Research Article A Rabbit Hole between Topology and .A Rabbit Hole between Topology and Geometry

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  • Hindawi Publishing CorporationISRN GeometryVolume 2013, Article ID 379074, 9 pages

    Research ArticleA Rabbit Hole between Topology and Geometry

    David G. Glynn

    CSEM, Flinders University, P.O. Box 2100, Adelaide, SA 5001, Australia

    Correspondence should be addressed to David G. Glynn;

    Received 10 July 2013; Accepted 13 August 2013

    Academic Editors: A. Ferrandez, J. Keesling, E. Previato, M. Przanowski, and H. J. Van Maldeghem

    Copyright 2013 David G. Glynn. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with differentaspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, propertiesinvariant under translations, rotations, or projections.The present paper shows away to go between them in an unexpected way thatuses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical propertiesare found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is thatthe most general geometry over noncommutative skewfields such as Hamiltons quaternions corresponds to planar graphs, whilegraphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers.

    1. Introduction

    The British/Canadian mathematician H.S.M. Coxeter (19072003) was one of most influential geometers of the 20thcentury. He learnt philosophy of mathematics from L.Wittgenstein at Cambridge, inspired M.C. Escher with hisdrawings, and influenced the architect R. Buckminster Fuller.See [1].When one looks at the cover of his book Introductionto Geometry [2], there is the depiction of the complete graph5on five vertices. It might surprise some people that such

    a discrete object as a graph could be deemed importantin geometry. However, Desargues 10-point 10-line theoremin the projective plane is in fact equivalent to the graph5: in mathematical terms the cycle matroid of

    5is the

    Desargues configuration in three-dimensional space, and aprojection from a general point gives the configurationaltheorem in the plane. Desargues theorem has long beenrecognised (by Hilbert, Coxeter, Russell, and so on) as one ofthe foundational theorems in projective geometry. However,there is an unexplained gap left in their philosophies: whydoes the graph give a theorem in space? Certainly, thematroids of almost all graphs are not theorems. The onlyother example known to the author of a geometrical theoremcoming directly from a graphic matroid is the completebipartite graph

    3,3, which gives the 9-point 9-plane theorem

    in three-dimensional space; see [3]. It is interesting that both


    3,3are minimal nonplanar (toroidal) graphs and

    both lead to configurational theorems in the same manner.In this paper, we explain how virtually all basic linear

    properties of projective space can be derived from graphsand topology. We show that any map (induced by a graphof vertices and edges) on an orientable surface of genus ,having V vertices, edges, and faces, where V + =2 2, is equivalent to a linear property of projective spaceof dimension V 1, coordinatized by a general commutativefield.This property is characterized by a configuration havingV+ points and hyperplanes.This leads to the philosophicaldeduction that topology and geometry are closely related, viagraph theory. If = 0 (and the graph is planar), the linearproperty is also valid for the most general projective spaces,which are over skewfields that in general have noncommuta-tivemultiplication.This is a powerful connection between thetopology of orientable surfaces and discrete configurationalproperties of the most general projective spaces.

    There are various fundamental theorems that pro-vide pathways between different areas of mathematics. Forexample, the fundamental theorem of projective geometry(FTPG) describes the group of automorphisms of projectivegeometries over fields or skewfields (all those of dimensionsgreater than two) as a group of nonsingular semilineartransformations. This most importantly allows the choice ofcoordinate systems in well-defined ways. Hence, the FTPG is

  • 2 ISRN Geometry

    a pathway between projective geometry and algebra, matrix,and group theory.

    Another example is the fundamental theorem of algebra.This provides another pathway between polynomials ofdegree over the real number field and multisets of roots,which are complex numbers. It explains why the complexnumbers are important for an understanding of the realnumbers.

    In a similar vein we show here how our rabbit holebetween topology and geometry can be used to obtain thebasic properties of the most general projective geometrydirectly from topological considerations.

    Here is an outline of the approach.

