nLab J-homomorphism

Idea

The $J$-homomorphism is traditionally a family of group homomorphisms

$$J_i \;\colon\; \pi_i(O(n)) \longrightarrow \pi_{n+i}(S^n)$$

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

$$J \;\colon\; O \longrightarrow GL_1(\mathbb{S})$$

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

$$B J \;\colon\; B O \to B GL_1(\mathbb{S})$$

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.

Definition

On groups

Definition

For $n \in \mathbb{N}$ regard the $n$-sphere (as a topological space) as the one-point compactification of the Cartesian space $\mathbb{R}^n$

$$S^n \simeq (\mathbb{R}^n)^\ast \,.$$

Remark

Since the process of one-point compactification is a functor on proper maps, hence on homeomorphisms, via def. the $n$-sphere inherits from the canonical action of the orthogonal group $O(n)$ on $\mathbb{R}^n$ an action

$$O(n) \times S^n \longrightarrow S^n$$

(by continuous maps) which preserves the base point (the “point at infinity”).

For definiteness we distinguish in the following notationally between

1. the $n$-sphere $S^n \in Top$ regarded as a topological space;

2. 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).

Definition

For $n \in \mathbb{N}$ write $H(n)$ for the automorphism ∞-group of homotopy self-equivalences $S^n \longrightarrow S^n$, hence

$$H(n) \coloneqq Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \,.$$

Remark

The ∞-group $H(n)$, def. , constitutes the two connected components of the $n$-fold based loop space $\Omega^n S^n$ corresponding to the homotopy classes $\pm 1 \in \pi_n(S^n)$.

Definition

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 , which by cartesian closure is equivalently a homomorphism of topological groups of the form

$$O(n) \longrightarrow Aut_{Top^{\ast/}}(S^n) \,,$$

induces a homomorphism of ∞-groups of the form

$$\Pi(O(n)) \longrightarrow Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \,.$$

This in turn induces for each $i \in \mathbb{N}$ homomorphisms of homotopy groups of the form

$$\pi_i(O(n)) \longrightarrow \pi_i(\Omega^n S^n) \simeq \pi_{n+i}(S^n) \,.$$

Remark

By construction, the homomorphisms of remark are compatible with suspension in that for all $n \in \mathbb{N}$ the diagrams

$$\array{ O(n) &\longrightarrow& Aut_{Top^{\ast/}}(S^n) \\ \downarrow && \downarrow \\ O(n+1) &\longrightarrow& Aut_{Top^{\ast/}}(S^{n+1}) }$$

in $Grp(Top)$ commute, and hence so do the diagrams

$$\array{ \Pi(O(n)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^n)) \\ \downarrow && \downarrow \\ \Pi(O(n+1)) &\longrightarrow& Aut_{\infty Grpd^{\ast/}}(\Pi(S^{n+1})) }$$

in $Grp(\infty Grpd)$, up to homotopy.

Therefore one can take the direct limit over $n$:

Definition

By remark there is induced a homomorphism

$$J_i \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})$$

from the homotopy groups of the stable orthogonal group to the stable homotopy groups of spheres. This is called the J-homomorphism.

Delooped: On classifying spaces and K-theory classes

Remark

Since the maps of def. are ∞-group homomorphisms, there exists their delooping

$$B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H \,.$$

Remark

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:

Definition

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:

Proposition

(spherical fibrations of vector bundles classified via J-homomorphism)

For $V \to X$ a vector bundle classified by a map $X \to B O$, the corresponding spherical fibration $\mathbb{S}^V$, def. , is classified by $X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S})$, def. .

This construction descends to a map

$$KO^0(X) \longrightarrow Sph(X)$$

from topological K-theory to spherical fibrations

(e.g. Clausen 11, Section 2, Westerland 14, slide 5, MO discussion)

Properties

Image of the J-homomorphism

Traditional formulation

Description of the image

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).

Remark

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

$n\;mod\; 8$	$n = 0$	0	1	2	3	4	5	6	7
$\pi_n(S O)$	0	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$

$n\;mod\; 8$	$n = 0$	0	1	2	3	4	5	6	7
${\vert\pi_n(S O)\vert}$	1	2	2	1	$\infty$	1	1	1	$\infty$

Theorem

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. , 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).

