Simplicial maps from the 3-sphere to the 2-sphere (Q2783469)

Let \(S^n\) be the \(n\)-sphere. It is well known that the homotopy classes of maps \(f:S^3 \to S^2\) are completely determined by the Hopf invariant of the map. In this paper the author considers the following question. Triangulate \(S^2\) as the boundary of the \(3\)-simplex. For each positive integer \(d\), find a minimal triangulation of \(S^3\) and a simplicial map \(g:S^3 \to S^2\) of Hopf invariant \(d\). Here ``minimal'' means with the smallest number of vertices and \(g\) must satisfy the further requirement that the preimage of any point is homeomorphic to a circle. In the article, the author gives a simplicial map of this type of Hopf invariant \(d\) and proves that it is minimal for \(d=1\) and \(d=2\). For \(d \geq 2\), the triangulation of \(S^3\) has \(6d\) vertices. For \(d=1\), the triangulation has \(12\) vertices. Minimality results from the assertion that if the triangulation of \(S^3\) has less than \(12\) vertices, then any simplicial map must be null-homotopic.