Mayer vietoris real projective plane pdf

Furthermore, we notice that the mayer vietoris sequence theorem can be easily proved using the excision theorem. The projective plane over r, denoted p2r, is the set of lines through the origin in r3. The real projective plane is a twodimensional manifold a closed surface. However this is not the only descriptions of the real projective plane. Recall that we can express the real projective plane rp2 as the quotient space of s2. M on f given by the intersection with a plane through o parallel to c, will have no image on c. 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. S1is closed if and only if a\snis closed for all n. As before, points in p2 can be described in homogeneous coordinates, but now there are three nonzero. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. Deformation retraction of plane rp2 physics forums.

Exercise 2 homology of the real projective planethe klein bottle. If you need to identify any of the maps in the long exact sequence, it helps to. A constructive real projective plane mark mandelkern abstract. Homology, problem sheet 2 the second assignment consists of questions 1,9,11,15. Quantum complex projective plane cp2 t as quotient space s5 h u3 2. The mayervietoris sequence, together with the homotopy invariance of. L, that is, p0 is p with one point added for each parallel class. A tangent vector at a point x can be viewed as a derivation of the algebra of the real di. As applications, we compute the homology of some spaces including the sphere, the wedge of two spaces, the torus, the klein bottle, and the projective plane. By using a mayervietoris sequence, compute the singular homology groups of the real projective plane.

A slightly more difficult application of the mayer vietoris sequence is the calculation of the homology groups of the klein bottle x. Problem h use the mayervietoris sequence to compute the homology of real projective nspace rpn. It is obtained by idendifying antipodal points on the boundary of a disk. Show there is no retraction of x s2 to its equator. I found tus book an introduction manifolds, where a computation is presented via mayervietoris sequences. From the toeplitz algebra to quantum projective spaces296 1. The complex algebraic geometry is the overlap of the complex geometry and algebraic geometry. The cycles and boundaries form subgroups of the group of chains. It is gained by adding a point at infinity to each line in the usual euklidean plane, the same point for each pair of opposite directions, so any number of parallel lines have exactly one point in common, which cancels the concept of parallelism. The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. As an application we compute the homology of the projective plane, rp2 which can. This result is obtained by specializing the following algebraic fact to a certain topological situation. And lines on f meeting on m will be mapped onto parallel lines on c.

A in such a way that there is a long exact sequence. Exactness of the mayervietoris sequence can be proved with the aid of the diagram. For a a subset of a topological space x, a retraction of x to a is a continuous map r. A from the mayervietoris sequence applied to x ca, where ca is the cone on a. It is closed and nonorientable, which implies that its image cannot be viewed in 3dimensions without selfintersections. Extensive use of figures, taken from page 150 hatcher. One may observe that in a real picture the horizon bisects the canvas, and projective plane. Topological embeddings of real projective space in euclidean space. Tu department of mathematics tufts university medford, ma 02155 loring. Homology 5 union of the spheres, with the equatorial identi. Mare subcomplexes of k, then we can form a long exact sequence of homology groups and homeomorphisms between them.

Using mayervietoris, compute the cohomology groups of complex projective space cpk. This video clip shows some methods to explore the real projective plane with services provided by visumap application. Suppose xisaspacewithabasepoint x 0,andx 1 and x 2 are path connected subspaces such that x 0 2x 1 \x 2, x x 1 x 2 and x 1 \x 2 is path connected. Topological embeddings of real projective space in. Explaining application of mayer vietoris to klein bottle and torus. Use the mayervietoris sequence to calculate the homology of the spaces below. More speci cally, if kis a simplicial complex and l. The projective plane, which is abbreviated as rp2, is the surface with euler characteristic 1. Look at the mayer vietoris sequence with z2z coefficients, in the unoriented case. A simplicial complex is constructible if it is a simplex, or, recursively, the union of two. Here, we show this directly via the mayervietoris sequence.

We often drop the subscript nfrom the boundary maps and just write c. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of x. The empty set is constructible, and, in dimension 0, every. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics. Furthermore, we notice that the mayervietoris sequence theorem can be easily proved using the excision theorem. Pictures of the projective plane by benno artmann pdf the fundamental group of the real projective plane by taidanae bradley read about more of my favorite spaces. Actions of compact hausdor groups on unital calgebras300 2. Visualizing real projective plane with visumap youtube. Given a real projective algebraic set xwe could hope that the equations describing it can give some information on its topology, e.

This plane is called the projective real plane the previous example suggests a way of turning any a. The projective plane we now construct a twodimensional projective space its just like before, but with one extra variable. B are homotopy equivalent to circles, so the nontrivial part of the sequence yields. This is true for example if 5n1 is a differentiably embedded sphere.

This sequence seems to give isomorphisms of the homology of the manifold and the punctured manifold in dimensions below n1 which makes sense. For the sake of clarity, this document is more detailed than what would be expected from a. Problem h use the mayer vietoris sequence to compute the homology of real projective nspace rpn. Simplicial homology of real projective space by mayervietoris. Algebraic mayervietoris sequence let us consider the following commutative diagram of abelian groups in which the rows are exact and all the f00 n are isomorphisms. For each n, construct a closed connected fourdimensional manifold x nwith h1x n 0 and h2x n. We start with the real projective spaces rpn, which we think of as obtained from sn by identifying antipodal points. This decomposes our manifold into the punctured manifold and an open ball, the intersection giving an n1sphere. We will use induction on the dimension nto show that 1. One uses the decomposition of x as the union of two mobius strips a and b glued along their boundary circle see illustration on the right. Think of the real projective plane as the union of a disc and a mobius strip, pasted together with the overlapping region an annulus. For each n, construct a closed connected fourdimensional manifold x n with h1x n 0 and h2x n. It cannot be embedded in standard threedimensional space without intersecting itself. Consider the projective plane p2 blown up at a point, which we denote by x.