The $J$-homomorphism is traditionally a family of group homomorphisms
from the homotopy groups of (the topological space underlying) the orthogonal group to the homotopy groups of spheres. This refines to a morphism of ∞-groups
from the stable orthogonal group (regarded as a group object in $L_{whe} Top \simeq$ ∞Grpd) to the ∞-group of units of the sphere spectrum, regarded as an E-∞ ring spectrum.
By postcomposition, the delooping of the J-homomorphism
sends real vector bundles to sphere bundles, namely to (∞,1)-line bundles with typical fiber the sphere spectrum $\mathbb{S}$. See also at Thom space for more on this.
The description of the image of the $J$-homomorphism in the stable homotopy groups of spheres was an important precursor to the development of chromatic homotopy theory, which is used to explain the periodicities seen in the image of the J-homomorphism (see also Lurie 10, remark 8). See also at periodicity theorem.
For $n \in \mathbb{N}$ regard the $n$-sphere (as a topological space) as the one-point compactification of the Cartesian space $\mathbb{R}^n$
Since the process of one-point compactification is a functor on proper maps, hence on homeomorphisms, via def. 1 the $n$-sphere inherits from the canonical action of the orthogonal group $O(n)$ on $\mathbb{R}^n$ an action
(by continuous maps) which preserves the base point (the “point at infinity”).
For definiteness we distinguish in the following notationally between
the $n$-sphere $S^n \in Top$ regarded as a topological space;
its homotopy type $\Pi(S^n) \in L_{whe} Top \simeq$ ∞Grpd given by its fundamental ∞-groupoid.
Similarly we write $\Pi(O(n))$ for the homotopy type of the orthogonal group, regarded as a group object in an (∞,1)-category in ∞Grpd (using that the shape modality $\Pi$ preserves finite products).
For $n \in \mathbb{N}$ write $H(n)$ for the automorphism ∞-group of homotopy self-equivalences $S^n \longrightarrow S^n$, hence
The ∞-group $H(n)$, def. 2, constitutes the two connected components of the $n$-fold based loop space $\Omega^n S^n$ corresponding to the homotopy groups $\pm 1 \in \pi_n(S^n)$.
Via the presentation of ∞Grpd by the cartesian closed model structure on compactly generated topological spaces (and using that $S^n$ and $O(n)$ and hence their product are compact) we have that for $n \in \mathbb{N}$ the continuous action of $O(n)$ on $S^n$ of remark 1, which by cartesian closure is equivalently a homomorphism of topological groups of the form
induces a homomorphism of ∞-groups of the form
This in turn induces for each $i \in \mathbb{N}$ homomorphisms of homotopy groups of the form
By construction, the homomorphisms of remark 3 are compatible with suspension in that for all $n \in \mathbb{N}$ the diagrams
in $Grp(Top)$ commute, and hence so do the diagrams
in $Grp(\infty Grpd)$, up to homotopy.
Therefore one can take the direct limit over $n$:
By remark 3 there is induced a homomorphism
from the homotopy groups of the stable orthogonal group to the stable homotopy groups of spheres. This is called the J-homomorphism.
Since the maps of def. 3 are ∞-group homomorphisms, there exists their delooping
Here $GL_1(\mathbb{S})$ is the ∞-group of units of the sphere spectrum.
This map $B J$ is the universal characteristic class of stable vector bundles with values in spherical fibrations:
For $V \to X$ a vector bundle, write $S^V$ for its fiber-wise one-point compactification. This is a sphere bundle/spherical fibration. Write $\mathbb{S}^V$ for the $X$-parameterized spectrum which is fiberwise the suspension spectrum of $S^V$.
It is immediate that:
For $V \to X$ a vector bundle classified by a map $X \to B O$, the corresponding spherical fibration $\mathbb{S}^V$, def. 5, is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. 4.
