Pdf planar, toroidal and projective commuting and non. The connected sum of two surfaces is obtained by deleting a disk from each, and then gluing the two surfaces together along their two boundary circles. Then gure out what you get when you glue two of these together. Show that the connected sum of the torus and the projective plane is the same as the connected sum of a klein bottle and a projective plane. Prove that the connected sum of a torus with the projective plane is homeomorphic to the connected sum of 3 projective planes. Yet theres another, much simpler method, relying on a single number. The connected sum of nprojective planes is homeomorphic with the connected sum of a torus with a projective plane if nis odd or with a klein bottle if nis even. 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. 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. As preparation for the proof, we shall describe what might be called a. This makes the torus a very fundamental and versatile manifold.
This g torus can also be constructed as the connected sum of gtori, where the connected sum of two surfaces is taken by cutting a disc out of each surface and gluing the surfaces together along the boundary of the discs figure 3. Then s is homeomorphic to exactly one of the following. Final project the classi cation of compact surfaces. This surface is nonorientable because the projective plane and klein bottle are both nonorientable. Such moves can be applied until the edge word can be recognised as the sphere or a connected sum of tori and projective planes. Cut along it and label the edges that were one with the same label. For example, the connected sum of two tori is the twohandled torus. Amazingly, every compact 2manifold is homeomorphic to either a sphere orientable, a connected sum of tori orientable, or a connected sum of projective planes nonorientable. Note that we have chosen in a specially convenient way the disks. The connected sum of two projective 3spaces has a s 2. The klein bottle is displayed as the connected sum of two copies of the real projective plane. The real projective plane is nonorientable surface that cannot be realized in r 3.
The product of a surface of negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. 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. When embedded in euclidean space, the klein bottle is onesided. Im not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. The classic examples are the sphere, the torus, the klein bottle and the projective plane. Pdf the fundamental group of an oriented surface of. Take out an open ball from both manifolds and glue them back up. Stretch the missing point from the torus until you get a hole in the torus. The only moves you require to perform this task are the following you should. We will classify compact, connected surfaces into three classes. 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.
To take the connected sum of two 2manifolds, remove the inside of a small disk from each of them and then glue the two boundary circles of these disks together. It is essentially the same as the set of all lines, passing through a given point in r3. Classification theorem for compact surfaces math images. This will allow us to show that these surfaces and s2 are all topologically distinct. Another description is the connected sum of a torus and a projective plane. A minimum starts and a maximum ends a family of conthe klein bottle is, and each connected nonorientable2manifoldis also diffeomorphicto theconnected sum of an orientable 2manifold. It is also homeomorphic to a sphere plus two cross caps. Visual proof that the connected sum of a real projective plane and a torus, and the connected sum of a real projective plane and a klein bottle are homeomorphic. The torus on the left is an orientable surface, while the klein bottle on the right is not, since it does not enclose any space, even though it is closed the torus on. Conjecture what is the connected sum of any surface s and the sphere. This can be viewed as lying in c 2 and is a subset of the 3sphere s 3 of radius v2. Note that the surfaces involving a projective plane explicitly or. The fundamental group of bordered surfaces theorem 1.
List of fundamental groups of common spaces mathonline. Pdf the fundamental group of an oriented surface of genus n. Connected sum with a torus is equivalent to adding a handle. While in this case doing so isnt terribly difficult, in general it can be quite a puzzle. Moreover, the union of n disjoint copies of a graph. The universal cover of both the torus and the klein bottle is the plane r 2. Then s is homeomorphic to either a sphere or a nite connected sum of tori and projective planes. Further, any two distinct surfaces from the above lists are not homeomorphic. Connected sums of real projective plane and torus or klein. This topological space is defined as the connected sum of two copies of the complex projective plane, where they are glued with the same orientation related facts. Show that the connected sum of two projective planes as in the proof of theorem 815 is a klein bottle. The classification of compact surfaces 1 connected sums. Further, any two distinct surfaces from the above lists are.
Surfaces manifolds and cw complexes 1 then w x y is the. Equivalently, we see that the torus is homeomorphic to the quotient space of i iwhere i denotes the. The fundamental group of an oriented surface of genus n. So, if a surface is homeomorphic to a connected sum of both tori and projective planes, then it is also homeomorphic to the connected sum of only projective planes.
The sphere, torus, klein bottle, and the projective plane are the. Donations go toward large size prints of my work to help me set up an exhibition. The real projective plane triangulated 2 pages the most efficient triangulation of the real projective plane is presented. Topologically, a torus is a closed surface defined as the product of two circles. In fact, s 3 is filled out by a family of nested tori in this manner with two degenerate circles, a fact which is important in the study of s 3 as a. Topology and geometry of 2 and 3 dimensional manifolds. However, there are other topological 3spaces, and in some of the nonorientable examples a klein bottle can be embedded such that it is twosided. Torus triangulated, in dvi format or torus triangulated, in pdf format. Application of the euler characterstic 14 references.
