Hello, I'm trying to construct an explicit map that takes the 4D torus to the 4-sphere such that the wrapping is non-trivial (i.e. homotopically, i.e. you can't shrink it continuously to zero). More concretely, I'm looking for [itex]\phi: T^4 \to S^4: (\alpha,\beta,\gamma,\delta) \mapsto ( x(\alpha,\beta,\cdots),y(\cdots),z(\cdots),t(\cdots),w(\cdots) )[/itex] where [itex]\alpha^2 + \beta^2 + \cdots + w^2 = 1[/itex]. I was thinking of embedding the 4D torus in 5D (i.e. it's natural representation) and then dividing each vector out by its norm, but I'm not sure how to ensure that it wraps around the sphere non-trivially.