Signature (topology) - Other Dimensions - Kervaire Invariant

Kervaire Invariant

When is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.

Read more about this topic:  Signature (topology), Other Dimensions

Other articles related to "kervaire, kervaire invariant, invariant":

Homotopy Groups Of Spheres - Applications
... This is an isomorphism unless n is of the form 2k−2, in which case the image has index 1 or 2 (Kervaire Milnor 1963) ... The Kervaire invariant problem, about the existence of manifolds of Kervaire invariant 1 in dimensions 2k − 2 can be reduced to a question about stable homotopy groups of ... knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension 26 − 2 = 62 (Barratt, Jones ...
The Arf Invariant in Topology - Examples
... 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 Arf invariant of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing ... 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 ...
Kervaire Invariant - Kervaire–Milnor Invariant
... The Kervaire–Milnor invariant is a closely related invariant of framed surgery of a 2, 6 or 14-dimensional framed manifold, that gives isomorphisms from the 2nd and ... n = 2, 6, 14 there is an exotic framing on Sn/2 x Sn/2 with Kervaire-Milnor invariant 1 ...