P1 any two distinct points are joined by exactly one line. Use the van kampen theorem to compute the fundamental group of ce. As before, points in p2 can be described in homogeneous coordinates, but now. The main result of the paper on van kampen diagrams, now known as the van kampen lemma can be. To make it a topological space, the following reinterpretation is. Once again, dover publications has done a service to the mathematical community by saving from extinction a classic, decadesold, text. Seifert and van kampens famous theorem on the fundamental group of a union of two spaces 66,71 has been sharpened and extended to other contexts in. The main purpose of this book is to give a systematic treatment of singular homology and cohomology theory. Finally we consider the projective plane, p, with cell structure e0 e1 e2, and with boundary of the 2cell egiven by e 2a, say. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. These allow for some precise nonabelian calculations of some homotopy types, obtained by a. P2 any two distinct lines meet in exactly one point.
More generally, if a line and all its points are removed from a projective plane, the result is an af. Cw complexes and see how to compute the fundamental group using the seifertvan kampen theorem. Part ii of the book chapters 712 introduces and explores a higher homotopy seifertvan kampen theorem. This is well known seifertvan kampen theorem for which i am asking any help in order to make a clear picture of it. Seifertvan kampen understanding mathematics stack exchange. For instance, two different points have a unique connecting line, and two different. The universal property expressed in the seifert van kampen theorem is an instance of the notion of a colimit of a diagram. This earlier book is definitely not a logical prerequisite for the present volume. P 2the real projective plane, t the twotorus, and k2 the klein bottle. The notion of a van kampen diagram was introduced by egbert van kampen in 1933. Projective planes a projective plane is an incidence system of points and lines satisfying the following axioms. Other articles where projective plane is discussed. Show that the real projective plane is not a nontrivial cover of any other space.
The projective plane over k, denoted pg2,k or kp 2, has a set of points consisting of all the 1dimensional subspaces in k 3. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. Use the seifertvan kampen theorem to determine the fundamental group of the connected sum of n projective planes. 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. Formally, this means that the set p consists of all antipodal pairs p. In particular, the 2dimensional seifertvan kampen theorem is given as theorem 2. The universal property expressed in the seifertvan kampen theorem is an instance of the notion of a colimit of a diagram. The projective plane takes care of that by declaring that the north and south poles are actually the same point. We can use the van kampen theorem to compute the fundamental groupoids of most basic spaces. H i2 i1 g 1 g 2 when is a pushout a free product with amalgamation. T2 is continuous for p2 the projective plane, and t the torus. We provide a new approach that takes on 2 time but is much easier to implement. A generalization of seifertvan kampen theorem for fundamental groups article in far east journal of mathematical sciences 612 june 2010 with 53 reads how we measure reads. In this section we will revisit the projective plane rp2, and show that it can be charac.
Mathematics 490 introduction to topology winter 2007 what is this. Also covered in that book is the monoidal closed structure on crossed complexes, giving an exponential law. In general, any two great circles lines that look like. This two volume book contains fundamental ideas of projective geometry such as the crossratio, perspective, involution and the circular points at in. List of fundamental groups of common spaces mathonline. Draw a projective plane which has four points on every line. There apparently is a slick proof of the seifertvan kampen theorem using covering spaces. Projective plane base point fundamental group simplicial complex euler characteristic these keywords were added by machine and not by the authors. Symmetries of nite projective planes g abor korchm aros universit a degli studi della basilicata italy 2016 phd summer school in discrete mathematics, rogla slovenia 26 june 2 july 2016 g abor korchm aros symmetries of nite projective planes. It is called playfairs axiom, although it was stated explicitly by proclus. Analogy with the seifertvan kampen theorem there is an analogy between the mayervietoris sequence especially for homology groups of dimension 1 and the seifertvan kampen theorem.
A linear time algorithm for projective planar embedding has been described by mohar 9. In the other direction, given a closed 4manifold whose. Monge view of a triangle in space invariant under projection. The rescued object this time is a slim and very inexpensive, about 10 dollars as i write this little book, first published in 1968, that gives a good introduction to some of the very interesting mathematics surrounding the theory of. In fact, mayer was initiated to topology by his colleague vietoris when attending his lectures in. This paper appeared in the same issue of american journal of mathematics as another paper of van kampen, where he proved what is now known as the seifertvan kampen theorem. Smstc geometry and topology 20112012 lecture 7 the classi cation of surfaces andrew ranicki edinburgh. Lifting criterion in terms of the fundamental group. Group trisections and smooth 4manifolds, preprint 2016. It cannot be embedded in standard threedimensional space without intersecting itself. Diagonal approximations for some fundamental crossed. So z is a twice punctured disk, and let az be the boundary component given. The integer q is called the order of the projective plane.
Learn about the abstract notions of categories and functors. Plane and solid geometry, universitext, springer verlag 2008. Axiom ap2 for the real plane is an equivalent form of euclids parallel postulate. This process is experimental and the keywords may be updated as the learning algorithm improves. State precisely the seifertvan kampen theorem and use it to compute the fundamental group of the connected sum of two projective planes, rp2 equivalently, the.
Finding the smallest possible structure in each case will make the following computations easier. Thanks for contributing an answer to mathematics stack exchange. Free products of groups, statement of seifertvan kampen without proof. Fundamental group of projective plane with g handles by van kampen. I example the m obius band, the projective plane rp2 and the klein bottle k are nonorientable.
The projective plane over r, denoted p2r, is the set of lines through the origin in r3. The standard example of a projective plane is the desarguesian plane of order q constructed from a 3dimensional vector space over a. Fundamental group of the real projective plane and its. This is a collection of topology notes compiled by math 490 topology students at the university of michigan in the winter 2007 semester. The reason is that in a nontrivial decomposition of s1 into two connected open sets, the intersection is not connected. It is in some sense a sequel to the authors previous book in this springerverlag series entitled algebraic topology.
A projective plane is equivalent to a disk with antipodal points identified. M345p21 algebraic topology imperial college london lecturer. Cw complexes and see how to compute the fundamental group using the seifert van kampen theorem. Introductory topics of pointset and algebraic topology are covered in a series of. The smallest projective plane has order 2 see figure 1. B \displaystyle a\cap b is pathconnected, the reduced mayervietoris sequence yields the isomorphism. There apparently is a slick proof of the seifert van kampen theorem using covering spaces. This theorem includes the 1 and 2dimensional theorems so far explained and, in a. An introduction to finite projective planes mathematical. The projective plane we now construct a twodimensional projective space its just like before, but with one extra variable. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. Algebraic topology 4 state the seifert and vankamp.
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