When we attach a disk to the boundary of the moebius strip we get the projective. Request pdf the 2extendability of graphs on the projective plane, the torus and the klein bottle a graph is said to be kextendable if any independent set of k edges extends to a perfect. The projective plane is of particular importance in relation to the. A cats game in tic tac toe is a game where neither side wins, even. Again it cannot be constructed in three dimensions, so is. This will allow us to show that these surfaces and s2 are all topologically distinct.
Orientable and nonorientable surfaces cornell university. Mathematics 490 introduction to topology winter 2007 what is this. The projective plane over k, denoted pg2,k or kp 2, has a set of points consisting of all the 1dimensional subspaces in k 3. For more information, see homology of real projective space. Using the method, some forbidden graphs on torus are obtained by the way. This two volume book contains fundamental ideas of projective geometry such as the crossratio, perspective, involution and the circular points at in. Request pdf toroidal and projective commuting and noncommuting graphs in this paper, all finite groups whose commuting noncommuting graphs can be embed on the plane, torus or projective. A different introduction to the projective plane by chris lambiehanson pictures of the projective plane by benno artmann pdf. Only one very important surface remains to be explored and of course we need a way to put surfaces together to make new surfaces. The final surface that can be obtained by identifying edges of a rectangle is even more complicated. We now consider one of the most important nonorientable surfaces the projective plane sometimes called the real projective plane. We construct a family of lagrangian submanifolds in the landauginzburg mirror to the projective plane equipped with a binodal cubic curve as anticanonical divisor.
In the simplest case a molecular space is a family of unit cubes in euclidean space e. From this point of view, the real projective plane is obtained by taking a 2sphere, cutting a hole, and pasting a moebius band on the edge of the hole. The author has recently proved that if the symmetry group of a compact selfdual manifold x is at least threedimensional, then x is either the complex projective plane or one of a handful of conformally flat manifolds 36. Anything that satisfies these rules is a projective plane, but when mathematicians refer to the projective plane, they generally mean a space more properly known as the real projective plane, or. To nd a triangulation of a klein bottle, the projective plane, etc. If a graphg is embedded in a manifoldm such that all faces are cells bounded by simple closed curves we say that this is a closed 2cell embedding ofg inm. 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. Rational real algebraic models of topological surfaces. Forbidden minors for projective plane are freetoroidal or. Deleting this band on the projective plane, we obtain a disk cf. Floer cohomology in the mirror of the projective plane and.
We embed this torus into the complex projective plane c geometry. Every surface is a connected sum of tori andor projective planes. We also describe the minimal graphs of a projective plane, a torus and a sphere. The klein bottle, obtained by gluing together two mobius bands, is similar in some ways to the torus, and is. Any nonzero vector z 1,z 2,z 3 in c 3 spans a complex line i. Triangulating the real projective plane 3 1 introduction the real projective plane p2 is in onetoone correspondence with the set of lines of the vector space r3. The connected sums of tori only are orientable the connected sums of pps only are. This paper will introduce these basic surfaces and provide two di erent proofs of the classi cation. Generating closed 2cell embeddings in the torus and the. We prove that every graph on the torus without triangles or quadrilaterals is 3colorable.
Another way to model the projective plane is to start with a hemisphere and connect each point on the rim to its corresponding point on the opposite side with a twist. Continuous, nonsingular transformations from the klein bottle to the torus rachel lash department of mathematics alfred university. The klein bottle and the projective plane are the basic nonorientable surfaces. Klein bottle and the torus bids fair to generalize to klein bottles and tori of. Axiom ap2 for the real plane is an equivalent form of euclids parallel postulate. Believe it or not, we have almost all the topological ingredients for making any surface whatsoever.
A subset l of the points of pg2,k is a line in pg2,k if there exists a 2dimensional subspace of k 3 whose set of 1dimensional subspaces is exactly l. An important result in topology, known as the classi cation theorem, is that any surface is a connected sum of the above examples. This settles a question raised in 1972 by kronk and white. In the projective plane, we have the remarkable fact that any two distinct lines meet in a unique point. There exists a projective plane of order n for some positive integer n.
Connected sums of real projective plane and torus or klein. We will focus on pagesmasked in chapter 8 which discusses betti numbers and pagesmasked in. The 2extendability of graphs on the projective plane, the. Coordinate system, chart, parameterization let mbe a topological space and u man open set. Quotient spaces and covering spaces city university of. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. These objects correspond under mirror symmetry to the powers of the twisting sheaf \\co1, and hence their floer cohomology groups form an algebra isomorphic to the homogeneous coordinate ring. In the case of torus general position the following transformations can be used. The projective plane is the space of lines through the origin in 3space. The confusing aspects of nonorientable 2manifolds have been captured in a delightful novel about the life within such a surface 1. Surfaces in the complex projective plane and their mapping.
Introducing the torus consider a circle in the xyplane with centre r,0 and radius a rotate. Continuous, nonsingular transformations from the klein. The 2sphere s2, torus, and projective plane play very fundamental roles in the classi. This was known earlier only for the sphere, the torus, the real projective plane and the klein bottle. More generally, if a line and all its points are removed from a. The classi cation of compact 2manifolds is sometimes credited to brahana 2. Like the klein bottle, the projective plane has no boundary and cannot be. Everything is taken out from references 1, 2 and 3, our contribution is just to give a selfcontained exposition and more details for certain parts of the original. It is called playfairs axiom, although it was stated explicitly by proclus. The complex projective plane, denoted cp2 is the set of complex lines in c3.
Minimal graphs of a torus, a projective plane and spheres. Such a diagram is called a polygonal presentation of a surface. The volume of a torus using cylindrical and spherical. Quotient spaces and covering spaces note the use of the moebius band described as a square with two opposite edges identi ed with reversed orientation. We shall show that every 5connected graph of even order embedded on the projective plane and every 6connected one embedded on the torus and the klein bottle is 2extendable and characterize the forbidden structures for 5connected toroidal graphs to be 2extendable. A sphere, klein bottle and projectiveplane can be constructed in the three following ways. It cannot be embedded in standard threedimensional space without intersecting itself. Connected sums of real projective plane and torus or klein bottle.
The sphere, the torus, and the projective plane are all examples of surfaces, or topological 2manifolds. Using mostly computer, we test how many edges may be added so that these graphs remain freetoroidal. The sphere, mobius strip, torus, real projective plane and klein bottle are all important ex. In section 2 we introduced it as the surface obtained from a rectangle by identifying each pair of opposite edges in opposite directions, as shown in figure 61. Betti numbers and the symmetry of the projective plane. I got as far as showing that it must be equivalent to a connected sum of projective planes, how can i argue though that i need precisely three projective planes.
Toroidal and projective commuting and noncommuting graphs. The only closed prime surfaces are the torus, the projective plane and the sphere. Monge view of a triangle in space invariant under projection. In particular, the second homology group is zero, which can be explained by the nonorientability of the real projective plane. Introductory topics of pointset and algebraic topology are covered in a series of. A graph is said to be kextendable if any independent set of k edges extends to a perfect matching. Van kampens theorem with torus and projective plane. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. After removing the darker mobius strip from the last two, we are left with a.