Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. A catalogue of simplicial arrangements in the real. Homology of simplicial complexes math 411, david perkinson introduction. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. This is in contrast with real projective plane rp2 and the complex projective plane cp2 which have unique triangulations on 6 vertices and 9 vertices respectively. Mthm014mas426 algebraic topology 20062007 exercises 3 simplicial complexes and simplicial homology 1. Simplicial complexes and deltacomplexes 4 pages this note compares simplicial complexes, ordered simplicial complexes, and deltacomplexes. Simplicial homology global optimisation an algorithm for optimising energy surfaces stefan endres march 29, 2018 institute of applied materials department of chemical engineering university of pretoria this presentation is intended for an audience of professional engineers from a diverse set of backgrounds.
A catalogue of simplicial arrangements in the real projective. The vertices of an arrangement are the intersection points of two or more lines, the edges are the segments into which the lines are parti. Consider the following simplicial complex of the real line. I think the projective plane for example is not in the image of the realization functor. These give an axiomatic characterization of homology for reasonable spaces. S n such that f x is orthogonal to x, for all x 2 s n. For manifolds, there are functions defining the \n\ sphere for any \n\, the torus, \n\ dimensional real projective space for any \n\, the complex projective plane, surfaces of arbitrary genus, and. A catalogue of simplicial arrangements in the real projective plane. Let xbe the abstract simplicial complex determined by the following list of maximal faces. The real projective plane p2p2 vp2r3 the sphere model. Finite simplicial complexes sage reference manual v9. Examples of simplicial complexes sage reference manual v9. Show that if l and m are subcomplexes of a simplicial complex k, then so are l.
If x is a dcomplex, let dnx be the free abelian group on nsimplices. Note that the notion of continuity of a mapping depends on the choice of topologies on both its source and its target. Simplicial homology global optimisation stefan endres. Many of the more advanced topics in algebraic topology involve. If i am not mistaken there are finite topological simplicial complexes which are not the geometric realization of a finite abstract simplicial complex.
How to triangulate real projective spaces as simplicial. For manifolds, there are functions defining the \n\sphere for any \n\, the torus, \n\dimensional real projective space for any \n\, the complex projective plane, surfaces of arbitrary genus, and some other manifolds, all as simplicial complexes. The real or complex projective plane and the projective plane of order 3 given above are examples of desarguesian projective planes. A quadrangle is a set of four points, no three of which are collinear. Enriched homology and cohomology modules of simplicial. The vertices of an arrangement are the intersec tion points of two or more lines, the edges are the segments into which the lines are parti. Z z for n 0, z 2 for n 1, 0 otherwise, to compute the homology h rp2. Homology of real projective plane using delta complex. Homology of real projective plane using delta complex youtube. As you read, note how the homology of a simplicial complex drawn on a manifold can measure the number.
This article describes the value and the process used to compute it of some homotopy invariant s for a topological space or family of topological spaces. Xof any open subset w2ty in y is an open subset in x, f 1w 2tx. If l is a subcollection of k that contains all faces of its elements, then l is a simplicial complex. Both the klein bottle and the projective plane shown above are nonorientable, and cannot be embedded in r3 without selfinteraction. Typically, results in algebraic topology focus on global, non.
Another construction of the projective plane is to take the twosphere s2 and identify opposite points. Algebraic topology lectures by haynes miller notes based on livetexed record made by sanath devalapurkar images created by john ni march 4, 2018 i. However this is not the only descriptions of the real projective plane. In mathematics, the complex projective plane, usually denoted p 2 c, is the twodimensional complex projective space. Rp 1 is called the real projective line, which is topologically equivalent to a circle. I have written a program in mathematica 7, which calculates for a finite abstract simplicial complex all its homology groups. The presentation of the homotopy theory and the account of duality in homology manifolds make the text ideal for a course on either homotopy or homology theory. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
We will use induction on the dimension nto show that 1. Projective geometry in a plane fundamental concepts undefined concepts. They are graded modules over kx 1, x n whose ranks are equal to the dimensions of the. A triangulation of the real projective plane p2 is a simplicial complex such that each face is bounded by a 3cycle, and each edge can be seen as a greater arc on the projective sphere. Note that for, all homology groups are zero, so we omit those cells for visual clarity. The invariant is homology group and the topological spacefamily is real projective space. A mapping is a homeomorphism if it is continuous and its inverse also is continuous. A manifold is a topological space that near each point resembles euclidean space. Identifying antipodal points in sn gives real projective space rpn s nxx. Simplicial complexes university of california, berkeley. It can however be embedded in r 4 and can be immersed in r 3. Jul 31, 2012 homology of real projective plane using delta complex.
Compute the homology of the klein bottle over z 2 using its integral homology groups. A triangulation of a topological space x is a simplicial complex k together with a homeomorphism jkj x. That is, these are homogeneous coordinates in the traditional. It is a complex manifold of complex dimension 2, described by three complex coordinates. The questions of embeddability and immersibility for projective nspace have been wellstudied.
Triangulating the real projective plane mridul aanjaneya monique teillaud macis07. Any two points p, q lie on exactly one line, denoted pq. This is an introduction to the homology of simplicial complexes suitable for a rst course in linear algebra. Homology 5 union of the spheres, with the equatorial identi.
Given a simplicial complex k, the collection of all simplices of k of dimension at most p is called the pskeleton of k and is denoted kp. Triangulation abstract simplicial complex set k and collection of s of abstract simplices subsets of k such that 1 for all v. The projective planes that can not be constructed in this manner are called nondesarguesian planes, and the moulton plane given above is an example of one. Computing homology groups algebraic topology nj wildberger. This is because the choice of the total order determines an orientation of the realization. This file constructs some examples of simplicial complexes. I would really like to test it on the projective spaces, but cannot find a way to triangulate them. We will also sometimes refer to a triangulation of the projective plane as a projective triangulation. Moreover, there are at least six combinatoriallydi. On modular homology in projective space article pdf available in journal of pure and applied algebra 1511. Quotient maps real projective line, modular arithmetic duration.
By fact 1, we note that the homology of real projective space is the same as the homology of the following chain complex, obtained as its cellular chain complex. Examples of simplicial complexes sage reference manual. Problem h use the mayervietoris sequence to compute the homology of real projective nspace rpn. P minimal triangulation of the real projective plane sage. While inspired by knots that appear in daily life in shoelaces and rope, a mathematicians knot differs in that the ends are joined together so that it cannot yopology undone.
The klein bottle and real projective plane cannot embed in r3. For example, to study singular homology, one considers the continuous maps. Clearly, as sets, jkj r, but not as topological spaces, e. Use the integral homology of the real projective plane rp2, h nrp2. Any two lines l, m intersect in at least one point, denoted lm. The invariant is cohomology group and the topological spacefamily is real projective space. For the sake of clarity, this document is more detailed than what would be expected from a. In part i of these notes we consider homology, beginning with simplicial homology theory. Homology of rpn now that we have the base cases, were ready to induct on the dimension. In this paper we follow some of the notations from 10.
Examples of simplicial complexes sage reference manual v6. Pdf enriched homology and cohomology modules of simplicial. So, lets suppose the homologies of rpk, k pdf format. Homology groups with integer coefficients in tabular form we illustrate how the homology groups work for small values of whereby the dimension of the corresponding complex projective space is. S1is closed if and only if a\snis closed for all n. Let v 0, v 1, and v 2 be three noncollinear points in rn.