Whitehead product について

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Whitehead product ：ウィキペディア英語版

Whitehead product
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 .
== Definition ==
Given elements

f \in \pi_k(X), g \in \pi_l(X)

, the Whitehead bracket
:

) \in \pi_(X) \,

is defined as follows:
The product

S^k \times S^l

can be obtained by attaching a

(k+l)

-cell to the wedge sum
:

S^k \vee S^l

;
the attaching map is a map
:

S^ \to S^k \vee S^l. \,

Represent

f

and

g

by maps
:

f\colon S^k \to X \,

and
:

g\colon S^l \to X, \,

then compose their wedge with the attaching map, as
:

S^ \to S^k \vee S^l \to X \,

The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of
:

\pi_(X). \,

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