Abstract
Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with different aspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. The present paper shows a way to go between them in an unexpected way that uses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical properties are found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is that the most general geometry over noncommutative skewfields such as Hamilton's quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers.
Highlights
Theorem 1 relates graphs or maps on orientable surfaces of any genus to configurational theorems in general projective space over any commutative field. This uses 2 × 2 determinants with the standard definition. For general skewfields this definition of determinant does not work, and so we use Lemma 2 to find an alternate way and find that there is a restriction to surfaces of genus zero
Any graph G embedded on an orientable surface of genus g ≥ 0, having V vertices, e edges, and f faces, where by Euler’s formula V − e + f = 2 − 2g, is equivalent to a certain configurational theorem in projective space PG(V − 1, F), where F is any commutative field
For the above embedding in the plane we find a natural Eulerian orientation that is determined by the faces
Summary
We show that any map (induced by a graph of vertices and edges) on an orientable surface of genus g, having V vertices, e edges, and f faces, where V − e + f = 2 − 2g, is equivalent to a linear property of projective space of dimension V − 1, coordinatized by a general commutative field This property is characterized by a configuration having V+f points and e hyperplanes. (4) A bonus is that when the surface has genus zero (i.e., the graph is planar), the commutative field restriction for the algebraic coordinates of the space can be relaxed to noncommutative skewfields including the quaternions This requires a different interpretation for a 2 × 2 determinant and another proof depending upon topological methods. In the case of 2-dimensional geometries (planes) there exist non-Desarguesian projective planes so these geometries do not appear to be produced topologically; see [5, page 120] and [6, Section 23]
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have