Whitehead product

From Wikipedia, the free encyclopedia

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

$S^k \vee S^l$ ;

the attaching map is a map

$S^{k+l-1} \to S^k \vee S^l$ .

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

$S^{k+l-1} \to S^k \vee S^l \to X$

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

π k + l - 1 (X).

[edit] Grading

Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so $π k (X)$ has degree $(k - 1)$ ; equivalently, $L k = π k + 1 (X)$ (setting L to be the graded quasi-Lie algebra). Thus $L 0 = π 1 (X)$ acts on each graded component.

[edit] Properties

The Whitehead product is bilinear, graded-symmetric, and satisfies the graded Jacobi identity, and is thus a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product.

If $f \in \pi_1(X)$ , then the Whitehead bracket is related to the usual conjugation action of $π 1$ on $π k$ by

[f, g] = g f - g

,

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

[f, g] = f g f - 1 g - 1

,

which is the usual commutator.

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

[edit] See also

[edit] References

Uehara, Hiroshi; Massey, W. S. (1957), "The Jacobi identity for Whitehead products", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton, N. J.,: Princeton University Press, pp. pp. 361–377, MR 0091473
[1] J. H. C. Whitehead, On adding relations to homotopy groups, Annals of Mathematics, 2nd Ser., Vol. 42, No. 2. (Apr., 1941), pp. 409 –428.
[2] George W. Whitehead, On products in homotopy groups, Annals of Mathematics, 2nd Ser., Vol. 47, No. 3. (Jul., 1946), pp. 460 –475.

Whitehead product

From Wikipedia, the free encyclopedia

Contents

[edit] Grading

[edit] Properties

[edit] See also

[edit] References

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