Mayer vietoris real projective plane pdf

Simplicial homology of real projective space by mayervietoris. The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. 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. Exactness of the mayervietoris sequence can be proved with the aid of the diagram.

Furthermore, we notice that the mayervietoris sequence theorem can be easily proved using the excision theorem. A tangent vector at a point x can be viewed as a derivation of the algebra of the real di. Use the mayervietoris sequence to calculate the homology of the spaces below. From the toeplitz algebra to quantum projective spaces296 1. Topological embeddings of real projective space in. For a a subset of a topological space x, a retraction of x to a is a continuous map r. This plane is called the projective real plane the previous example suggests a way of turning any a.

Here, we show this directly via the mayervietoris sequence. The cycles and boundaries form subgroups of the group of chains. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics. Show there is no retraction of x s2 to its equator. Furthermore, we notice that the mayer vietoris sequence theorem can be easily proved using the excision theorem. It is obtained by idendifying antipodal points on the boundary of a disk. This sequence seems to give isomorphisms of the homology of the manifold and the punctured manifold in dimensions below n1 which makes sense. Recall that we can express the real projective plane rp2 as the quotient space of s2.

For each n, construct a closed connected fourdimensional manifold x n with h1x n 0 and h2x n. S1is closed if and only if a\snis closed for all n. 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. We will use induction on the dimension nto show that 1. Exercise 2 homology of the real projective planethe klein bottle. 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.

We start with the real projective spaces rpn, which we think of as obtained from sn by identifying antipodal points. Topological embeddings of real projective space in euclidean space. The projective plane we now construct a twodimensional projective space its just like before, but with one extra variable. Quantum complex projective plane cp2 t as quotient space s5 h u3 2. And lines on f meeting on m will be mapped onto parallel lines on c. 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. We often drop the subscript nfrom the boundary maps and just write c. Problem h use the mayer vietoris sequence to compute the homology of real projective nspace rpn. This is true for example if 5n1 is a differentiably embedded sphere. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. Think of the real projective plane as the union of a disc and a mobius strip, pasted together with the overlapping region an annulus. Consider the projective plane p2 blown up at a point, which we denote by x. 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. Explaining application of mayer vietoris to klein bottle and torus.

Mare subcomplexes of k, then we can form a long exact sequence of homology groups and homeomorphisms between them. Homology 5 union of the spheres, with the equatorial identi. The complex algebraic geometry is the overlap of the complex geometry and algebraic geometry. The mayervietoris sequence, together with the homotopy invariance of. This video clip shows some methods to explore the real projective plane with services provided by visumap application.

Deformation retraction of plane rp2 physics forums. The projective plane, which is abbreviated as rp2, is the surface with euler characteristic 1. One may observe that in a real picture the horizon bisects the canvas, and projective plane. M on f given by the intersection with a plane through o parallel to c, will have no image on c. Extensive use of figures, taken from page 150 hatcher. 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. It cannot be embedded in standard threedimensional space without intersecting itself. 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 projective plane over r, denoted p2r, is the set of lines through the origin in r3. By using a mayervietoris sequence, compute the singular homology groups of the real projective plane. Homology, problem sheet 2 the second assignment consists of questions 1,9,11,15. 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.

For each n, construct a closed connected fourdimensional manifold x nwith h1x n 0 and h2x n. A simplicial complex is constructible if it is a simplex, or, recursively, the union of two. More speci cally, if kis a simplicial complex and l. L, that is, p0 is p with one point added for each parallel class. A from the mayervietoris sequence applied to x ca, where ca is the cone on a. B are homotopy equivalent to circles, so the nontrivial part of the sequence yields. Using mayervietoris, compute the cohomology groups of complex projective space cpk. Given a real projective algebraic set xwe could hope that the equations describing it can give some information on its topology, e. I found tus book an introduction manifolds, where a computation is presented via mayervietoris sequences. Tu department of mathematics tufts university medford, ma 02155 loring. However this is not the only descriptions of the real projective plane. The real projective plane is a twodimensional manifold a closed surface. It is closed and nonorientable, which implies that its image cannot be viewed in 3dimensions without selfintersections. A slightly more difficult application of the mayer vietoris sequence is the calculation of the homology groups of the klein bottle x.

Actions of compact hausdor groups on unital calgebras300 2. Visualizing real projective plane with visumap youtube. Problem h use the mayervietoris sequence to compute the homology of real projective nspace rpn. The empty set is constructible, and, in dimension 0, every. If you need to identify any of the maps in the long exact sequence, it helps to. As before, points in p2 can be described in homogeneous coordinates, but now there are three nonzero. A constructive real projective plane mark mandelkern abstract. This result is obtained by specializing the following algebraic fact to a certain topological situation.