nLab Samelson product

Idea

The Samelson product associated with a grouplike space (such as a topological group or a based loop space) is the group commutator operation descended from the Cartesian product to the smash product of the space with itself.

Definition

Let $\mathcal{G}$ be a grouplike space with:

• neutral element$\;$ $\mathrm{e} \,\colon\, \ast \longrightarrow \mathcal{G}$,

• binary operation$\;$ $(-)\cdot(-) \,\colon\, \mathcal{G} \times \mathcal{G} \longrightarrow \mathcal{G}$,

• inverse-assigning map $(-)^{-1} \,\colon\, \mathcal{G} \longrightarrow \mathcal{G}$,

and regarded as a pointed topological space with base point $\mathrm{e}$.

Consider the “group commutator”

(1)$$\begin{array}{ccc} \mathcal{G} \times \mathcal{G} &\overset{[-,-]}{\longrightarrow}& \mathcal{G} \\ (g_1 , g_2) &\mapsto& (g_1 \cdot g_2) \cdot (g_1^{-1} \cdot g_2^{-1}) \end{array}$$

(or any other way of bracketing it) and observe that this is a pointed map which as such vanishes on the wedge sum

$$\begin{array}{ccc} \mathcal{G} \vee \mathcal{G} &\xhookrightarrow{\phantom{--}}& \mathcal{G} \times \mathcal{G} \end{array}$$

in that $[g,\mathrm{e}] = [\mathrm{e}, g] = \ast$ for all $g \in \mathcal{G}$. Therefore this map descends to the quotient space which is the smash product

$$\mathcal{G} \wedge \mathcal{G} \,\coloneqq\, \frac{ \mathcal{G} \times \mathcal{G} }{ \mathcal{G} \vee \mathcal{G} } \,.$$

Definition

This descended map is, up to homotopy, the Samelson product on $\mathcal{G}$, which we shall denote by the same symbols:

(2)$$\mathcal{G} \wedge \mathcal{G} \xrightarrow{ \langle -,- \rangle } \mathcal{G} \mathrlap{\,.}$$

Induced from this, is the Samelson product on homotopy groups

(3)$$\pi_{n_1}(\mathcal{G}) \times \pi_{n_2}(\mathcal{G}) \xrightarrow{ \langle-,-\rangle } \pi_{n_1 + n_2}(\mathcal{G})$$

for $n_1, n_2 \in \mathbb{N}$, given by sending representative maps

$$g_i \colon S^{n_i} \longrightarrow \mathcal{G}$$

to the homotopy class of the following composite (their cup product under (2)):

$$S^{n_1 + n_2} \simeq S^{n_1} \wedge S^{n_2} \xrightarrow{ g_1 \wedge g_2 } \mathcal{G} \wedge \mathcal{G} \xrightarrow{ \langle - , - \rangle } \mathcal{G} \mathrlap{\,.}$$

Properties

Relation to the Whitehead bracket

For $X$ a connected pointed topological space, consider its loop space

$$\mathcal{G} \coloneqq \Omega X \mathrlap{\,,}$$

regarded as a grouplike space with a Samelson bracket $\langle-,-\rangle$ (2), whence (by the looping and delooping relation)

$$X \simeq B \mathcal{G} \mathrlap{\,.}$$

Then we have:

1. the Samelson product (3) on on the homotopy groups of $\mathcal{G}$:

   $$\langle -, - \rangle \;\colon\; \pi_n(\mathcal{G}) \times \pi_m(\mathcal{G}) \longrightarrow \pi_{n+m}(\mathcal{G}) \mathrlap{\,.}$$

2. the Whitehead bracket on the homotopy groups of $X$:

   $$[- , - ] \;\colon\; \pi_n(X) \times \pi_m(X) \longrightarrow \pi_{n+m-1}(X) \mathrlap{\,,}$$

3. the canonical isomorphism

   (4)$$\tau \,\colon\, \pi_{n+1}(X) \xrightarrow{ \sim } \pi_n(\mathcal{G}) \mathrlap{\,.}$$

Proposition

Under the isomorphism (4) the Samelson product on the homotopy groups of $\mathcal{G}$ coincides up to a sign with the Whitehead bracket on the homotopy groups of $X \simeq B \mathcal{G}$, in that for pairs $\alpha_p \in \pi_{p+1}(X)$, $\beta_q \in \pi_{q+1}(X)$ we have

$$\tau[\alpha_p, \beta_q] \;=\; (-1)^p \big\langle \tau \alpha_p ,\, \tau \beta_q \big\rangle \mathrlap{\,.}$$

(cf. Whitehead 1978 §X.7 Thm 7.10 (p. 476))

Related concepts

• Whitehead product

• Pontrjagin product

References

Original reference, proving that the Whitehead product is the commutator of the Pontrjagin product:

• Hans Samelson, A Connection Between the Whitehead and the Pontryagin Product, American Journal of Mathematics 75 4 (1953) 744–752 [doi:10.2307/2372549]

• George W. Whitehead: On mappings into group-like spaces, Commentarii Mathematici Helvetici 28 (1954) 320–328 [doi:10.1007/BF02566938]

Review:

• George Whitehead: The Samelson Product, §X.5 in: Elements of homotopy theory, Springer (1978) [doi:10.1007/978-1-4612-6318-0]

• Joseph Neisendorfer: Samelson products and exponents of homotopy groups, Journal of Homotopy and Related Structures 8 2 (2013) 239-277 [doi:10.1007/s40062-012-0021-4]

• Joseph Neisendorfer: Samelson products, Chapter 6 in: Algebraic Methods in Unstable Homotopy Theory, 158-220. Cambridge University Press. [doi:10.1017/cbo9780511691638.008]

History:

• Ioan Mackenzie James: Products between homotopy groups, Compositio Mathematica, 23 3 (1971) 329-345 [numdam:CM_1971__23_3_329_0]

In the context of gauge groups (automorphism groups) of principal bundles:

• Christoph Wockel: The Samelson Product and Rational Homotopy for Gauge Groups, Abh. Math. Sem. Univ. Hamburg 77 (2007) 219-228 [arXiv:math/0511404]