In mathematics, the Whitehead product is a
graded quasi-Lie algebra structure on the
homotopy groups of a space. It was defined by
J. H. C. Whitehead in .
The relevant
MSC code is: 55Q15, Whitehead products and generalizations.
Definition
Given elements
, the Whitehead bracket
:
is defined as follows:
The product
can be obtained by attaching a
-cell to the
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
:
;
the
attaching map In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ...
is a map
:
Represent
and
by maps
:
and
:
then compose their wedge with the attaching map, as
:
The
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
:
Grading
Note that there is a shift of 1 in the grading (compared to the indexing of
homotopy groups), so
has degree
; equivalently,
(setting ''L'' to be the graded quasi-Lie algebra). Thus
acts on each graded component.
Properties
The Whitehead product satisfies the following properties:
* Bilinearity.