Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

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Paths and cylinders

Homotopy groups

Basic facts


Stable Homotopy theory



The JJ-homomorphism is traditionally a family of group homomorphisms

J i:π i(O(n))π n+i(S n) 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:OGL 1(𝕊) J \;\colon\; O \longrightarrow GL_1(\mathbb{S})

from the stable orthogonal group (regarded as a group object in L wheTopL_{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

BJ:BOBGL 1(𝕊) 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 𝕊\mathbb{S}. See also at Thom space for more on this.

The description of the image of the JJ-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.


On groups


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

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

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

O(n)×S nS n 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 nn-sphere S nTopS^n \in Top regarded as a topological space;

  2. its homotopy type Π(S n)L wheTop\Pi(S^n) \in L_{whe} Top \simeq ∞Grpd given by its fundamental ∞-groupoid.

Similarly we write Π(O(n))\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 Π\Pi preserves finite products).


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

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

The ∞-group H(n)H(n), def. , constitutes the two connected components of the nn-fold based loop space Ω nS n\Omega^n S^n corresponding to the homotopy groups ±1π n(S n)\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 nS^n and O(n)O(n) and hence their product are compact) we have that for nn \in \mathbb{N} the continuous action of O(n)O(n) on S nS^n of remark , which by cartesian closure is equivalently a homomorphism of topological groups of the form

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

induces a homomorphism of ∞-groups of the form

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

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

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

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

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

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

Π(O(n)) Aut Grpd */(Π(S n)) Π(O(n+1)) Aut Grpd */(Π(S n+1)) \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(Grpd)Grp(\infty Grpd), up to homotopy.

Therefore one can take the direct limit over nn:


By remark there is induced a homomorphism

J i:π (O)π (𝕊) 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


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

BJ:BOBGL 1(𝕊)=BH. B J \;\colon\; B O \longrightarrow B GL_1(\mathbb{S}) = B H \,.

Here GL 1(𝕊)GL_1(\mathbb{S}) is the ∞-group of units of the sphere spectrum.

This map BJB J is the universal characteristic class of stable vector bundles with values in spherical fibrations:


For VXV \to X a vector bundle, write S VS^V for its fiber-wise one-point compactification. This is a sphere bundle/spherical fibration. Write 𝕊 V\mathbb{S}^V for the XX-parameterized spectrum which is fiberwise the suspension spectrum of S VS^V.

It is immediate that:


For VXV \to X a vector bundle classified by a map XBOX \to B O, the corresponding spherical fibration 𝕊 V\mathbb{S}^V, def. , is classified by XBOBJBGL 1(𝕊)X \to B O \stackrel{B J}{\longrightarrow} B GL_1(\mathbb{S}), def. .

This construction descends to a map

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

from topological K-theory to spherical fibrations

(…) (MO discussion)


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).


By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are

nmod8n\;mod\; 801234567
π n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}

Because all groups appearing here and in the following are cyclic groups, we instead write down the order

nmod8n\;mod\; 801234567
|π n(O)|{\vert\pi_n(O)\vert}221\infty111\infty

For the following statement it is convenient to restrict to J-homomorphism to the stable special orthogonal group SOS O, which removes the lowest degree homotopy group in the above

nmod8n\;mod\; 801234567
π n(SO)\pi_n(S O)0 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z}
nmod8n\;mod\; 801234567
|π n(SO)|{\vert\pi_n(S O)\vert}121\infty111\infty

The stable homotopy groups of spheres π n(𝕊)\pi_n(\mathbb{S}) are the direct sum of the (cyclic) image im(J| SO)im(J|_{SO}) of the J-homomorphism, def. , applied to the special orthogonal group and the kernel of the Adams e-invariant.


  • for n=0mod8n = 0 \;mod \;8 and n=1mod8n = 1 \;mod \; 8 and nn positive the J-homomorphism π n(J):π n(SO)π n(𝕊)\pi_n(J) \colon \pi_n(S O) \to \pi_n(\mathbb{S}) is injective, hence its image is 2\mathbb{Z}_2,

  • for n=3mod8n = 3\; mod\; 8 and n=7mod8n = 7 \; mod \; 8 hence for n=4k1n = 4 k -1, the order of the image is equal to the denominator of B 2k/4kB_{2k}/4k in its reduced form, where B 2kB_{2k} is the Bernoulli number

  • for all other cases the image is necessarily zero.

