Elementary constructions of locally compact affine planes
A well known method to construct topological affine planes is the following: Given \(n \in \{1,2,4,8\}\) and a planar function \(f : \mathbb{R}^n \to \mathbb{R}^n\), i.e. a continuous function such that the mapping \(f_d : \mathbb{R}^n \to \mathbb{R}^n\); \(x \mapsto f(x + d) - f(x)\) is bijective for each \(d \in \mathbb{R}^n \backslash \{0\}\), let \(L(a,b) : = \{(x, f(x - a) + b) : x \in \mathbb{R}^n\}\) \((a,b \in \mathbb{R}^n)\), \(L(c) = \{(c,x) : x \in \mathbb{R}^n\}\) \((c \in \mathbb{R}^n)\); then the point set \(\mathbb{R}^{2n} = \mathbb{R}^n \times \mathbb{R}^n\) with the set \(\{L(a,b) : a,b \in \mathbb{R}^n\} \cup \{L(c) : c\in \mathbb{R}^n\}\) of lines is a topological affine plane; a so-called shift plane. Many of the known 2 and 4 dimensional topological affine planes are shift planes. Unfortunately, it is known that neither 8 or 16 dimensional shift planes exist. In the paper under review, the author presents a generalization of the above shift construction which is able to construct 8 and 16 dimensional planes. We call a continuous map \(f : \mathbb{R}^{2n} \to \mathbb{R}^n\) a multiplication, if for each \(a_1\), \(a_2 \in \mathbb{R}^n\) with \(a_1 \neq a_2\) the mappings \(f^{(a_1, a_2)}\), \(f_{(a_1, a_2)} : \mathbb{R}^n \to \mathbb{R}^n\) given by \(f^{(a_1, a_2)} (x) = f(a_1, x) - f(a_2, x)\) and \(f_{(a_1, a_2)} (x) = f(x, a_1) - f(x, a_2)\), respectively, are both bijective. With the lines \(L(a,b) : = \{(x, f(a,x) + b) : x \in \mathbb{R}^n\}\) we get again a topological affine plane. Classical examples of multiplications are the multiplication maps of the classical real division algebras, i.e. \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) and \(\mathbb{O}\). The author presents the following construction of new multiplication maps. Let \(n \in \{2,4,8\}\) and denote the multiplication in the \(n\)-dimensional classical real division algebra by \(m\). For \(1 \leq i \leq n\) the \(i\)-th component of \(m\) has the form \(m_i (x,y) = \sum^n_{j = 1} s_{ij} x_{t (i,j)} y_j\) with a permutation \(t(i,-)\) of \(1, \dots, n\) and \(s_{ij} \in \{- 1,1\}\). For \(1 \leq i\), \(j \leq n\) let \(m_{ij} : \mathbb{R}^2 \to \mathbb{R}\) be a multiplication map on \(\mathbb{R}\). Then we define a new mapping \(m_M : \mathbb{R}^{2n} \to \mathbb{R}^n\) by replacing the term \(x_{t(i,j)} y_j\) in the above sum with \(m_{ij} (x_{t(i,j)}, y_j)\). The author provides some criterions to decide whether the map \(m_M\) is a multiplication on \(\mathbb{R}^n\). Using these criterions he is able to construct new topological affine planes.
scientific article; zbMATH DE number 806253
scientific article