Arf invariant

From Wikipedia, the free encyclopedia

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 the 2-element field F₂ is 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 Arf invariant. The invariant was essentially known to Dickson (1901) and rediscovered by Cahit Arf (1941). It is used in algebraic topology to define the Arf-Kervaire invariant and the Arf invariant of a knot.

[edit] Structure of quadratic forms over F₂

Up to isomorphism, there are two nonsingular 2-dimensional quadratic forms over F₂, namely $A (x, y) = x 2 + x y + y 2$ and $B (x, y) = x y$ . Here A has three elements of norm 1, and B has one element of norm 1.

Already in 1901 it was known^[1] that every nonsingular quadratic form over F₂ is isomorphic to an orthogonal sum A^m + Bⁿ for some nonnegative integers m and n. Since A + A is isomorphic to B + B, the integers m and n are not uniquely determined by the form, but they are determined modulo 2. The value of m mod 2 is the Arf invariant of the quadratic form.

A quadratic form C of dimension 2k has 2^2k elements. A non-singular C has 2^2k−1 + 2^k−1 elements of norm 1 if its Arf invariant is 1, and 2^2k−1 − 2^k−1 elements of norm 1 if its Arf invariant is 0. William Browder has called the Arf invariant the democratic invariant^[2] because it is the norm of the majority of its elements^[3]. Another characterization: C 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 C is identically 0.

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.

[edit] The Arf invariant in Topology

Let M be a compact, connected 2k-dimensional manifold with a boundary $\partial M$ such that the induced morphisms in $\mathbb{Z}_2$ -coefficient homology

$H_k(M,\partial M;\mathbb{Z}_2) \to H_{k-1}(\partial M;\mathbb{Z}_2)$ , $H_k(\partial M;\mathbb{Z}_2) \to H_k(M;\mathbb{Z}_2)$

are both zero (e.g. if $M$ is closed). The intersection form

$\lambda\colon H_k(M;\mathbb{Z}_2)\times H_k(M;\mathbb{Z}_2)\to \mathbb{Z}_2$

is non-singular. (Topologists usually write F₂ as $\mathbb{Z}_2$ .) A quadratic refinement for $λ$ is a function $\mu \colon H_k(M;\mathbb{Z}_2) \to \mathbb{Z}_2$ which satisfies

$\mu(x+y) + \mu(x) + \mu(y) \equiv \lambda(x,y) \pmod 2 \; \forall \,x,y \in H_k(M;\mathbb{Z}_2)$

Let ${x, y}$ be any 2-dimensional subspace of $H_k(M;\mathbb{Z}_2)$ , such that $λ(x, y) = 1$ . Then there are two possibilities. Either all of $μ(x + y),μ(x),μ(y)$ are 1, or else just one of them is 1, and the other two are 0. Call the first case $H 1,1$ , and the second case $H 0,0$ . Since every form is equivalent to a symplectic form, we can always find subspaces ${x, y}$ with x and y being $λ$ -dual. We can therefore split $H_k(M;\mathbb{Z}_2)$ into a direct sum of subspaces isomorphic to either $H 0,0$ or $H 1,1$ . Furthermore, by a clever change of basis, $H^{0,0} \oplus H^{0,0} \cong H^{1,1} \oplus H^{1,1}$ . We therefore define the Arf invariant

$Arf(H_k(M;\mathbb{Z}_2);\mu)$ = (number of copies of

H 1,1

in a decomposition Mod 2) $\in \mathbb{Z}_2$ .

[edit] Examples

Let $M$ be a compact, connected, oriented 2-dimensional manifold, i.e. a surface, of genus $g$ such that the boundary $\partial M$ is either empty or is connected. Embed $M$ in $S m$ , where $m \geq 4$ . Choose a framing of M, that is a trivialization of the normal (m-2)-plane vector bundle. (This is possible for $m = 3$ , so is certainly possible for $m \geq 4$ ). Choose a symplectic basis $x_1,x_2,\dots,x_{2g-1},x_{2g}$ for $H_1(M)=\mathbb{Z}^{2g}$ . Each basis element is represented by an embedded circle $x_i:S^1 \subset M$ . The normal (m-1)-plane vector bundle of $S^1 \subset M \subset S^m$ has two trivializations, one determined by a standard framing of a standard embedding $S^1 \subset S^m$ and one determined by the framing of M, which differ by a map $S^1 \to SO(m-1)$ i.e. an element of $\pi_1(SO(m-1)) \cong \mathbb{Z}_2$ for $m \geq 4$ . This can also be viewed as the framed cobordism class of $S 1$ with this framing in the 1-dimensional framed cobordism group $\Omega^{framed}_1 \cong \pi_m(S^{m-1}) \, (m \geq 4) \cong \mathbb{Z}_2$ , which is generated by the circle $S 1$ with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.) Define $\mu(x)\in \mathbb{Z}_2$ to be this element. The Arf invariant of the framed surface is now defined

$\Phi(M) = Arf(H_1(M,\partial M;\mathbb{Z}_2);\mu) \in \mathbb{Z}_2$

Note that $\pi_1(SO(2)) \cong \mathbb{Z}$ , so we had to stabilise, taking $m$ to be at least 4, in order to get an element of $\mathbb{Z}_2$ . The case $m = 3$ is also admissible as long as we take the residue modulo 2 of the framing.