This construction descends to a map
from topological K-theory to spherical fibrations
(…) (MO discussion)
The following characterization of the image of the J-homomorphism on homotopy groups derives from a statement that was first conjectured in (Adams 66) – and since called the Adams conjecture – and then proven in (Quillen 71, Sullivan 74).
By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
$\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |
Because all groups appearing here and in the following are cyclic groups, we instead write down the order
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
${\vert\pi_n(O)\vert}$ | 2 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |
For the following statement it is convenient to restrict to J-homomorphism to the stable special orthogonal group $S O$, which removes the lowest degree homotopy group in the above
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
$\pi_n(S O)$ | 0 | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ |
$n\;mod\; 8$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
${\vert\pi_n(S O)\vert}$ | 1 | 2 | 1 | $\infty$ | 1 | 1 | 1 | $\infty$ |
The stable homotopy groups of spheres $\pi_n(\mathbb{S})$ are the direct sum of the (cyclic) image $im(J|_{SO})$ of the J-homomorphism, def. 4, applied to the special orthogonal group and the kernel of the Adams e-invariant.
Moreover,
for $n = 0 \;mod \;8$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism $\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S})$ is injective, hence its image is $\mathbb{Z}_2$,
for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the order of the image is equal to the denominator of $B_{2k}/4k$ in its reduced form, where $B_{2k}$ is the Bernoulli number
for all other cases the image is necessarily zero.
This characterization of the image of $J$ is due to (Adams 66, Quillen 71, Sullivan 74). Specifically the identification of $J(\pi_{4n-1}(S O))$ is (Adams 65a, theorem 3.7 and the direct summand property is (Adams 66, theorems 1.1-1.6.). That the image is a direct summand of the codomain is proven for instance in (Switzer 75, end of chapter 19).
A modern version of the proof, using methods from chromatic homotopy theory, is surveyed in some detail in (Lorman 13).
The statement of the theorem is recalled for instance as (Ravenel, chapter 1, theorem 1.1.13). Another computation of the image of $J$ is in (Ravenel, chapter 5, section 3).
The order of $J(\pi_{4k-1} O)$ in theorem 1 is for low $k$ given by the following table
k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
$\vert J(\pi_{4k-1}(O))\vert$ | 24 | 240 | 504 | 480 | 264 | 65,520 | 24 | 16,320 | 28,728 | 13,200 |
See for instance (Ravenel, Chapt. 1, p. 5).
Therefore we have in low degree the following situation
$n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Whitehead tower of orthogonal group | orientation | spin | string | fivebrane | ninebrane | |||||||||||||
homotopy groups of stable orthogonal group | $\pi_n(O)$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}$ | 0 | 0 | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ |
stable homotopy groups of spheres | $\pi_n(\mathbb{S})$ | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{24}$ | 0 | 0 | $\mathbb{Z}_2$ | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_6$ | $\mathbb{Z}_{504}$ | 0 | $\mathbb{Z}_3$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ | $\mathbb{Z}_{480} \oplus \mathbb{Z}_2$ | $\mathbb{Z}_2 \oplus \mathbb{Z}_2$ |
image of J-homomorphism | $im(\pi_n(J))$ | 0 | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{24}$ | 0 | 0 | 0 | $\mathbb{Z}_{240}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | 0 | $\mathbb{Z}_{504}$ | 0 | 0 | 0 | $\mathbb{Z}_{480}$ | $\mathbb{Z}_2$ |
The following tables show the p-primary components of the stable homotopy groups of spheres for low values, the image of J appears as the bottom row.
Here the horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a direct summand primary group of order $p^k$. See example 1 below for illustration.
(The tables are taken from (Hatcher), where in turn were they were generated based on (Ravenel 86).
at $p = 2$
at $p = 3$
at $p = 5$
The finite abelian group $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into primary groups as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.
The finite abelian group $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into primary groups as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.
(…)
We indicate how the Adams conjecture/Adams-Quillen-Sullivan theorem serves to identify the image of the J-homomorphism. We follow the modern account as reviewed in (Lorman 13).