This characterization of the image of JJ is due to (Adams 66, Quillen 71, Sullivan 74). Specifically the identification of J(π 4n1(SO))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 JJ is in (Ravenel, chapter 5, section 3).


The order of J(π 4k1O)J(\pi_{4k-1} O) in theorem is for low kk given by the following table

|J(π 4k1(O))|\vert J(\pi_{4k-1}(O))\vert2424050448026465,5202416,32028,72813,200

See for instance (Ravenel, Chapt. 1, p. 5).


Therefore we have in low degree the following situation

Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\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 nn of the stable homotopy group π n\pi_n. The appearance of a string of kk connected dots vertically above index nn means that there is a direct summand primary group of order p kp^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=2p = 2

stable homotopy groups of spheres at 2

at p=3p = 3

stable homotopy groups of spheres at 3

at p=5p = 5

stable homotopy groups of spheres at 5


The finite abelian group π 3(𝕊) 24\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24} decomposes into primary groups as 8 3\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3. Here 8=2 38 = 2^3 corresponds to the three dots above n=3n = 3 in the first table, and 3=3 13 = 3^1 to the single dot over n=3n = 3 in the second.

The finite abelian group π 7(𝕊) 24\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{24} decomposes into primary groups as 16 3 5\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5. Here 16=2 416 = 2^4 corresponds to the four dots above n=7n = 7 in the first table, and 3=3 13 = 3^1 to the single dot over n=7n = 7 in the second and 5=5 15 = 5^1 to the single dot over n=7n = 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 ψ k\psi^k for the kkth Adams operation on complex K-theory.

Let pp be a prime. Consider kk coprime to pp.

The Adams conjecture implies that completed at pp, the J-homomorphism factors through the homotopy fiber of 1ψ k1 - \psi^k.


We have a homotopy-commuting diagram

BU p 1ψ k BU p * 0 BH p. \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 1ψ k1 - \psi^k induces via the universal property of the loop space object a canonical map fib(1ψ k)H pfib(1-\psi^k) \longrightarrow H_p:

fib(1ψ k) * BU p 1ψ k BU p * 0 BH pfib(1ψ k) * H p * * 0 BH 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)E(1) for the first Morava E-theory spectrum at given prime number pp. Write L E(1)𝕊L_{E(1)}\mathbb{S} for the Bousfield localization of spectra of the sphere spectrum at E(1)E(1).


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

π nL E(1)𝕊{ ifn=0 p/ p ifn=2 /p k+1 ifn+1=2(p1)p kmwithm0modp 0 otherwise. \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)𝕊L_{K(1)} \mathbb{S} rather than L E(1)𝕊L_{E(1)} \mathbb{S}, and second, it should be n+1=2(p1)p kmn+1 = 2(p-1)p^k m but the 22 is missing.)


Write 𝕊 p\mathbb{S}_p for the p-localization of the sphere spectrum. For nn \in \mathbb{Z}, write im(J) nim(J)_n for the image of the pp-localized J-homomorphism

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

For nn \in \mathbb{N}, the further Bousfield localization at Morava E(1)-theory 𝕊 (p)L E(1)𝕊\mathbb{S}_{(p)} \longrightarrow L_{E(1)}\mathbb{S} induces a isomorphism

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

between the image of the JJ-homomorphism and the E(1)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)E(1)-localization map is surjective on non-negative homotopy groups:

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

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


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



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).

Lecture notes include

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 JJ-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 JJ is due to

  • John Adams, On the groups J(X)J(X) I, Topology 2 (3) (1963) (pdf)

  • John Adams, On the groups J(X)J(X) II, Topology 3 (2) (1965) (pdf)

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

see there also around theorem 3.4.16.

The details of the proof are surveyed in

Tables showing the image of JJ at low primes are in

Other reviews include

Discussion from the point of view of chromatic homotopy theory is in

Relation to OO-action on general spectra

Similarly there is a canonical O(n)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 JJ-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:

Last revised on August 27, 2018 at 16:28:17. See the history of this page for a list of all contributions to it.