J homomorphism
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In mathematics, the ''J''-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the
homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
. It was defined by , extending a construction of .


Definition

Whitehead's original
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
is defined geometrically, and gives a homomorphism :J \colon \pi_r (\mathrm(q)) \to \pi_(S^q) of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s for
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''q'', and r \ge 2. (Hopf defined this for the special case q = r+1.) The ''J''-homomorphism can be defined as follows. An element of the special orthogonal group SO(''q'') can be regarded as a map :S^\rightarrow S^ and the homotopy group \pi_r(\operatorname(q))) consists of
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
classes of maps from the ''r''-sphere to SO(''q''). Thus an element of \pi_r(\operatorname(q)) can be represented by a map :S^r\times S^\rightarrow S^ Applying the Hopf construction to this gives a map :S^= S^r*S^\rightarrow S( S^) =S^q in \pi_(S^q), which Whitehead defined as the image of the element of \pi_r(\operatorname(q)) under the J-homomorphism. Taking a limit as ''q'' tends to infinity gives the stable ''J''-homomorphism in stable homotopy theory: : J \colon \pi_r(\mathrm) \to \pi_r^S , where \mathrm is the infinite special orthogonal group, and the right-hand side is the ''r''-th stable stem of the stable homotopy groups of spheres.


Image of the J-homomorphism

The image of the ''J''-homomorphism was described by , assuming the Adams conjecture of which was proved by , as follows. The
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
\pi_r(\operatorname) is given by
Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
. It is always
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
; and if ''r'' is positive, it is of order 2 if ''r'' is 0 or 1 modulo 8, infinite if ''r'' is 3 modulo 4, and order 1 otherwise . In particular the image of the stable ''J''-homomorphism is cyclic. The stable homotopy groups \pi_r^S are the direct sum of the (cyclic) image of the ''J''-homomorphism, and the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
of the Adams e-invariant , a homomorphism from the stable homotopy groups to \Q/\Z. If ''r'' is 0 or 1 mod 8 and positive, the order of the image is 2 (so in this case the ''J''-homomorphism is injective). If ''r'' is 3 mod 4, the image is a cyclic group of order equal to the denominator of B_/4n, where B_ is a
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, ...
. In the remaining cases where ''r'' is 2, 4, 5, or 6 mod 8 the image is
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
because \pi_r(\operatorname) is trivial. :


Applications

introduced the group ''J''(''X'') of a space ''X'', which for ''X'' a sphere is the image of the ''J''-homomorphism in a suitable dimension. The
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of the ''J''-homomorphism J \colon \pi_n(\mathrm) \to \pi_n^S appears in the group Θ''n'' of ''h''-cobordism classes of oriented homotopy ''n''-spheres ().


References

* * * * *. * * * * * * * {{Citation , last=Whitehead , first=George W. , author-link=George W. Whitehead, title=Elements of homotopy theory , publisher=
Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...
, location=Berlin , year=1978 , isbn=0-387-90336-4 , mr= 0516508 Homotopy theory Topology of Lie groups