Pdf matroid duality from topological duality in surfaces. If two smooth plane curves have the same degree, hence the same topology, they are linearly equivalent, so by the exact sheaf sequence of a divisor on a surface also have the same chio. The search for a finite projective plane of order 10. Note that an euler characteristic of 1 corresponds to the nonorientable projective plane. The euler characteristic of a space with finitely generated homology, denoted, is defined as a signed sum of its betti numbers, viz. Embeddings of 3connected 3regular planar graphs on surfaces. An introduction to topology the classification theorem for surfaces. Milnor numbers of projective hypersurfaces and the chromatic.
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. Manifolds with odd euler characteristic and higher. Yet the euler characteristic is 2 for the sphere and 2n for the connected sum of n protective planes. Article pdf available in combinatorics probability and computing 115. 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. The euler characteristic is another major invariant for groups which are virtually fp. The euler characteristic of is a homology, homotopy and topological invariant of. Euler characteristic we can also see something special in the table if we look along any row. Self intersections in immersions of the projective plane. Geometry of algebraic curves university of chicago. Note that the largest euler characteristic is 2, and it corresponds to a sphere. Euler characteristic of the projective plane and sphere. The euler characteristic can be defined for connected plane graphs by the same. Could be that it is the euler characteristic of a projectiv plane, but i cant jugde on that one.
Common and familiar examples include triangles, squares, rectangles. This as somewhat of a surprise, since on the may come. We label all the vertices, edges and faces, using the. The euler characteristic of a connected sum of two surfaces is given by the relation loss of two open disks. In degree one, both 1genus and chio are equal to 1. The sphere, mobius strip, torus, real projective plane and klein bottle are all important ex. Then for euler characteristic of 0, we have two surfaces, one orientable, one not. Eulers formula by adam sheffer plane graphs a plane graph is a drawing of a graph in the plane such that the edges are noncrossing curves. Using the relation between genus and euler characteristic we have. The proof given is elementary in the sense that only geometric techniques are used. Euler characteristic of the projective plane using. Note that corresponding cwspace is a space with finitely generated homology and the euler characteristic of that topological space equals the euler characteristic of the cwcomplex. Triangulations and the euler characteristic let sbe a compact connected surface.
The euler characteristic is a topological invariant that means that if two objects are topologically the same, they have the same euler characteristic. This is somewhat difficult to picture, so other representations were developed. One of the most important numerical invariants of a germ of an analytic function f. Real projective plane 1 mobius strip 0 klein bottle 0. We found a cell structure with two 2cells, six 1cells, and four 0cells, so. Worksheet on euler characteristic for surfaces this worksheet accompanies todays lecture on euler characteristic for surfaces. The earlier examples now enable us to conclude that the euler characteristic of the sphere is 2, of the closed disc is 1, of the torus is 0, of the projective plane is 1, of the torus with 1 hole is. Logically, we should explore next nonsome orientable surfaces of euler characteristic.
The vanishing of the top wu class is in fact a stronger condition than having. Observe that there are one edge in a projective plane and two edges in a torus. Find the euler characteristic of the following surfaces. Is it possible to draw 5 points on the plane and connect each pair of points with a line segment in such a way that the line segments do not cross. The class of projective planes intersects the class of 3configurations in the fano plane pg2, 2, as we have seen. The euler characteristic of any plane connected graph g is 2. Steiners roman surface is a more degenerate map of the projective plane into 3space. An projective algebraic variety xis a subset of a complex projective space pn of form x fx2pn. Span tree projective plane simplicial complex euler characteristic klein bottle these keywords were added by machine and not by the authors.
In particular, it does not depend on the way in which the space is partitioned into cells. The founders of calculus understood that some algebraic functions could be integrated using elementary functions logarithms and. Self intersections in immersions of the projective plane paulo henrique renato f. Euler characteristic an overview sciencedirect topics. 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.
Consequently one can speak, for example, of the euler characteristic of an arbitrary compact polyhedron, meaning by it the euler. The projective plane is of particular importance in relation to. In this note, we shall consider sas a topological surface, meaning a hausdor topological space such that each point pin s has an open neighbourhood u u p homeomorphic to an open disc in r2. Euler number of a smooth embedding of the real projective plane in 4space. The torus t can be constructed from a rectangular sheet of paper by identifyinggluing opposite sides of the sheet. But it turns out that there exist no regular maps on singlesided surfaces of this genus. The vanishing of the top wu class is in fact a stronger condition than having an even euler. Geometry of algebraic curves lectures delivered by joe harris notes by akhil mathew fall 2011, harvard contents lecture 1 92 x1 introduction 5 x2 topics 5 x3 basics 6 x4 homework 11 lecture 2 97 x1 riemann surfaces associated to a polynomial 11 x2 ious from last time. A presentation of the projective plane is a aa and a presentation of the sphere is b bb1 yet the euler characteristic is 2 for the sphere and 2n for the connected. Matroid duality from topological duality in surfaces of nonnegative euler characteristic.
