact, connected, oriented 2-dimensional manifoldManifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, i.e. a surfaceSurfaceIn mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
, of genusGenusIn biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
such that the boundary 
is either empty or is connected. Embed 
in 
, where 
. Choose a framing of M, that is a trivialization of the normal (m-2)-plane vector bundleVector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
. (This is possible for
, so is certainly possible for 
). Choose a symplectic basisSymplectic vector spaceIn mathematics, a symplectic vector space is a vector space V equipped with a bilinear form ω : V × V → R that is...
for 
. Each basis element is represented by an embedded circle 
. The normal (m-1)-plane vector bundleVector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
of
has two trivializations, one determined by a standard framing of a standard embedding 
and one determined by the framing of M, which differ by a map 
i.e. an element of 
for 
. This can also be viewed as the framed cobordism class of 
with this framing in the 1-dimensional framed cobordism group 
, which is generated by the circle 
with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.) Define 
to be this element. The Arf invariant of the framed surface is now defined
Note that
, so we had to stabilise, taking 
to be at least 4, in order to get an element of 
. The case 
is also admissible as long as we take the residue modulo 2 of the framing.- The Arf invariant
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 
does not bound. 
represents a torus 
with a trivialisation on both generators of 
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 
. 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 structureSpin structureIn differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
on our surface.) Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed cobordismCobordismIn mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
group
, which is generated by the torusTorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.
- Let
be a Seifert surfaceSeifert surfaceIn mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
for a knot,
, which can be represented as a disc 
with bands attached. The bands will typically be twisted and knotted. Each band corresponds to a generator 
. 
can be represented by a circle which traverses one of the bands. Define 
to be the number of full twists in the band modulo 2. Suppose we let 
bound 
, and push the Seifert surface 
into 
, so that its boundary still resides in 
. Around any generator 
, we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding 
for 2 of the sections required. For the third, choose a section which remains normal to 
, whilst always remaining tangent to 
. This trivialisation again determines an element of 
, which we take to be 
. Note that this coincides with the previous definition of 
.
- The Arf invariant of a knotArf invariant (knot)In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface...
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
direct summand), and so is a knot invariantKnot invariantIn the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...
. It is additive under connected sumConnected sumIn mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...
, and vanishes on slice knots, so is a knot concordance invariant.
- The intersection formIntersection formIntersection form may refer to:*Intersection theory *intersection form...
on the 2k+1-dimensional
-coefficient homology 
of a framed 4k+2-dimensional manifold M has a quadratic refinement 
, which depends on the framing. For 
and 
represented by an embeddingEmbeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
the value 
is 0 or 1, according as to the normal bundle of 
is trivial or not. The Kervaire invariant of the framed 4k+2-dimensional manifold M is the Arf invariant of the quadratic refinement 
on 
. The Kervaire invariant is a homomorphism 
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 theorySurgery theoryIn mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
, for any
-dimensional normal map 
there is defined a nonsingular quadratic form 
on the 
-coefficient homology kernel
refining the homological intersection form 
. The Arf invariant of this form is the Kervaire invariant of (f,b). In the special case 
this is the Kervaire invariant of M. The Kervaire invariant features in the classification of exotic sphereExotic sphereIn differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s by Kervaire and Milnor, and more generally in the classification of manifolds by surgery theorySurgery theoryIn mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
. BrowderWilliam Browder (mathematician)William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
defined
using functional Steenrod squares, and Wall defined 
using framed immersionImmersionImmersion may refer to:* Immersion therapy overcoming fears through confrontation* Baptism by immersion* Immersion Games a developer of video games...
s. The quadratic enhancement
crucially provides more information than 
: it is possible to kill x by surgery if and only if 
. The corresponding Kervaire invariant detects the surgery obstruction of 
in the L-groupL-theoryAlgebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...
.