In mathematics, the Kervaire invariant is an invariant of a
framed -dimensional
manifold 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.
The Kervaire invariant is defined as the
Arf invariant 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, 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 manifolds, 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 of the
quadratic form 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 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 in 1950 to compute the
homotopy group 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, 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 ...
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 dif ...
, 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 spheres (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 of dimension
,
is the ''n''th
stable homotopy group of spheres, and ''J'' is the image of the
J-homomorphism, 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, 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. This form thus has
Arf invariant 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 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 th ...
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 planes (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, a (4''k'' + 1)-dimensional invariant
References
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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''
* {{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