    (1) Consider the properties of fundamental configura-tions in (V 1)-dimensional projective geometry,which are collections of points and hyperplanes withincidences between them. The most important haveV points on each hyperplane, and these points form aminimal dependent set (a circuit inmatroid theory).

    (2) Inmost of these configurations, the algebraic propertythat corresponds to a configurational theorem is that aset of subdeterminants of size two in a general V matrix over a field has a linear dependency; that is,the vanishing of any 1 subdeterminants impliesthe vanishing of the remaining subdeterminant.

    (3) The condition for such a set of subdeterminants istopological: the dependency amongst the subdeter-minants happens if and only if there exists a graphhaving V vertices and edges embedded on anorientable surface of genus and inducing faces(certain circuits of the graph that can be contractedto a point on the surface).

    (4) A bonus is that when the surface has genus zero (i.e.,the graph is planar), the commutative field restrictionfor the algebraic coordinates of the space can berelaxed to noncommutative skewfields including thequaternions. This requires a different interpretationfor a 2 2 determinant and another proof dependingupon topological methods.

    (5) Since the latter method of planar graphs produces themain axiom for projective geometry (the bundle theo-rem or its dual Pasch axiom; see [4, page 24]) and theformer one for standard determinants over commu-tative fields produces the Pappus theorem, we see thatall bases are covered, and a topological explanationfor standard projective geometry, that is, embeddableinto space of dimension greater than two, is obtained.In the case of 2-dimensional geometries (planes) thereexist non-Desarguesian projective planes so thesegeometries do not appear to be produced topologi-cally; see [5, page 120] and [6, Section 23].

    2. Definitions and Concepts

    Let us summarize the topological and geometrical conceptsthat are used in this paper. A graph is a collection of verticeswith a certain specified multiset of edges, each of which is

    amultiset containing two vertices. If a vertex is repeated, thenthe edge is a loop. The graph is simple if it contains no loopand nomultiple edges, edges that are repeated.

    An orientable surface is a surface in real three-dimensional space that can be constructed from the sphere byappending handles; see [2, Section 21.1]. This surface has holes, andwe say that it has genus. One classical use for sucha surface is to parametrize the points on an algebraic curve inthe complex plane, but we have another application in mind.

    A skewfield or division ring is an algebraic structure(, +, ), where is a set containing distinct elements 0 and1, for which (, +) is an abelian (i.e., commutative) group,with identity 0, and ( := \ {0}, ) is a group (nonabelian ifthe skewfield is proper). The left and right distributive laws( + ) = + and ( + ) = + hold, for , , .The classical example of a proper skewfield is the quaternionsystem of Hamilton (four-dimensional over the reals). If themultiplicative group is abelian (i.e., commutative), iscalled a field. Thus a field is a special case of skewfield.Classical examples of a field are the rational numbers, thereal numbers, and the complex numbers. It is known (byWedderburns theorem and elementary field theory) that theonly finite skewfields are the Galois fields GF(), where is apower of a prime.

    A projective geometry of dimension over a skewfieldis the set of subspaces of a (left or right) vector spaceof rank + 1 over the skewfield. Points are subspaces ofprojective dimension zero, while hyperplanes are subspacesof projective dimension 1. It is well-known (or by theFTPG) that every projective space of dimension at least threehas a coordinatization involving a skewfield and comes fromthe relevant vector space.There are some incidence propertiesfor geometries over fields that are not valid for those over themore general skewfields. For example, the bundle theorem isvalid for skewfields (and fields), but Pappus 9

    3theorem only

    holds for geometries over fields.It is known that certain of the configurational theorems

    are in some sense equivalent in that assuming any one ofthem implies the remaining ones.These include the theoremsof Pappus,Mobius, andGallucci.These latter theorems are allexplained by the present topological theory. Desargues theo-rem and the bundle theorem (or its dual, the configurationof Pasch) are also in some sense equivalent in the case of themore general geometries over skewfields; see [6]. We showthat the bundle the