In mathematics, the Kervaire invariant is an invariant of a
framed -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 ...
that measures whether the manifold could be
surgically
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pa ...
converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after
Michel Kervaire
Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra.
He introduced the Kervaire semi-characteristic. He was the first to show the existence of topologi ...
who built on work of
Cahit Arf
Cahit Arf (; 24 October 1910 – 26 December 1997) was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasse–Arf theorem ...
.
The Kervaire invariant is defined as the
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf i ...
of the
skew-quadratic form on the middle dimensional
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
. It can be thought of as the simply-connected ''quadratic''
L-group , and thus analogous to the other invariants from L-theory: the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
, a
-dimensional invariant (either symmetric or quadratic,
), and the
De Rham invariant In geometric topology, the de Rham invariant is a mod 2 invariant of a (4''k''+1)-dimensional manifold, that is, an element of \mathbf/2 – either 0 or 1. It can be thought of as the simply-connected ''symmetric'' L-group L^, and thus analogous t ...
, a
-dimensional ''symmetric'' invariant
.
In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.
The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open.
Definition
The Kervaire invariant is the
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf i ...
of the
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 a ...
determined by the framing on the middle-dimensional
-coefficient homology group
:
and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly,
skew-quadratic form) is 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 ...
of the usual
ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.
The quadratic form ''q'' can be defined by algebraic topology using functional
Steenrod squares, and geometrically via the self-intersections
of
immersions determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings
(for
) and the mod 2
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the ''Hopf map''
:\eta\colon S^3 \to S^ ...
of maps
(for
).
History
The Kervaire invariant is a generalization of the Arf invariant of a framed surface (that is, a 2-dimensional manifold with stably trivialized tangent bundle) which was used by
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 due ...
in 1950 to compute the
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of maps
(for
), which is the cobordism group of surfaces embedded in
with trivialized normal bundle.
used his invariant for ''n'' = 10 to construct the
Kervaire manifold In mathematics, specifically in differential topology, a Kervaire manifold K^ is a piecewise-linear manifold of dimension 4n+2 constructed by by plumbing together the tangent bundles of two (2n+1)-spheres, and then gluing a ball to the result. ...
, a 10-dimensional
PL manifold
PL, P.L., Pl, or .pl may refer to:
Businesses and organizations Government and political
* Partit Laburista, a Maltese political party
* Liberal Party (Brazil, 2006), a Brazilian political party
* Liberal Party (Moldova), a Moldovan political pa ...
with no
differentiable structure In mathematics, an ''n''-dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for diff ...
, the first example of such a manifold, by showing that his invariant does not vanish on this PL manifold, but vanishes on all smooth manifolds of dimension 10.
computes the group of
exotic sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...
s (in dimension greater than 4), with one step in the computation depending on the Kervaire invariant problem. Specifically, they show that the set of exotic spheres of dimension ''n'' – specifically the monoid of smooth structures on the standard ''n''-sphere – is isomorphic to the group
of
''h''-cobordism classes of oriented
homotopy ''n''-spheres. They compute this latter in terms of a map
:
where
is the cyclic subgroup of ''n''-spheres that bound a
parallelizable manifold
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equi ...
of dimension
,
is the ''n''th
stable homotopy group of spheres, and ''J'' is the image of the
J-homomorphism
In mathematics, the ''J''-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .
Definition
Whitehead's original homomorphism is d ...
, which is also a cyclic group. The groups
and
have easily understood cyclic factors, which are trivial or order two except in dimension
, in which case they are large, with order related to the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s. The quotients are the difficult parts of the groups. The map between these quotient groups is either an isomorphism or is injective and has an image of index 2. It is the latter if and only if there is an ''n''-dimensional framed manifold of nonzero Kervaire invariant, and thus the classification of exotic spheres depends up to a factor of 2 on the Kervaire invariant problem.
Examples
For the standard embedded
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 not tou ...
, the skew-symmetric form is given by
(with respect to the standard
symplectic basis In linear algebra, a standard symplectic basis is a basis _i, _i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega(_i, _j) = 0 = \omega(_i, _j), \omega(_i, _j) = \delta_. A ...
