Paths and cylinders
Stable Homotopy theory
The -homomorphism is traditionally a family of group homomorphisms
\pi 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 ∞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 . See also at Thom space for more on this.
The description of the image of the -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 definiteness we distinguish in the following notationally between
the -sphere regarded as a topological space;
its homotopy type ∞Grpd given by its fundamental ∞-groupoid.
Similarly we write for the homotopy type of the orthogonal group, regarded as a group object in an (∞,1)-category in ∞Grpd (using that the shape modality preserves finite products).
For write for the automorphism ∞-group of homotopy self-equivalences , hence
Therefore one can take the direct limit over :
Delooped: On classifying spaces and K-theory classes
This map is the universal characteristic class of stable vector bundles with values in spherical fibrations:
It is immediate that:
For a vector bundle classified by a map , the corresponding spherical fibration , def. 5, is classified by , def. 4.
This construction descends to a map
from topological K-theory to spherical fibrations
Image of the J-homomorphism
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).
For the following statement it is convenient to restrict to J-homomorphism to the stable special orthogonal group , which removes the lowest degree homotopy group in the above
The stable homotopy groups of spheres are the direct sum of the (cyclic) image of the J-homomorphism, def. 4, applied to the special orthogonal group and the kernel of the Adams e-invariant.
for and and positive the J-homomorphism is injective, hence its image is ,
for and hence for , the order of the image is equal to the denominator of in its reduced form, where is the Bernoulli number
for all other cases the image is necessarily zero.
This characterization of the image of is due to (Adams 66, Quillen 71, Sullivan 74). Specifically the identification of 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 is in (Ravenel, chapter 5, section 3).
See for instance (Ravenel, Chapt. 1, p. 5).
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 of the stable homotopy group . The appearance of a string of connected dots vertically above index means that there is a direct summand primary group of order . See example 1 below for illustration.
(The tables are taken from (Hatcher), where in turn were they were generated based on (Ravenel 86).
The finite abelian group decomposes into primary groups as . Here corresponds to the three dots above in the first table, and to the single dot over in the second.
The finite abelian group decomposes into primary groups as . Here corresponds to the four dots above in the first table, and to the single dot over in the second and to the single dot over 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 for the th Adams operation on complex K-theory.
Let be a prime. Consider coprime to .
The Adams conjecture implies that completed at , the J-homomorphism factors through the homotopy fiber of .
We have a homotopy-commuting diagram
The pasting composite with the homotopy pullback that witnesses the homotopy fiber of induces via the universal property of the loop space object a canonical map :
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 for the first Morava E-theory spectrum at given prime number . Write for the Bousfield localization of spectra of the sphere spectrum at .
The homotopy groups of the -localized sphere spectrum are
This appears as (Lurie 10, theorem 6)
Write for the p-localization of the sphere spectrum. For , write for the image of the -localized J-homomorphism
In this form this appears as (Lurie 10, theorem 7). See also (Behrens 13, section 1).
The -localization map is surjective on non-negative homotopy groups:
For review see also (Lorman 13). That factors through is in (Lorman 13, p. 4)
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 -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 is due to
John Adams, On the groups I, Topology 2 (3) (1963) (pdf)
John Adams, On the groups II, Topology 3 (2) (1965) (pdf)
- John Adams, On the groups III, Topology 3 (3) (1965) (pdf)
- John Adams, On the groups 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 at low primes are in
Other reviews include
Discussion from the point of view of chromatic homotopy theory is in
- Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the -local sphere, talk at 2013 Pre-Talbot Seminar (pdf)