

Homotopy Groups

When the domain
X
is a suspension of a
CW complex
with few cells, the based
homotopy set
[
X,Y
]
_{
*
}
is called a homotopy group. Although homotopy groups
are as fundamental as
homology groups
in algebraic topology, most elementary textbooks only
treat the
fundamental groups
. It should be noted, however, homotopy groups (
homotopy sets
) are more fundamental than homology groups, since
homology groups can be described in terms of homotopy
groups by the
DoldThom theorem
.
One of the most famous textbooks on algebraic
topology which contains enough materials on homotopy
groups is the book
[
Whi78
]
by G.W. Whitehead. And Gray’s book
[
Gra75
]
. There are two good Japanese books, an old book by
Komatsu, Nakaoka, and Sugawara and more recent book by
Nishida.
Both Whitehead’s and Gray’s books are
entitled “Homotopy Theory”. One of the
central objects of study in homotopy theory has been
homotopy groups, and especially
homotopy groups of spheres
.
Many people tried to compute homotopy groups of
spheres. One of the ﬁrst systematic attempts
was Adams’ work
[
Ada58
]
, but Adam’s method initially only worked for
stable homotopy groups. Toda’s method
[
Tod62
]
is more elementary and provide a powerful
computational method.
One of the most popular method to generalize homotopy
groups is to replace the spheres in the
deﬁnition of homotopy groups by other spaces.
And many more variations have been introduced.
[Ada58]
J. F. Adams. On the structure and
applications of the Steenrod algebra.
Comment. Math. Helv.
, 32:180–214, 1958.
[Gra75]
Brayton Gray.
Homotopy theory
. Academic Press [Harcourt Brace Jovanovich
Publishers], New York, 1975.
[Tod62]
Hirosi Toda.
Composition methods in homotopy groups of spheres
. Annals of Mathematics Studies, No. 49. Princeton
University Press, Princeton, N.J., 1962.
[Whi78]
George W. Whitehead.
Elements of homotopy theory
, volume 61 of
Graduate Texts in Mathematics
. SpringerVerlag, New York, 1978.