However, on the right we have a different drawing of the same graph, which is a plane graph. Hence, the doubling operation constructs the sphere from the projective plane, the torus from the klein bottle, etc. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. Vianna y scott northrup z july 26, 2007 this paper was written during the ires 2007 program, from june 29th.
We claimed, but did not really prove, the seemingly obvious fact that these. This process is experimental and the keywords may be updated as the learning algorithm improves. Such a surface is known to be projective algebraic and it is the quotient of the open unit ball bin c2 bis the symmetric space of pu2,1 by a torsionfree cocompact discrete subgroup of pu2,1 whose eulerpoincar. A closed surface embeds in the 3dimensional real projective space if and only if it is orientable or of odd euler characteristic. It cannot be embedded in standard threedimensional space without intersecting itself. So the euler characteristic is a number intrinsic to the underlying topology of an object, not its speci. Similarly, we have seen a subdivision of the torus with euler characteristic 0.
Since the the euler characteristic of the projective plane is one and bancho s theorem states that the number of triple points for any immersion of the projective plane must be odd, the. This is easily proved by induction on the number of faces determined by g, starting with a tree as the base case. A general cubic curve of 7r is any set of points ca, a. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. 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. If the euler characteristics of s is odd, then we may choose the projective plane p2 in s so that the surface s in the splitting 1 is orientable. Euler and algebraic geometry burt totaro eulers work on elliptic integrals is a milestone in the history of algebraic geometry.
It is equal to 2 2n for the ntorus and 2 n for the sphere with n crosscaps. All cell structures on the projective plane will give this same euler characteristic. However, it is possible for a cwcomplex with infinitely many cells either infinitely many cells at a given dimension, or arbitrarily large dimensions that still. In these papers, they analysed reembedding structures of nonplanar graphs. The founders of calculus understood that some algebraic functions could be integrated using elementary functions logarithms and inverse trigonometric functions. The following problems are designed to lead to the discovery of the euler characteristic and to the understanding of why it is a topological invariant. The main proof was presented here the paper is behind a paywall, but there is a share link from elsevier, for a few days january 19, 2020. There is an important topological invariant called the euler characteristic. So one projective plane should have euler characteristic of 1. Euler characteristic euler characteristic does not depend on the tiling of the.
Complex ball quotients and line arrangements in the. But objects with the same euler characteristic need not be topologically equivalent. An example of fake projective plane was rst constructed. Consequently one can speak, for example, of the euler characteristic of an arbitrary compact polyhedron, meaning by it the euler characteristic of any of its triangulations. Euler characteristic of the projective plane using embedding. We now consider one of the most important nonorientable surfaces the projective plane sometimes called the real projective plane. The projective plane can be immersed local neighbourhoods of the source space do not have selfintersections in 3space. Looking at these manifolds as equivalences on the closed disk, it seems that their euler characteristic should be the same. Definition the euler characteristic of a finite cell complex.
Classically, the real projective plane is defined as the space of lines through the origin in euclidean threespace. Mathematics 490 introduction to topology winter 2007 what is this. Milnor numbers of projective hypersurfaces and the. Projective plane euler characteristic klein bottle connected neighborhood finite complex these keywords were added by machine and not by the authors. The real projective plane is the unique nonorientable surface with euler characteristic equal to 1. Euler characteristic 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. Smstc geometry and topology 201220 lecture 9 the seifert.
For example, every subdivision of the sphere has euler characteristic 2. We will also state and prove old and new results of the type that. Cn c with an isolated singularity at the origin is the sequence. Part xix euler characteristic and topology the goal for this part is to classify topological surfaces based on their euler characteristic and orientability. On cubic curves in projective planes of characteristic two. Pdf matroid duality from topological duality in surfaces of. Embeddings of 3connected 3regular planar graphs on. Dont ask me why they did call it characteristic on the homework sheet.
This book examines the explicit computation of this proportionality deviation for. Faces given a plane graph, in addition to vertices and edges, we also have faces. In section 3, we shall construct the complete list of reembedding structures of a planar graph gembedded on the projectiveplane, the torus or the klein bottle when gis 3connected and 3regular. Now lets see if the euler characteristic can ever be a nontwo number. An abstract graph that can be drawn as a plane graph is called a planar graph.
This notion coincides with the topological euler characteristic if the group g has a finite kg, 1 which requires that g be torsion free. So chio is a topological invariant for smooth plane curves. The hattoristallings rank see brw1 makes it possible to define euler characteristic using complexes of projective modules rather than free modules. Moreover, as ag2, 2 has been shown to be a planar geometry, ag2, 3 is the first candidate for serious. Euler characteristic of some familiar surfaces find the euler characteristic for. The characteristic of the projective plane is 1 open mobius strip plus a point. The only affine plane which is also a 3configuration is ag2, 3. Find the euler characteristic of the subdivision of the projective plane given by figure 90.
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