An introduction to topology the classification theorem for surfaces. We will also prove that if n 5, and there is an arithmetic subgroup of gof covolume a submultiple of 15r, then k q, so r6 1. Let rp2 be the real projective plane and pgl3, r the group of projective transformations rp2 rp2. To verify that this is in fact a projective plane with one handle, we compute its euler characteristic. Once we have proven this result, we invest chapters 3 and 4 to a systematic study of two important types of characteristic classes associated to real vector bundles, namely, the stiefelwhitney classes and the euler class. The following examples of real vector bundles will be of vital importance throughout the rest of our work. In mathematics, the real projective plane is an example of a compact non orientable. Euler and algebraic geometry burt totaro eulers work on elliptic integrals is a milestone in the history of algebraic geometry. The tube is of euler characteristic 0, so the resulting surface has euler characteristic 2 less than the euler characteristic of m, hence the surface is m plus a handle. Homogeneous coordinates of points and lines both points and lines can be represented as triples of numbers, not all zero. Topologically, it has euler characteristic 1, hence a demigenus nonorientable genus, euler genus of 1. The real projective space, denoted by rpn, is the set of all unordered pairs fx.
Yet the euler characteristic is 2 for the sphere and 2n for the connected sum of n protective planes. The real projective plane is the unique nonorientable surface with euler characteristic equal to 1. As is an irreducible maximal arithmetic subgroup of g, there exist a totally real number eld k, an absolutely simple. Foundations of projective geometry bernoulli institute. The euler poincar e characteristic of pn 1 c, and so also of any arithmetic fake pn 1 c, is n. We now consider one of the most important nonorientable surfaces the projective plane sometimes called the real projective plane. Mobius band, there are two nonisotopic unknotted projective planes, p2.
Tightness for smooth and polyhedral immersions of the real. This is somewhat difficult to picture, so other representations were developed. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics, pascals theorem, poles and polars. Lecture 14 031120 homology with field coefficients and computational examples the homology of the real projective plane. We conclude with an application of the euler characteristic as an approach to solving the mapcoloring problem in. There is also a projective plane over the octonians. In comparison the klein bottle is a mobius strip closed into a cylinder. In an immersed projective plane with only one triple point, these arcs are joined pairwise by the double curve dotted lines. We show that the clifford torus and the totally geodesic real projective plane rp2 in the complex projective plane cp2 are the unique hamiltonian stable minimal lagrangian compact surfaces of cp2 with genus less than or equal to 4, when the surface is orientable, and with euler characteristic greater than or equal to 1, when the surface is nonorientable. Euler characteristic of the projective plane using. The set whose elements are straight lines going through the origin in threedimensional euclidean space is known as the real projective plane. Deleting this band on the projective plane, we obtain a disk cf. Euler characteristic does not depend on the tiling of the surface or deformations of the surface but it does depend on the overall shape of the surface.
This is easily proved by induction on the number of faces determined by g, starting with a tree as the base case. And lines on f meeting on m will be mapped onto parallel lines on c. One of the most important numerical invariants of a germ of an analytic function f. Euler characteristic wikipedia, the free encyclopedia. The euler characteristic, explored in section 5, is used to prove thm 5. As before, points in p2 can be described in homogeneous coordinates, but now there are three nonzero. Cn c with an isolated singularity at the origin is the sequence. For the cylinder, since we identity awith a0, there are two vertices aa0 and bb0, four edges and two triangles. The euler characteristic, betti numbers, barcodes and.
The totally geodesic real projective plane rp2 is the unique hamiltonian stable minimal lagrangian compact nonorientable surface of cp2 with euler characteristic. A surface in threespace is tight provided that any plane cuts it into at. The vanishing of the top wu class is in fact a stronger condition than having an even euler characteristic. Hamiltonian stability and index of minimal lagrangian. Looking at these manifolds as equivalences on the closed disk, it seems that their euler characteristic should be the same. The euler characteristic of any plane connected graph g is 2. Similarly, we have seen a subdivision of the torus with euler characteristic 0. One may observe that in a real picture the horizon bisects the canvas, and projective plane. Euler characteristic of the projective plane and sphere. Euler and algebraic geometry university of california. The projective plane is of particular importance in relation to the. Pdf average volume, curvatures, and euler characteristic.