Remark

The order of $J(\pi_{4k-1} O)$ in theorem 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).

Remark

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 group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
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 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 $p = 5$

Example

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.

Characterization via the Adams operations

(…)

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

$$\array{ B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p \\ \downarrow &\swArrow_\simeq& \downarrow \\ \ast &\stackrel{0}{\longrightarrow}& B H_p } \,.$$

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 $fib(1-\psi^k) \longrightarrow H_p$:

$$\array{ fib(1-\psi^k) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ B U_p &\stackrel{1 - \psi^k}{\longrightarrow}& B U_p \\ \downarrow &\swArrow_\simeq& \downarrow \\ \ast &\stackrel{0}{\longrightarrow}& B H_p } \;\;\; \simeq \;\;\; \array{ fib(1-\psi^k) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ H_p &\stackrel{}{\longrightarrow}& \ast \\ \downarrow &\swArrow_\simeq& \downarrow \\ \ast &\stackrel{0}{\longrightarrow}& B H_p } \,.$$

(…)

The J-spectrum

The J-spectrum is a spectrum whose homotopy groups are close to being the image of the J-homomorphism.

(…)

Formulation in chromatic homotopy theory

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 $E(1)$.

Theorem

The homotopy groups of the $E(1)$-localized sphere spectrum are

$$\pi_n L_{E(1)} \mathbb{S} \simeq \left\{ \array{ \mathbb{Z} & if\; n = 0 \\ \mathbb{Q}_p/\mathbb{Z}_p & if\; n= -2 \\ \mathbb{Z}/p^{k+1}\mathbb{Z} & if\; n+1 = 2(p-1)p^k m \;with\; m \neq 0\;mod\;p \\ 0 & otherwise } \right. \,.$$

This appears as (Lurie 10, theorem 6) (note that there are two typos: first, the theorem is stated with $L_{K(1)} \mathbb{S}$ rather than $L_{E(1)} \mathbb{S}$, and second, it should be $n+1 = 2(p-1)p^k m$ but the $2$ is missing.)

Definition

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

$$J \;\colon\; \pi_n(O) \longrightarrow \pi_n(\mathbb{S}) \longrightarrow \pi_n(\mathbb{S}_{(p)}) \,.$$

Theorem

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

$$im(J)_n \stackrel{\simeq}{\longrightarrow} \pi_n (L_{E(1)} \mathbb{S})$$

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).

Corollary

The $E(1)$-localization map is surjective on non-negative homotopy groups:

$$\pi_n(\mathbb{S}_{(p)}) \longrightarrow \pi_n(L_{E(1)} \mathbb{S}) \,.$$

For review see also (Lorman 13). That $J$ factors through $L_{K(1)}\mathbb{S}$ is in (Lorman 13, p. 4)

Remark

Hence: the image of $J$ is essentially the first chromatic layer of the sphere spectrum.

Related concepts

• e-invariant

• Thom spectra, MO

• orientation in generalized cohomology

• J-homomorphism and chromatic homotopy

• scanning map equivalence

• twisted cohomotopy

References

General

The J-homomorphism was introduced in

• George Whitehead, On the homotopy groups of spheres and rotation groups, Annals of Mathematics. Second Series 43 (4): 634–640 (1942) (jstor:1968956).

Textbook accounts:

• Antoni Kosinski, chapter IX 6.3 of: Differential manifolds, Academic Press 1993 (pdf, ISBN:978-0-12-421850-5)

• Dai Tamaki, Akira Kono, Section 4.4 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

Lecture notes:

• Akhil Mathew, The Adams conjecture I (web)

• Akhil Mathew, Notes on the J-homomorphism, 2012 (pdf, pdf)

• Michael Hopkins (notes by Akhil Mathew), Lecture 13 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

• Arpon Raksit, Vector fields and the J-homomorphism, 2014 (pdf, pdf)

