Thomas Banchoff

The 9-vertex complex projective plane

In the early days of topology, most of the objects of interest were defined in terms of triangulations, describing a topological space as a union of finitely many vertices, edges, triangles, and higher dimensional simplexes identified in certain ways along their boundaries. A triangulation with a relatively small number of simplexes, symmetrically placed, could make computations easier and suggest new properties of the object itself. Although subsequent approaches to algebraic topology have stressed other ways of defining properties of topological spaces, the discovery of a new particularly nice triangulation of an important space can once again bring out relationships that lead to new insights in different branches of mathematics. In this article we describe such a triangulation for one of the most significant objects in topology, the complex projective plane. In a triangulation each edge has two distinct vertices and no two vertices determine more than one edge so the minimum number of vertices in a triangulation of a circle is three. In a triangulation of a surface or surface-with-boundary, each triangle has three distinct vertices and no three vertices determine more than one triangle. Moreover, at most two triangles come together at every edge. Since a cylinder has two edge curves a minimal triangulation requires at least six vertices and we can present a triangulation with precisely six vertices. On the other hand, it is possible to give a triangulation of the M6bius band with only five vertices (Figure 1, on next page) so in a sense the twist in the band makes it possible to triangulate using fewer vertices than in the case of the untwisted cylindrical band. The boundary of the M6bius band is a pentagon and every vertex is connected to every other vertex. By adding a cone over the pentagon from a sixth point we obtain a surface without boundary called the real projective plane with six vertices, fifteen edges (connecting all distinct pairs of vertices) and ten triangles. We denote this special triangulation by RP 2. It can be described by taking the icosahedral triangulation of the 2-sphere and then identifying opposite vertices, edges and faces. This triangulation RP 2 has numerous special properties. Although it is not possible to construct any nonself-intersecting real projective plane in ordinary Euclidean 3-space, we can construct a one-to-one mapping of RP 2 into Euclidean 4-space by first building a

Triple points and surgery of immersed surfaces

For a sufficiently general immersion of a smooth or polyhedral closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper involves elementary notions of modification of surfaces by surgery. Let f: M2 2E3 be an immersion in general position of a closed surface M2 into Euclidean 3-space so that there are finitely many N(f) triple points of /. The purpose of this note is to give a direct and elementary proof of the fact that N(f) x(M) (mod 2). The techniques used in this paper are related to the notion of surgery for surfaces, by which a pair of discs is replaced by a cylinder having the same boundary. The approach used in this paper was reported on at the American Mathematical Society Annual Meeting in Las Vegas, January 1972. With the added assumption of differentiability, it is possible to give a new proof of this same result using normal characteristic classes and singularities of projections, as in [l]. For an immersion /: M2 -. E3, set Gr(f) = lx in E3 f| -1(x) consists of precisely r points of M2}. The condition that f is in general position implies that if x e Gr(f) and f1(x) = tp1, P2,..., P,i then there are disjoint disc neighborhoods D(p1), D(p2),.9., D(pr) of these points in M and a homeomorphism w: B D3 of a ball neighborhood B of x to the unit ball in E3 so that (w of) (D(pi)) is the intersection of D3 with the plane orthogonal to the ith coordinate vector. If the immersion f is from a particular category, for example, differentiable or piecewise-linear, then we may assume that the homeomorphism w is in the same category, and in fact all the constructions which will be described can be altered in fairly standard ways so Presented to the Society January 18, 1972; received by the editors September 10, 1973. AMS(MOS) subject classfications(1970). Primary 57C35, 57D40, 57D65; Secondary 55A20.

Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions

From the Publisher: Beyond The Third Dimension investigates ways of picturing and understanding dimensions below and above our own. Ranging from Egyptian pyramids to the nineteenth-century satire Flatland to the paintings of Salvador Dali.

Topology and mechanics with computer graphics

The global topology of constant energy surfaces of Hamiltonian systems in four dimensions is investigated using techniques of computer graphics.

Equilibrium triangulations of the complex projective plane

Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of ℂP2 which fits the equilibrium decomposition of ℂP2 in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that ℂP2 modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation ℝP27of the real projective plane. We also determine all tight simplicial embeddings of ℂP210and ℝP27.

COUNTING TRITANGENT PLANES OF SPACE CURVES

LET C be a smooth simple closed curve in Iw3. A tritangent plane of C is a plane in W3 which is tangent to C at exactly three points. A stall x of C is a point of C at which the torsion of C is zero. We will say that a stall x is transoerse if the curvature of C is non-zero at x, the derivative of the torsion of C is non-zero at x, and the osculating plane P of C at x is transverse to C away from x. If x is a transverse stall of C then an interval of C about x lies on one side of the osculating plane P of C at x, so P intersects Cat an even number 2n of points other than x. The integer n = n(x, C) is the index of the transverse stall x of C. Let Coc(S’, rW3) be the space of C” maps ~1: S’ -+ W3 with the Whitney topology.

Triple points and singularities of projections of smoothly immersed surfaces

For a transversal smooth immersion of a closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper includes an introduction to normal Euler classes of smoothly immersed manifolds by means of singularities of projections. In this note we prove that for a (transversal) smooth immersion of a closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach used here includes an introduction to the normal Euler class of a smoothly immersed manifold and is related to the theory of Stiefel-Whitney classes in terms of singularities of projections as developed in [2]. The main result of this paper is correct also for manifolds which are not smooth, and such a proof has been carried out using surgery techniques in [3]. 1. Singularities of projections and the Euler characteristic. Let f: M2 -_ R3 be a smooth immersion of a closed 2-dimensional surface into real 3-dimensional space with a coordinate system (x, y, z). Let rrz: R3 + R(z) denote the orthogonal projection into the z-axis and let SP) be the set of critical points of ruz 0 f: M2 -_ R(z), i.e. the set of points where the tangent plane to f(M) is orthogonal to the z-axis. Let rxz : R3 -p R(x, z) denote the orthogonal projection into the xz-plane, and let SXZQ) be the set of singularities of 0xz ? f: M2 -_ R(x, z), i.e. the set of points where the tangent plane to f(M) is orthogonal to the xz-plane. We assume that the coordinate system in 3-space is so chosen that the set SP) consists of a finite number of points, consisting of mo(rz 0 f) local minima, m1(7z ? f) nondegenerate saddle points, and m 2(Z Tf) local maxima. Received by the editors November 15, 1973. AMS (MOS) subject classifications (1970). Primary 57D45, 57D20, 57D35.

