Paths and cylinders
Stable Homotopy theory
Homotopy groups of spheres
The homotopy groups of spheres …
The stable homotopy groups of the sphere spectrum …
The first stable homotopy groups of the sphere spectrum
The following tables for the p-primary components of in low degrees are taken from (Hatcher), where in turn they were generated based on (Ravenel 86).
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 . The bottom rows in each case are given by the image of the J-homomorphism. See example 1 below for illustration.
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.
The Serre finiteness theorem
The homotopy group is a finite group for except when and in which case
for a finite group.
J-homomorphism and Adams e-invariant
The following characterizes the image of the J-homomorphism
from the homotopy groups of the stable orthogonal group to the stable homotopy groups of spheres. This was first conjectured in (Adams 66) (since called the Adams conjecture) and then proven in (Quillen 71).
|Whitehead tower of orthogonal group||orientation||spin||string||fivebrane||ninebrane|
|homotopy groups of stable orthogonal group||0||0||0||0||0||0||0||0|
|stable homotopy groups of spheres||0||0||0|
|image of J-homomorphism||0||0||0||0||0||0||0||0||0|
Introductions and surveys include
Alex Writght, Homotopy groups of spheres: A very basic introduction (pdf)
Alan Hatcher, Stable homotopy groups of spheres (html)
A tabulation of stable homotopy groups of spheres is in
Original articles on basic properties include
- Jean-Pierre Serre _ Groupes d’homotopie et classes de groupes abelien_, Ann. of Math. 58 (1953), 258–294.
Image of the J-homomorphism
Discussion of the image of the J-homomorphism is due to
- John Adams, On the groups IV, Topology 5: 21,(1966) Correction, Topology 7 (3): 331 (1968)
- Daniel Quillen, The Adams conjecture, Topology. an International Journal of Mathematics 10: 67–80 (1971)
Formalization in homotopy type theory
For formalization in homotopy type theory see at