First let us consider the following torus and the projective plane. There is a 21 covering map from the torus to the klein bottle, because two copies of the fundamental region of the klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. Every surface is the connected sum of either the sphere, projective plane, or klein bottle with zero or more tori. Jul 18, 2011 additionally, a theorem exists which details how any closed surface is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes. The only closed prime surfaces are the torus, the projective plane and the sphere. This topological torus is also often called the clifford torus. Apply the above propostition iteratively until you get either a single projective plane nodd or two projective planes, i.
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 moe. R geometry, and is also the connected sum of two pieces with s 3 geometry. Let pbe an identi cation polygon representing a surface s. S 1, and the latter is taken to be the definition in that context. 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.
Certainly, this is the modern formulation of his theorem, given that dyck proved his result in 1888 the citation that i have seen for this theorem is usually given as. The torus t2 is a subset of r3 which is obtained by rotating a circle of radius 1 centred at 2. Realworld objects that approximate a solid torus include orings, noninflatable lifebuoys, ring doughnuts, and bagels. It can be shown that connected sum does not depend on the choice of. After removing the darker m obius strip from the last two, we are left with a disk in the case of the projective plane. Project report on classification theorem and fundamental. Yet theres another, much simpler method, relying on a single. Project report on classification theorem and fundamental group. In summary, any triangulated surface is homeomorphic to one of three kinds of 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. The 2sphere s2, torus, and projective plane play very fundamental roles in the classi. The sphere, the connected sum of n tori and the connected sum of n real projective planes are denoted by s 2, t n and p n, respectively. In particular, these surfaces are all distinguished by their fundamental groups. Surfaces manifolds and cw complexes 1 then w x y is the connected sum of the from mat 4855 at eastern illinois university.
The idea is to apply simplifying moves to the edge word which do not change the surface. The 2sphere, torus, projective plane, and double torus are topologically distinct. The fundamental group of an oriented surface of genus n with. Of course, the connected sum of two surfaces is a surface. Connect sum a 1holed torus to a 2holed torus, and you get a 3holed torus. The projective plane contains a circle c whose complement is orientable. A solid torus is a torus plus the volume inside the torus. Forming the connected sum with the sphere does not change the manifold since it just means replacing one disk by another. A theorem of rado asserts that this is a complete list of compact connected 2manifolds. The connected sum of two tori or torus with two holes will be denoted t2, and, more generally, the connected sum of n tori will be denoted tn, for any n. So far we have arrived at the following methods for determining the fundamental group of a space. The klein bottle is the union of two mobius bands along the boundary, and hence the connected sum of two projective planes.
It can be proved that any compact connected 2dimensional manifold can. Connected sum of two complex projective planes with same. One way is to try deforming it until you can make it look like an nholed torus or a connected sum of projective planes. Thefundamentalgroup ofthe2sphereisthetrivial group sinces2 issimply connected by theorem 59. Every surface is a connected sum of tori andor projective planes. The sphere, mobius strip, torus, real projective plane and klein bottle are all. A klein bottle is homeomorphic to the connected sum of two projective planes. Homotopy type of connected sum depends on choice of gluing map. In topology, a ring torus is homeomorphic to the cartesian product of two circles.
You will eventually get two open washers connected at a point. In particular, this connected sum is of a different homotopy type than the connected sum of two complex projective planes with opposite orientation. Since a klein bottle is homeomorpic to the connected sum of two projective planes not demonstrated, the connected sum of a real projective plane and a. Connected sums of real projective plane and torus or klein bottle. Denoting the sphere, the torus, and the projective plane by the number of arcs that share the node. Another thing this tells us is that connected sum is not cancellative, that is, just.
Second of all, the connected sum of a torus and a projective plane is homeomorphic to the connected sum of three projective planes. A compact surfaces without boundary is either a sphere or a connected sum of tori or a connected sum of real projective planes. Adding the torus is the same as attaching the cylinder at both boundary circles after removing two open disks. That number is the euler characteristic of a surface. Then figure out what you get when you glue two of these together.
Rough diagram of how the connected sum of a real projective plane and a torus is homeomorphic to that of a real projective plane and a klein bottle. Dycks theorem in topology is sometimes stated as follows. The connected sum of two projective planes is the klein bottle. By removing one simplex, one obtains the moebius band. P2 \ b2 perform the following steps and draw a picture for each one.
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