Connected sum torus projective plane pdf

First of all, the klein bottle is the connected sum of two projective planes. Show that the connected sum of the torus and the projective plane is. Algorithmical determination of the topology of a real. More generally, a topological surface with boundary is a hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closure of the upper half plane h. Any triangulated surface is homeomorphic to either the sphere or a connected sum of tori and projective planes. 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. Recall that topology, like euclidean geometry, is a study of the properties of spaces that. After the great success of thurstons geometrisation of 3manifolds, the classi. So in this section, we will give an idea of how to.

A connected sum of ntori is called the nfold torus. The fundamental group of the projective plane is isomorphic to z2 by corollary 60. Connected sums of real projective plane and torus or klein bottle. Apply the above propostition iteratively until you get either a single projective plane nodd or two projective planes, i.

The projective plane rp2 is the quotient of the unit square q 0,12. If the axis of revolution is tangent to the circle, the surface is a horn torus. Every surface is a connected sum of tori andor projective planes. In geometry, a torus plural tori is a surface of revolution generated by revolving a circle in threedimensional space about an axis that is coplanar with the circle if the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. Emille davie lawrences favorite theorem scientific.

Then figure out what you get when you glue two of these together. Manifolds and surfaces city university of new york. Believe it or not, we have almost all the topological ingredients for making any surface whatsoever. Question if you make a connected sum of two spaces, what is the dimension of the resulting space equal to. We therefore get another representation of it by adjoining two open crosscaps. The only closed prime surfaces are the torus, the projective plane and the sphere. Hard show that the connected sum of a torus and a projective plane is the same as the connected sum of a klein bottle and a projective plane. Every closed 2manifold that cannot be embedded in r3 is homeomorphic to the connected sum of n projective planes for some n 1. Then s is homeomorphic to exactly one of the following. Connect sum a 1holed torus to a 2holed torus, and you get a 3holed torus. The nonorientable surface n 3 of euler characteristic 1 can be described as the connected sum of a sphere with three crosscaps the sphere model, or as the connected sum of a torus and one crosscap the torus model. The sphere is an identity for the connected sum operation. A connected sum of mprojective planes is called the mfold projective plane. Two tori glued along a slit the empty set the menger sponge the connected sum.

List of fundamental groups of common spaces mathonline. It can be shown that connected sum does not depend on the choice of. So, we can eliminate any klein bottles from the description. This is easily proved by induction on the number of faces determined by g, starting with a tree as the base case.

The classi cation of closed compact surfaces every closed compact connected surface is homeomorphic to a sphere or a connected sum of tori or a connected sum of projective planes. The connected sum of n projective planes is home omorphic with the connected sum of a torus with a projective plane if n is odd or with a klein bottle if n is even. Give two proofs that the klein bottle k is the union of two mobius strips. The sphere and the torus are what we call orientable surfaces because they have a distinct inside and outside, while the klein bottle and projective plane are nonorientable. Such a diagram is called a polygonal presentation of a surface. Again if the manifold is not orientable this is not true.

Klein bottle and projectiveplane can be constructed in the three. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, hausdorff, second countable, and connected. Theorem 1 let s be a compact connected 2dimensional manifold, formed from a polygon in the plane by gluing corresponding sides of the boundary together. What is the general pattern for an n torus, the connected sum of n tori. It can be shown that connected sum does not depend on the choice of disks that are cut out from each surface, and so it. The problem of the algorithmic determination of the topological type of a nonsingular real projective algebraic surface has already been solved in fortuna et al.

T 0 we see that adding a handle decreases the euler characteristic of any surface by 2. Visual proof that the connected sum of a real projective plane represented here as a moebius strip, but remember that glueing a disc to the border of a moebius strip produces a real projective plane and a torus, and the connected sum of a real projective plane and a klein bottle are homeomorphic. In particular, this connected sum is of a different homotopy type than the connected sum of two complex projective planes with opposite orientation. Use mathematical induction to prove that a connected sum of projective planes is homeomorphic to the connected sum of tori in connected sum with either a projective plane or a klein bottle.

For example joining a torus and a projective plane gives a nonorientable surface with. A sphere, klein bottle and projective plane can be constructed in the three following. Introduction to computational topology notes stanford graphics. Thus, the connected sum of three real projective planes is homeomorphic to the connected. Two more exotic surfaces are the projective plane and the klein bottle fig.

A 2manifold with boundary can be derived by removing an open disk from a 2manifold without boundary. The euler characteristic can be defined for connected plane graphs by the same. Only one very important surface remains to be explored and of course we need a way to put surfaces together to make new surfaces. To connect sum two surfaces you pull out a disc from each, creating holes, and then sew the two surfaces together along the boundaries of the holes. Assume the theorem is true for a connected sum of g copies of projective. Connected sum of two complex projective planes with same. Since the connected sum of two projective planes is obtained by removing a disc from each and gluing the resulting spaces together along the boundary circles and since the result of removing a disc from the projective plane is the m obius band proposition 1. What surface in our catalog do we obtain from the connected sum of a torus and a projective plane. However, it is not obvious that all real surfaces are obtained by connected sums starting only with the sphere, the torus and the real projective plane. Prove that the connected sum of a torus with the projective plane is homeomorphic to the connected sum of 3 projective planes.

Homotopy type of connected sum depends on choice of gluing map. The simplest nonorientable surface is the real projective plane. For n 1, the nhole torus is the connected sum of n tori, denoted by nt2. Given two surfaces m and n, we move an open disk d on both of them. Visual proof that the connected sum of a real projective plane represented here as a moebius strip, but remember that glueing a disc to the border of a moebius strip produces a real projective. Pictures of the projective plane by benno artmann pdf. For instance, if you view real projective space as the connected sum of itself and a small sphere then the bounding circle is not homologous to zero over z in the plane minus the disc.