Simplicial degree $d$ self-maps on $n$-spheres

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces. In this paper, we focus on the construction of non-degenerate simplicial maps of degree $d\in \mathbb{Z}$ on $n$-spheres for $n\geq 2$. We develop a general method, based on connected sums and facet orientations, to construct simplicial maps of any prescribed degree $d \in \mathbb{Z}$ between triangulated spheres. We investigate the asymptotic behavior of $\Lambda(n,d)$, defined as the minimum number of vertices required for a triangulated $n$-sphere to admit a simplicial map of degree $d$ to $\mathbb{S}^n_{n+2}$, for $n \geq 3$ and $d \geq 1$. As a consequence, we answer a question posed by Ryabichev in [22]. In addition to vertex-minimal constructions, we obtain facet-minimal degree maps for large degrees. Specifically, for each $d \geq n^2 + 1$, we construct a simplicial map of degree $d$ from a triangulated $n$-sphere with $d(n+2)$ facets to $\mathbb{S}^n_{n+2}$, for $n \geq 3$. As an application of the constructions, we derive improved bounds on the covering type of Moore spaces, refining results from [8]. Finally, we conclude with several open questions that may be of independent interest.