Global geometry of polygons. I: The theorem of Fabricius-Bjerre

Deformation methods provide a direct proof of a polygonal analogue of a theorem proved by Fabricius-Bjerre and by Halpern relating the numbers of crossings, pairs of inflections, and lines of double tangency for smooth closed plane curves. Let X: \m, b] —> F describe a closed curve with a finite number C of crossings, a finite number F of inflection points (or inflection intervals), and a finite number of double support lines (i.e. lines containing two points of the curve each with a neighborhood lying to one side of the line, I being the number for which the neighborhoods lie on opposite sides of the line and II being the number with both neighborhoods to the same side). For a convex curve all these numbers are zero. Examination of a few examples leads to the conjecture that C + ViF + I II = 0, and in [2] Halpern announced a proof of this result for smooth curves satisfying certain regularity conditions. The proof described there uses techniques of critical points for vector fields and winding numbers and requires that the curve be of differentiability class C . The result was discovered by Fabricius-Bjerre [3], but again the proof made use of regularity slightly stronger than C . In this note we present an elementary proof of this result for polygonal curves in the plane. As in [2], this result generalizes to the case of mappings of an arbitrary 1-manifold into the plane (Theorem 1 ). The techniques developed in this paper will be used in subsequent investigations of polygonal analogues of a number of other global theorems on plane and space curves. There is also a sense in which these techniques fit in with the polyhedral analogue of catastrophe theory as developed in [l]. Definition. By a polygon we mean a mapping X: [a, b] —> E of a closed interval into the Euclidean plane such that for some finite subdivision, a = ?0 < Zj < • • • < t = b, the mapping X restricted to the interval t. < i < £.+ . Presented to the Society, April 27, 1973 under the title Polygonal methods in global curve theory; received by the editors January 17, 1973 and, in revised form, August 29, 1973. AMS (MOS) subject classifications (1970). Primary 53A05, 57C35; Secondary 55A25.

Normal curvatures and Euler classes for polyhedral surfaces in 4-space

Using the approach of singularities of projections into lower dimensional spaces it is possible to define nonintrinsic local curvature quantities at each vertex of a polyhedral surface immersed in 4-space which add up to the normal Euler number of the immersion. Related uniqueness results for lattice polyhedra have been established by B. Yusin. For a surface immersed in Euclidean n-space the total (or tangential) curvature can be expressed in terms of singularities of projections into lines, and this interpretation makes it possible to give a unified treatment of curvature for smooth and polyhedral embeddings. In this note we show that for a 2-dimensional surface immersed in 4-space we can carry out a similar construction for the normal curvature in terms of singularities of projections into oriented 3-spaces, recapturing the standard definition for smooth immersions and leading to a new curvature quantity for a polyhedral immersion f: M2 -v R4. We show how to assign to each vertex v of u2 a real number ivf (v) such that the normal Euler class v(f) of the immersion is the sum of the normal curvature at the vertices of M2. In contrast to the case of tangential curvatures where the quantities involved are intrinsic, depending only on the metric of the surface in a neighborhood of a vertex, the normal curvature will depend on the immersion f. This is to be expected since the (tangential) Euler characteristic is a topological invariant but the normal Euler class depends on the immersion. The constructions depend on the authors paper [2] defining the normal Euler class in terms of singularities of projections. Working independently, B. Yusin has constructed curvature quantities for vertex stars of lattice polyhedral surfaces in 4-space, with all edges parallel to the four coordinate axes [4]. His values agree with those described in this note and they establish a uniqueness result, showing that the curvature quantities described here are the only ones which can sum to the normal Euler class of an immersed polyhedral surface. Construction of the curvature quantities. For a smooth immersion f: M2 R4 we may define curvature quantities by using moving frames {ei, e2, e3, e4} with el, e2 tangent to f(M2) and e3, e4 normal to the surface. We define the 1forms Jij by dej = Ej=1 wijej. The tangential curvature of an open set U of M2 Received by the editors December 29, 1979 and, in revised form, August 12, 1983. 1980 Mathematics S*ect Classifiation. Primary 57R20, 57Q35, 53C42; Secondary 52A25.

TOTAL CENTRAL CURVATURE OF CURVES

Total central curvature refers to the measure of curvedness of a space curve contained in a ball (bounded by a sphere) obtained by averaging the total absolute curvatures of the image curves under central projection from all points on the sphere. The major object of this paper is to show that this total curvature coincides with the classical total absolute curvature of the original space curve. This result generalizes immediately to curves in n-space. As a corollary we show that a curve on S in E with total absolute curvature 4 in E can be unknotted in S3. We begin by studying, from an elementary standpoint, the specialization of this theorem to plane curves, and illustrate at the same time the methods toe be used in the general case. 1. Total central curvature of plane curves. Let f: S --, E be a continuous map of the circle S into the plane. A local support line to ] at x is a line containing x and bounding a closed half-plane which contains the image of a neighborhood of x in S1. Let r(]) be the number of local support lines to ] passing through the point p of E.