), and the skew-quadratic refinement is given by
with respect to this basis:
: the basis curves don't self-link; and
: a (1,1) self-links, as in the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
. This form thus has
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf i ...
0 (most of its elements have norm 0; it has
isotropy index 1), and thus the standard embedded torus has Kervaire invariant 0.
Kervaire invariant problem
The question of in which dimensions ''n'' there are ''n''-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem. This is only possible if ''n'' is 2 mod 4, and indeed one must have ''n'' is of the form
(two less than a power of two). The question is almost completely resolved; only the case of dimension 126 is open: there are manifolds with nonzero Kervaire invariant in dimension 2, 6, 14, 30, 62, and none in all other dimensions other than possibly 126.
The main results are those of , who reduced the problem from differential topology to
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
and showed that the only possible dimensions are
, and those of , who showed that there were no such manifolds for
(
). Together with explicit constructions for lower dimensions (through 62), this leaves open only dimension 126.
It was conjectured by
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the ...
that there is such a manifold in dimension 126, and that the higher-dimensional manifolds with nonzero Kervaire invariant are related to well-known exotic manifolds two dimension higher, in dimensions 16, 32, 64, and 128, namely the
Cayley projective plane (dimension 16, octonionic projective plane) and the analogous
Rosenfeld projective plane
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea indep ...
s (the bi-octonionic projective plane in dimension 32, the
quateroctonionic projective plane in dimension 64, and the octo-octonionic projective plane in dimension 128), specifically that there is a construction that takes these projective planes and produces a manifold with nonzero Kervaire invariant in two dimensions lower.
comment
by André Henriques Jul 1, 2012 at 19:26, on
Kervaire invariant: Why dimension 126 especially difficult?
, ''MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a ...
''
History
* proved that the Kervaire invariant is zero for manifolds of dimension 10, 18
* proved that the Kervaire invariant can be nonzero for manifolds of dimension 6, 14
* proved that the Kervaire invariant is zero for manifolds of dimension 8''n''+2 for ''n''>1
* proved that the Kervaire invariant can be nonzero for manifolds of dimension 30
* proved that the Kervaire invariant is zero for manifolds of dimension ''n'' not of the form 2''k'' − ''2''.
* showed that the Kervaire invariant is nonzero for some manifold of dimension 62. An alternative proof was given later by .
* showed that the Kervaire invariant is zero for ''n''-dimensional framed manifolds for ''n'' = 2''k''− 2 with ''k'' ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
**The coefficient groups Ω''n''(point) have period 28 = 256 in ''n''
**The coefficient groups Ω''n''(point) have a "gap": they vanish for ''n'' = -1, -2, and -3
**The coefficient groups Ω''n''(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension ''n'' is nonzero then it has a nonzero image in Ω−''n''(point)
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 6th stable homotopy group of spheres to ,
and a homomorphism from the 14th stable homotopy group of spheres onto . For ''n'' = 2, 6, 14 there is an
exotic framing on with Kervaire–Milnor invariant 1.
See also
* Signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
, a 4''k''-dimensional invariant
* De Rham invariant In geometric topology, the de Rham invariant is a mod 2 invariant of a (4''k''+1)-dimensional manifold, that is, an element of \mathbf/2 – either 0 or 1. It can be thought of as the simply-connected ''symmetric'' L-group L^, and thus analogous t ...
, a (4''k'' + 1)-dimensional invariant
References
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
Slides and video of lecture by Hopkins at Edinburgh, 21 April, 2009
April 23, 2009, blog post by John Baez and discussion, The n-Category Café
Exotic spheres
at the manifold atlas
Popular news stories
Hypersphere Exotica: Kervaire Invariant Problem Has a Solution! A 45-year-old problem on higher-dimensional spheres is solved–probably
by Davide Castelvecchi, August 2009 ''Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
''
* {{Cite journal , last1 = Ball , first1 = Philip , doi = 10.1038/news.2009.427 , title = Hidden riddle of shapes solved , journal = Nature , year = 2009
Mathematicians solve 45-year-old Kervaire invariant puzzle
Erica Klarreich, 20 Jul 2009
Differential topology
Surgery theory