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A sphere map $f:S^n\to X$ is sent to $f_{\ast }[S^n]\in H_n(X;\mathbb{Z})$. After tensoring with $\mathbb{Q}$ this gives the rational Hurewicz homomorphism. It is natural in $X$, sends Whitehead products to zero, and is an isomorphism in the first nonzero homotopy degree of a simply connected space by the Hurewicz theorem.

For based maps $\alpha :S^p\to Y$ and $\beta :S^q\to Y$, use the standard cell decompositionThe attaching map $w:S^{p+q-1}\to S^p\vee S^q$ is obtained from the boundary of $D^p\times D^q$, collapsing the appropriate coordinate boundaries. The geometric Whitehead product is the homotopy classOrient the product cell consistently with the ordinary product orientation. Its class depends only on the two homotopy classes, and it is natural: $f_{\ast }[\alpha ,\beta {]}_W=[f_{\ast }\alpha ,f_{\ast }\beta {]}_W$.

For $p,q\ge 2$ it is additive in both variables:Thus it is bilinear over $\mathbb{Z}$, and after rationalization of a topological space over $\mathbb{Q}$. In the geometric convention its symmetry isFor a genuine graded Lie convention, put a class in ${\pi }_{p}Y$ in shifted degree $p-1$ and set $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$. Equivalently transport the Samelson product through reduced suspension and loop space adjunction. Writing $|\alpha |=p-1$, this bracket has degree zero, satisfiesand has the graded Jacobi identityThese are two equivalent sign presentations of the same Whitehead product. In particular the square of an odd-dimensional geometric class is two-torsion, whereas an even-dimensional sphere class can have a nonzero rational square. Reduced suspension kills Whitehead products, since the suspended attaching map of the product cell is null-homotopic. No proofs of these remaining algebraic properties are needed here.

Bilinearity has a specific fundamental-group exception. If both inputs lie in ${\pi }_{1}Y$, the attaching loop of the torus is the commutator, so the operation is $gh{g}^{-1}{h}^{-1}$, with the possible inverse corresponding to reversal of its boundary orientation. This is not a homomorphism in either input in a general nonabelian group.

If one input is $g\in {\pi }_{1}Y$ and the other is $\beta \in {\pi }_{q}Y$, $q\ge 2$, the product records the action difference $g\cdot \beta -\beta$, up to the fixed overall boundary sign. Choose the action convention matching that boundary orientation and write ${\delta }_g\beta =g\cdot \beta -\beta$. ThenThe second formula is a crossed-additivity law, not ordinary additivity in $g$. It follows by expanding $gh\cdot \beta -\beta =g\cdot (h\cdot \beta -\beta )+(g\cdot \beta -\beta )$. Consequently the higher-homotopy variable remains additive, while the fundamental-group variable is generally not. If the action is trivial the mixed product is zero. This explains exactly where the usual bilinearity statement fails.

For based continuous maps $\alpha :S^p\to G$ and $\beta :S^q\to G$ into a topological group, take the pointwise commutatorIt is exactly the identity on $S^p\vee S^q\subseteq S^p\times S^q$, so it descends to the smash product $S^p\wedge S^q\cong S^{p+q}$. Its homotopy class is the Samelson product $⟨\alpha ,\beta ⟩\in {\pi }_{p+q}(G)$. This measures failure of the two maps to commute; it vanishes for an abelian group.^[ref]

For an ordinary loop space, reversal is an inverse up to homotopy, and a retraced loop need not be the constant loop. Literal descent therefore requires a strict topological group model or the corresponding coherent contractions, rather than treating loop reversal as a strict inverse. Equivalently, suspension-loop adjunction defines the operation through the sign-adjusted Whitehead product in the original space.

In positive degrees these operations give the graded antisymmetry and Jacobi identity of the loop-space homotopy Lie algebra. Under the Hurewicz homomorphism, the product becomes the graded commutator in the Pontryagin ring.

The geometric Whitehead product of based maps $\alpha :S^p\to Y$, $\beta :S^q\to Y$ is their wedge sum map composed with the attaching map $S^{p+q-1}\to S^p\vee S^q$ of the top cell of $S^p\times S^q$. With the usual geometric signs it satisfies $[\alpha ,\beta {]}_W=(-1{)}^{pq}[\beta ,\alpha {]}_W$. Regrading ${\pi }_{p}Y$ to degree $p-1$ and setting $\{\alpha ,\beta \}=(-1{)}^p[\alpha ,\beta {]}_W$ gives the graded Lie bracket transported from the Samelson product on the loop space. In degrees $p,q\ge 2$ it is bilinear; fundamental-group inputs instead record commutators and the action of ${\pi }_1$ on higher homotopy.