Day Convolution and Homotopy Groups on the Pointed Homotopy Category of CW Complexes

Abstract

We freely enrich the pointed homotopy category of CW complexes over abelian groups and equip the resulting functor category with the enriched Day convolution induced by the smash product. After identifying the convolution unit and recording the coherence maps, we compute the convolution of the homotopy-group functors. For integers $n,m\geq 2$ we construct a natural isomorphism $π_n\starπ_m\congπ_{n+m}$ induced by the external smash product; equivalently, the external smash product is the universal natural pairing of homotopy groups. Under this identification, the symmetry interchanging the two factors acts on $π_{n+m}$ by multiplication by $(-1)^{nm}$.