M on f given by the intersection with a plane through o parallel to c, will have no image on c. Geometry and topology shp fall 16 columbia mathematics. The two disks are each of euler characteristic 1, so their removal lowers the euler characteristic of m by 2. Euler characteristic of the projective plane using embedding. We determine that the deformation space of convex real projective structures, that is, projectively. The euler characteristic of a space with finitely generated homology, denoted, is defined as a signed sum of its betti numbers, viz. Remember that the points and lines of the real projective plane are just the lines and planes of euclidean xyzspace that pass through 0, 0, 0. A constructive real projective plane mark mandelkern abstract. It is an immediate consequence of the hirzebruch proportionality principle, see se1, proposition 23, that the orbifold euler poincar e characteristic i. Characteristic points, fractional monodromy and euler. Classically, the real projective plane is defined as the space of lines through the origin in euclidean threespace. The invariant thus obtained depends only on the homotopy class of f rel n.
This considerably extends previously known results on the number of roots, the volume, and. We know that the euler characteristic of a solid plane recangle is 1. Planar graphs the euler characteristic can be defined for connected planar graphs by the same formula as for polyhedral surfaces, where f is the number of faces. Pdf average volume, curvatures, and euler characteristic of. For via stereographic projection the plane maps to the twodimensional sphere, such that the graph maps to a polygonal decomposition of. Tightness for smooth and polyhedral immersions of the. Another example of a projective plane can be constructed as follows. The euler characteristic can be defined for connected plane graphs by the same. That fact was only established twentyfive years later, using very sophisticated techniques from algebraic topology 15, 171.
Sampling the klein bottle, the projective plane of. The real projective plane in homogeneous coordinates plus. For example, every subdivision of the sphere has euler characteristic 2. Both the klein bottle and the real projective plane contain m. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics. In mathematics, the real projective plane is an example of a compact nonorientable. In particular, the expected euler characteristic of such random real projective varieties is found. Euler characteristic an overview sciencedirect topics. The sphere, mobius strip, torus, real projective plane and klein bottle are all. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in r 3 passing through the origin. The torus t can be constructed from a rectangular sheet of paper by identifyinggluing opposite sides of the sheet.
An embedded deformation of a projective line in the projecti ve plane r p 2 has at least 3 in ection points. In mathematics, the real projective plane is an example of a compact non orientable twodimensional. All ends in a structure of nite volume are cusps in the same manner that noncompact complete hyperbolic surfaces of nite volume have cusps at their ends. In the case where m is a compact, connected manifold without boundary and n is a compact, connected surface without boundary different from the 2sphere and the real projective plane, we formulate this property in terms of the pure and full 2string braid groups of n, and of the fundamental groups of m and the orbit space of m with respect to. The borsukulam property for homotopy classes of selfmaps. Arithmetic fake projective spaces and grassmannians 3 6 1nr, and if n 7, there does not exist an arithmetic subgroup whose covolume is a submultiple of 17r. Manifolds with odd euler characteristic and higher. Euler characteristic does not depend on the tiling of the surface or deformations of the surface but it does depend on. Faces given a plane graph, in addition to vertices and edges, we also have faces. The projective plane we now construct a twodimensional projective space its just like before, but with one extra variable. An abstract graph that can be drawn as a plane graph is called a planar graph. Locally plane every point has a neighbourhood homeomorphic to an open disc of the real euclidean plane. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold.
A tight polyhedral immersion of the twisted surface of. The purpose of this paper is to investigate convex real projective structures on compact surfaces. The projective plane over r, denoted p2r, is the set of lines through the origin in r3. Cervone october 15, 2003 1 introduction a longstanding open problem in the study of tight surfaces centered around a question posed by nicolaas kuiper asking whether the surface with euler characteristic. A closed surface embeds in the 3dimensional real projective space if and only if it is orientable or of odd euler characteristic. The others have curvatures which lie between 1 and 4. The simplest nonorientable surface is the real projective plane. The euler characteristic can be defined for planar graphs by the same v. It cannot be embedded in standard threedimensional space without intersecting itself. Whats wrong with the growth of simple closed geodesics on. A tight polyhedral immersion of the twisted surface of euler. If the euler characteristics of s is odd, then we may choose the projective. The mobius strip with a single edge, can be closed into a projective plane by gluing opposite open edges together. Formally, this means that the set p consists of all antipodal pairs p.
Projective polyhedra all have euler characteristic 1, corresponding to the real projective plane, while toroidal polyhedra all have euler characteristic 0, corresponding to the torus. The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. However, on the right we have a different drawing of the same graph, which is a plane graph. For regular polyhedra, arthur cayley derived a modified form of euler s formula using the density d, vertex figure density d v, and face density. So one projective plane should have euler characteristic of 1. Euler number of a smooth embedding of the real projective plane in 4space. Milnor numbers of projective hypersurfaces and the. Here is my diagram of the identified square of the projective plane with the embedded graph on it. A convex real projective manifold convex rp2manifold is a quotient m.
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