Arf Invariant - The Arf Invariant in Topology - Examples

Examples

  • Let be a compact, connected, oriented 2-dimensional manifold, i.e. a surface, of genus 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 bundle. (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 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 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 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 embedding 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 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 Kervaire and Milnor, and more generally in the classification of manifolds by surgery theory. Browder defined using functional Steenrod squares, and 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 .

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