(…)
Write $\psi^k$ for the $k$th Adams operation on complex K-theory.
Let $p$ be a prime. Consider $k$ coprime to $p$.
The Adams conjecture implies that completed at $p$, the J-homomorphism factors through the homotopy fiber of $1 - \psi^k$.
proof:
We have a homotopy-commuting diagram
The pasting composite with the homotopy pullback that witnesses the homotopy fiber of $1 - \psi^k$ induces via the universal property of the loop space object a canonical map $fib(1-\psi^k) \longrightarrow H_p$:
(…)
The J-spectrum is a spectrum whose homotopy groups are close to being the image of the J-homomorphism.
(…)
In terms of chromatic homotopy theory the nature of the image of the J-homomorphism can be formulated more succinctly as follows.
Write $E(1)$ for the first Morava E-theory spectrum at given prime number $p$. Write $L_{E(1)}\mathbb{S}$ for the Bousfield localization of spectra of the sphere spectrum at $E(1)$.
The homotopy groups of the $E(1)$-localized sphere spectrum are
This appears as (Lurie 10, theorem 6)
Write $\mathbb{S}_p$ for the p-localization of the sphere spectrum. For $n \in \mathbb{Z}$, write $im(J)_n$ for the image of the $p$-localized J-homomorphism
For $n \in \mathbb{N}$, the further Bousfield localization at Morava E(1)-theory $\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S}$ induces a isomorphism
between the image of the $J$-homomorphism and the $E(1)$-local stable homotopy groups of spheres.
In this form this appears as (Lurie 10, theorem 7). See also (Behrens 13, section 1).
The $E(1)$-localization map is surjective on non-negative homotopy groups:
For review see also (Lorman 13). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in (Lorman 13, p. 4)
Hence: the image of $J$ is essentially the first chromatic layer of the sphere spectrum.
The J-homomorphism was introduced in
Lecture notes include
Akhil Mathew, The Adams conjecture I (web)
Akhil Mathew, Notes on the J-homomorphism (pdf)
Discussion in higher algebra in term of (∞,1)-module bundles is in
The complex J-homomorphism is discussed in
Victor Snaith, The complex J-homomorphism, Proc. London Math. Soc. (1977) s3-34 (2): 269-302 (journal)
Victor Snaith, Infinite loop maps and the complex $J$-homomorphism, Bull. Amer. Math. Soc. Volume 82, Number 3 (1976), 508-510. (Euclid)
A p-adic J-homomorphism is described in
The analysis of the image of $J$ is due to
John Adams, On the groups $J(X)$ I, Topology 2 (3) (1963) (pdf)
John Adams, On the groups $J(X)$ II, Topology 3 (2) (1965) (pdf)
The statement of the theorem about the characterization of the image is reviewed in
see there also around theorem 3.4.16.
The details of the proof are surveyed in
Tables showing the image of $J$ at low primes are in
Other reviews include
Mark Mahowald, The Image of J in the EHP Sequence, Annals of Mathematics Second Series, Vol. 116, No. (JSTOR)
Johannes Ebert, The Adams conjecture after Edgar Brown, (pdf)
Discussion from the point of view of chromatic homotopy theory is in
Jacob Lurie, The image of $J$, lecture 35 in Chromatic Homotopy Theory (pdf)
Mark Behrens, section 1 of Introduction talk at Talbot 2013: Chromatic Homotopy Theory (pdf, pdf)
[[Benjamin Knudsen, First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere, talk at 2013 Pre-Talbot Seminar (pdf)
Vitaly Lorman, Chromatic homotopy theory at height 1 and the image of $J$, talk at Talbot 2013: Chromatic Homotopy Theory (pdf)
Similarly there is a canonical $O(n)$-∞-action on an n-fold loop space, not just on the sphere spectrum. But the general case is closely related to the J-homomorphism. Discussion includes
and in the context of the cobordism hypothesis: