Whitehead product

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 an Annals of Mathematics paper from 1941.

Given elements $f\; in\; pi\_k(X),\; g\; in\; pi\_l(X)$, the Whitehead bracket

:$[f,g]\; in\; pi\_\{k+l-1\}(X)$

is defined as follows:

The product $S^k\; imes\; S^l$ can be obtained by attaching a $(k+l)$-cell to the wedge sum

:$S^k\; vee\; S^l$;

the attaching map is a map

:$S^\{k+l-1\}\; o\; S^k\; vee\; S^l$.

Represent $f$ and $g$ by maps

:$fcolon\; S^k\; o\; X$

and:$gcolon\; S^l\; o\; X$,

then compose their wedge with the attaching map, as

:$S^\{k+l-1\}\; o\; S^k\; vee\; S^l\; o\; X$

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

:$pi\_\{k+l-1\}(X).$

Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so $pi\_k(X)$ has degree $(k-1)$; equivalently, $L\_k\; =\; pi\_\{k+1\}(X)$ (setting "L" to be the graded quasi-Lie algebra). Thus $L\_0\; =\; pi\_1(X)$ acts on each graded component.

Properties

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra.

If $f\; in\; pi\_1(X)$, then the Whitehead bracket is related to the usual conjugation action of $pi\_1$ on $pi\_k$ by

:$[f,g]\; =g^f-g$,

where $g^f$ denotes the conjugation of $g$ by $f$.For $k=1$, this reduces to

:$[f,g]\; =fgf^\{-1\}g^\{-1\}$,

which is the usual commutator.

The relevant MSC code is:55Q15, Whitehead products and generalizations.

References

* [http://links.jstor.org/sici?sici=0003-486X%28194104%292%3A42%3A2%3C409%3AOARTHG%3E2.0.CO%3B2-5] J. H. C. Whitehead, "On adding relations to homotopy groups", Annals of Mathematics, 2nd Ser., Vol. 42, No. 2. (Apr., 1941), pp. 409 –428.
* [http://links.jstor.org/sici?sici=0003-486X%28194607%292%3A47%3A3%3C460%3AOPIHG%3E2.0.CO%3B2-T] George W. Whitehead, "On products in homotopy groups", Annals of Mathematics, 2nd Ser., Vol. 47, No. 3. (Jul., 1946), pp. 460 –475.