The Arf invariant $Φ(M)$ 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 $H 1,1$ does not bound. $H 1,1$ represents a torus $T 2$ with a trivialisation on both generators of $H_1(T^2;\mathbb{Z}_2)$ 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 $π 1 (S O (3))$ . 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 $\Omega^{framed}_2 \cong \pi_m(S^{m-2}) \, (m \geq 4) \cong \mathbb{Z}_2$ , which is generated by the torus $T 2$ with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.

Let $(M^2,\partial M) \subset S^3$ be a Seifert surface for a knot, $\partial M = K \colon S^1 \hookrightarrow S^3$ , which can be represented as a disc $D 2$ with bands attached. The bands will typically be twisted and knotted. Each band corresponds to a generator $x \in H_1(M;\mathbb{Z}_2)$ . $x$ can be represented by a circle which traverses one of the bands. Define $μ(x)$ to be the number of full twists in the band modulo 2. Suppose we let $S 3$ bound $D 4$ , and push the Seifert surface $M$ into $D 4$ , so that its boundary still resides in $S 3$ . Around any generator $x \in H_1(M,\partial M)$ , we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding $M \hookrightarrow D^4$ for 2 of the sections required. For the third, choose a section which remains normal to $x$ , whilst always remaining tangent to $M$ . This trivialisation again determines an element of $π 1 (S O (3))$ , which we take to be $μ(x)$ . Note that this coincides with the previous definition of $μ$ .

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 $H 0,0$ 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 $\mathbb{Z}_2$ -coefficient homology $H_{2k+1}(M;\mathbb{Z}_2)$ of a framed 4k+2-dimensional manifold M has a quadratic refinement $μ$ , which depends on the framing. For $k \neq 0,1,3$ and $x \in H_{2k+1}(M;\mathbb{Z}_2)$ represented by an embedding $x:S^{2k+1}\subset M$ the value $\mu(x)\in \mathbb{Z}_2$ is 0 or 1, according as to the normal bundle of $x$ is trivial or not. The Kervaire invariant of the framed 4k+2-dimensional manifold M is the Arf invariant of the quadratic refinement $μ$ on $H_{2k+1}(M;\mathbb{Z}_2)$ . The Kervaire invariant is a homomorphism $\pi_{4k+2}^S \to \mathbb{Z}_2$ 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 $4 k + 2$ -dimensional normal map $(f,b):M \to X$ there is defined a nonsingular quadratic form $(K_{2k+1}(M;\mathbb{Z}_2),\mu)$ on the $\mathbb{Z}_2$ -coefficient homology kernel

$K_{2k+1}(M;\mathbb{Z}_2)=ker(f_*:H_{2k+1}(M;\mathbb{Z}_2)\to H_{2k+1}(X;\mathbb{Z}_2))$ refining the homological intersection form

λ

. The Arf invariant of this form is the Kervaire invariant of (f,b). In the special case

X = S 4 k + 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

using functional Steenrod squares, and Wall defined

μ

using framed immersions. The quadratic enhancement

μ(x)

crucially provides more information than

λ(x, x)

: it is possible to kill x by surgery if and only if

μ(x) = 0

. The corresponding Kervaire invariant detects the surgery obstruction of

(f, b)

in $L_{4k+2}(\mathbb{Z})=\mathbb{Z}_2$ .

[edit] See also

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

[edit] Notes

^ Dickson, Chapter VIII.
^ Martino and Priddy, p.61
^ Browder, Proposition III.1.8

[edit] References

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

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

A.V. Chernavskii (2001), "Arf invariant", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

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

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

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, http://arxiv.org/pdf/0711.1536, retrieved on 2009-05-24

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

[0] Dickson, Chapter VIII.

[1] Martino and Priddy, p.61

[2] Browder, Proposition III.1.8

[1]

[2]

[3]

Arf invariant

From Wikipedia, the free encyclopedia

Contents

[edit] Structure of quadratic forms over F₂

[edit] The Arf invariant in Topology

[edit] Examples

[edit] See also

[edit] Notes

[edit] References

Views

Personal tools

Navigation

Search

Interaction

Toolbox

Arf invariant

From Wikipedia, the free encyclopedia

Contents

[edit] Structure of quadratic forms over F2

[edit] The Arf invariant in Topology

[edit] Examples

[edit] See also

[edit] Notes

[edit] References

Views

Personal tools

Navigation

Search

Interaction

Toolbox

[edit] Structure of quadratic forms over F₂