A note on twisted Cartesian products with base a sphere

We prove that, for a twisted Cartesian product S m × τ F {Sᵐ×τF} with base a sphere, the corresponding twisted tensor product C ⁢ ( S m ) ⊗ t C ⁢ ( F ) {C(Sᵐ)⊗ₜC(F)} induced by the Twisted Eilenberg–Zilber Theorem can be endowed with a differential graded coalgebra structure. Furthermore, we show that the corresponding injection C ⁢ ( S m ) ⊗ t C ⁢ ( F ) → C ⁢ ( S m × τ F ) C(Sᵐ)⊗ₜC(F)→ C(Sᵐ×τF) reduces to the Eilenberg–Mac Lane shuffle map while preserving the condition of morphism of differential graded coalgebras. These computations allow us to explicitly exhibit the differences and resemblances with respect to Szczarba’s approach to the study of fibre bundles.