• Craig Westerland, Views on the J-homomorphism, MSRI talk, 2014 (recording, pdf, pdf)

• Gereon Quick, The $e$-invariant and the $J$-homomorphism, lecture notes in: Advanced algebraic topology, 2014 (pdf)

Discussion in higher algebra in term of (∞,1)-module bundles is in

• Matthew Ando, Andrew Blumberg, David Gepner, Twists of K-theory and TMF (arXiv:1002.3004)

and in higher chromatic homotopy theory:

• Craig Westerland, A higher chromatic analogue of the image of J, Geometry & Topology 21 (2017) 1033–1093 (doi:10.2140/gt.2017.21.1033)

Discussion in p-adic geometry:

• Dustin Clausen, $p$-adic J-homomorphisms and a product formula (arXiv:1110.5851)

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

• Dustin Clausen, p-adic J-homomorphisms and a product formula (arXiv:1110.5851)

The image of J

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)

• John Adams, On the groups $J(X)$ III, Topology 3 (3) (1965) (pdf)

• John Adams, On the groups $J(X)$ IV, Topology 5: 21,(1966) Correction, Topology 7 (3): 331 (1968) (pdf)

• Daniel Quillen, The Adams conjecture, Topology. an International Journal of Mathematics 10: 67–80 (1971) (pdf)

• Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100 (1974), 1–79.

• Robert Switzer, Algebraic topology–homotopy and homology, Springer-Verlag, New York, 1975.

The statement of the theorem about the characterization of the image is reviewed in

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, chapter 1, An introduction to the homotopy groups of spheres (pdf)

see there also around theorem 3.4.16.

The details of the proof are surveyed in

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, chapter 5, The chromatic spectral sequence (pdf)

Tables showing the image of $J$ at low primes are in

• Allen Hatcher, Stable homotopy groups of spheres (html)

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)

• Ben 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)

Generalization to equivariant cohomology (equivariant K-theory) is discussed in

• Zbigniew Fiedorowicz, Henning Hauschild, Peter May, Equivariant algebraic K-theory, in: Algebraic K-Theory, Lecture Notes in Mathematics 967, Springer (1982) 23-80 [doi:10.1007/BFb0061898, pdf]

• Henning Hauschild, Stefan Waner, The equivariant Dold theorem mod $k$ and the Adams conjecture, Illinois J. Math. 27 1 (1983) 52-66. [euclid:1256065410&rbrack

• Kuzuhisa Shimakawa, Note on the equivariant $K$-theory spectrum, Publ. RIMS, Kyoto Univ. 29 (1993), 449-453 (pdf, doi)

• Christopher French, theorem 2.4 in The equivariant $J$–homomorphism for finite groups at certain primes, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (euclid:1513797069)

Equivariant enhancement

The generalization to equivariant cohomology (equivariant K-theory and the equivariant Adams conjecture) is discussed in

• Tammo tom Dieck, theorem 11.3.8 in Transformation Groups and Representation Theory Lecture Notes in Mathematics 766 Springer 1979

• Z. Fiedorowicz, H. Hauschild, Peter May, theorem 0.4 of Equivariant algebraic K-theory, Equivariant algebraic K-theory, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 (pdf)

• Henning Hauschild, Stefan Waner, theorem 0.1 of The equivariant Dold theorem mod $k$ and the Adams conjecture, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. (euclid:1256065410)

• Kuzuhisa Shimakawa, Note on the equivariant $K$-theory spectrum, Publ. RIMS, Kyoto Univ. 29 (1993), 449-453 (pdf, doi)

• Christopher French, theorem 2.4 in The equivariant $J$–homomorphism for finite groups at certain primes, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 (euclid:1513797069)

Relation to $O$-action on general spectra

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

• Gerald Gaudens, Luc Menichi, section 5 of Batalin-Vilkovisky algebras and the $J$-homomorphism, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365–374 (arXiv:0707.3103)

and in the context of the cobordism hypothesis:

• Jacob Lurie, example 2.4.15 of On the Classification of Topological Field Theories, Current Developments in Mathematics Volume 2008 (2009), 129-280 (arXiv:0905.0465)