Graded homotopy groups

Observation 16.6.7 (Graded homotopy groups). The functor \[ \pi _*\colon \Sp \longrightarrow \Ab ^{\Z } \] admits a lax symmetric monoidal refinement when the target is equipped with the Koszul symmetry.

Indeed, for all \(n,m\in \Z \) the tensor product of representatives defines natural pairings \[ \pi _n(X)\otimes \pi _m(Y)\longrightarrow \pi _{n+m}(X\otimes Y), \qquad [x]\otimes [y]\longmapsto \bigl [\S ^{n+m}\simeq \S ^n\otimes \S ^m\xrightarrow {x\otimes y}X\otimes Y\bigr ]. \] Associativity and unitality are inherited from the symmetric monoidal structure on \(\Sp \). Interchanging \(x\) and \(y\) introduces the symmetry \[ \S ^n\otimes \S ^m\longrightarrow \S ^m\otimes \S ^n, \] which acts by \((-1)^{nm}\) on \(\S ^{n+m}\). For \(n,m\geq 0\), this is the sign of the block permutation interchanging \(n\) and \(m\) suspension coordinates; the general case follows by invertibility of suspension. This iterates the basic sign calculation of Warning 4.2.11. Thus the symmetry compatibility is exactly the Koszul rule. Associativity, unitality, and symmetry give all coherence conditions because the target is a \(1\)-category, so the pairings assemble into the claimed lax symmetric monoidal structure.

With the ordinary, unsigned Day symmetry, the same construction is only lax monoidal. Consequently, the homotopy groups of an associative ring spectrum form a graded ring, while those of a commutative ring spectrum form a graded-commutative ring: \[ xy=(-1)^{nm}yx \qquad (x\in \pi _nR,\ y\in \pi _mR). \]