In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Arf invariant of a nonsingular
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
over a
field of
characteristic 2 was defined by
Turkish mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
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 F
2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F
2 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 Arf proved it for an arbitrary
perfect field.
The Arf invariant is particularly
applied in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originate ...
, where it is primarily used to define an invariant of -dimensional manifolds (
singly even-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s: 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 4''k''-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 2''k''-dimensional manifolds.
Definitions
The Arf invariant is defined for a
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
''q'' over a field ''K'' of characteristic 2 such that ''q'' is nonsingular, in the sense that the associated bilinear form
is
nondegenerate. The form
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
with
in ''K''. The Arf invariant is defined to be the product
. If the form
is equivalent to
, then the products
and
differ by an element of the form
with
in ''K''. These elements form an additive subgroup ''U'' of ''K''. Hence the coset of
modulo ''U'' is an invariant of
, which means that it is not changed when
is replaced by an equivalent form.
Every nonsingular quadratic form
over ''K'' is equivalent to a direct sum
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(
) is defined to be the sum of the Arf invariants of the
. By definition, this is a coset of ''K'' modulo ''U''. Arf showed that indeed
does not change if
is replaced by an equivalent quadratic 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.
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 ''H''
1(''K'', F
2). 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 ''H''
1(''F'', F
2) 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 F
2. In this case, the subgroup ''U'' above is zero, and hence the Arf invariant is an element of the base field F
2; it is either 0 or 1.
If the field ''K'' of characteristic 2 is not perfect (that is, ''K'' is different from its subfield ''K''
2 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 2">'K'': ''K''2is at most 2, then every quadratic form over ''K'' 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 F
2, the 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 has called the Arf invariant the ''democratic invariant'' because it is the value which is assumed most often by the quadratic form.
[Browder, Proposition III.1.8] Another characterization: ''q'' has Arf invariant 0 if and only if the underlying 2''k''-dimensional vector space over the field F
2 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 2''k'' 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
Let ''M'' be a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
,
connected 2''k''-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
with a boundary
such that the induced morphisms in
-coefficient homology
:
are both zero (e.g. if
is closed). The
intersection form
:
is non-singular. (Topologists usually write F
2 as
.) A
quadratic refinement
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''.
Mathematics ...
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
:
Examples
* Let
be a compact, connected,
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
2-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, i.e. a
surface, of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
such that the boundary
is either empty or is connected.
Embed
Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
* Embedding, in mathematics, one instance of some mathematical object contained within another instance
** Graph embedding
* Embedded generation, a distributed ...
in
, where
. Choose a framing of ''M'', that is a trivialization of the normal (''m'' − 2)-plane
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
. (This is possible for
, so is certainly possible for
). Choose a
symplectic basis for
. Each basis element is represented by an embedded circle
. The normal (''m'' − 1)-plane
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
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 structure on our surface.)
Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed
cobordism group
, which is generated by the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
with the Lie group framing. The isomorphism here is via the
Pontrjagin-Thom construction.
* Let
be 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 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
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 -dimensional
-coefficient homology
of a
framed -dimensional manifold ''M'' has a quadratic refinement
, which depends on the framing. For
and
represented by an
embedding the value
is 0 or 1, according as to the normal bundle of
is trivial or not. The
Kervaire invariant of the framed -dimensional manifold ''M'' is the Arf invariant of the quadratic refinement
on
. The Kervaire invariant is a homomorphism
on the -dimensional stable homotopy group of spheres. The Kervaire invariant can also be defined for a -dimensional manifold ''M'' which is framed except at a point.
* In
surgery theory, 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 spheres by
Michel Kervaire and
John Milnor, and more generally in the classification of manifolds by
surgery theory.
William Browder defined
using functional
Steenrod square In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, c ...
s, and
C. T. C. Wall defined
using framed
immersions. 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-group .
See also
*
de Rham invariant, a mod 2 invariant of
-dimensional manifolds
Notes
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.
*
*
Glen Bredon
Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000, in North Fork, California) was an American mathematician who worked in the area of topology.
Education and career
Bredon received a bachelor's degree from Stanford Univer ...
: ''Topology and Geometry'', 1993, .
*
* J. Scott Carter: ''How Surfaces Intersect in Space'', Series on Knots and Everything, 1993, .
*
*
*
*
W. B. Raymond Lickorish, ''An Introduction to Knot Theory'', Graduate Texts in Mathematics, Springer, 1997,
*
*
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely d ...
, ''Smooth manifolds and their applications in homotopy theory'' American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959)
Further reading
*
*
{{DEFAULTSORT:Arf Invariant
Quadratic forms
Surgery theory