DOCSLIB.ORG

Actions of Compact Groups on Spheres and on Generalized Quadrangles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Actions of Compact Groups on Spheres and on Generalized Quadrangles Actions of Compact Groups on Spheres and on Generalized Quadrangles Von der Fakult¨at Mathematik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Harald Biller aus Frankfurt am Main Hauptberichter: Privatdozent Dr. Markus Stroppel Mitberichter: Prof. Dr. Hermann H¨ahl Prof. Dr. Theo Grundh¨ofer (Universit¨at Wurzburg)¨ Tag der mundlichen¨ Prufung:¨ 8. Juni 1999 Mathematisches Institut B der Universit¨at Stuttgart 1999 ii Harald Biller Fachbereich Mathematik Technische Universit¨at Darmstadt Schloßgartenstraße 7 D-64289 Darmstadt Germany [email protected] Mathematics Subject Classification (1991): 51H10 Topological linear incidence structures 51E12 Generalized quadrangles, generalized polygons 57S10 Compact groups of homeomorphisms 57S25 Groups acting on specific manifolds 57P99 Generalized manifolds Keywords: topological geometry, generalized quadrangle, compact transformation group, homology sphere, cohomology manifold This thesis is available online: http://elib.uni-stuttgart.de/opus Abstract The actions of sufficiently high-dimensional compact connected groups on spheres and on two types of compact Tits buildings are classified explicitly. The result for spheres may be summarized as follows: every effective continuous action of a compact connected group whose di- mension exceeds 1 + dim SOn−2R on an n-sphere is linear, i.e. it is equivalent to the natural action of a subgroup of SOn+1R . Under sim- ilar hypotheses, we study actions on finite-dimensional compact gener- alized quadrangles whose point rows have dimension either 1 or 4. We find that every effective action of a sufficiently high-dimensional com- pact group is equivalent to an action on a Moufang quadrangle, i.e. on a coset geometry associated to a BN-pair in a simple Lie group. Both for spheres and for generalized quadrangles, the classification arises from an explicit description of the actions. One main source for this thesis is the investigation of compact pro- jective planes and, recently, other compact generalized polygons by Salzmann and his school. They developed the specific hypothesis of a sufficiently large group dimension, which here is applied to gener- alized quadrangles for the first time. Compactness of the group is a strong additional assumption which allows us to introduce the sophisti- cated theory of actions of compact groups on (cohomology) manifolds further into topological incidence geometry. Conversely, the results about spheres, which lie completely within the scope of the classical theory, are rendered possible by Salzmann’s specific question. When combined with a thorough exploitation of the classification of compact Lie groups, it essentially reduces the problem to the consideration of a small number of series of groups. To obtain the results about gen- eralized quadrangles, we first show transitivity of the action and then use, and partly re-prove, recent classification results. Zusammenfassung Alle Wirkungen kompakter zusammenh¨angender Gruppen von genu-¨ gend großer Dimension auf Sph¨aren und auf zwei Arten von verallge- meinerten Vierecken werden im einzelnen beschrieben. Fur¨ Sph¨aren l¨aßt sich das Ergebnis wie folgt zusammenfassen: Jede treue steti- ge Wirkung einer kompakten zusammenh¨angenden Gruppe, deren Di- mension 1 + dim SOn−2R ubersteigt,¨ auf einer n-Sph¨are ist linear, also ¨aquivalent zur naturlichen¨ Wirkung einer Untergruppe von SOn+1R . Unter ¨ahnlichen Voraussetzungen untersuchen wir Wirkungen auf endlichdimensionalen kompakten verallgemeinerten Vierecken, deren Punktreihen Dimension 1 oder 4 haben. Hier zeigen wir, daß jede treue Wirkung einer kompakten Gruppe von genugend¨ großer Dimen- sion ¨aquivalent ist zu einer Wirkung auf einem Moufang-Viereck, also auf einer Nebenklassengeometrie einer einfachen Lie-Gruppe, die durch ein BN-Paar beschrieben wird. Die vorliegende Arbeit steht in der Tradition der Untersuchung kompakter projektiver Ebenen und neuerdings anderer kompakter ver- allgemeinerter Polygone durch Salzmann und seine Schule. Der dabei entstandene Leitgedanke, nur die Wirkung einer Gruppe von genugend¨ großer Dimension vorauszusetzen, wird in dieser Arbeit erstmals fur¨ verallgemeinerte Vierecke durchgefuhrt.¨ Wir setzen zus¨atzlich voraus, daß die Gruppe kompakt ist, um die hochentwickelte Theorie der Wir- kungen kompakter Gruppen auf (Kohomologie-) Mannigfaltigkeiten fur¨ die topologische Inzidenzgeometrie weiter zu erschließen. Umge- kehrt erm¨oglicht erst die spezifische Salzmannsche Fragestellung die Ergebnisse uber¨ Sph¨aren, die ja dem Bereich der klassischen Theo- rie angeh¨oren. Indem die Klassifikation der kompakten Lie-Gruppen konsequent ausgenutzt wird, l¨aßt sich das Problem auf die Behand- lung weniger Serien von Gruppen zuruckf¨ uhren.¨ Bei verallgemeinerten Vierecken zeigt man dagegen zuerst die Transitivit¨at der Wirkung und benutzt dann die bestehende (teilweise hier neu bewiesene) Klassifika- tion. Contents List of Tables vii Preface ix Deutschsprachige Kurzfassung xv 1 Foundations 1 1.1 Locally compact groups .................. 2 1.2 Cohomology manifolds . .................. 4 1.3 Generalized spheres . .................. 12 1.4Compactquadrangles................... 14 2 Actions of compact groups 23 2.1 General theory . .................. 23 2.2 Actions on cohomology manifolds . 36 2.3 Consequences of a theorem of Mann’s . 38 2.4 Calculating homology . .................. 41 2.5 Large subalgebras of compact Lie algebras . 49 3 Spheres 63 3.1Verylargeorbits...................... 63 3.2Complexunitarygroups.................. 73 3.3 A homogeneity property of torus groups . 83 3.4Otherparticulargroups.................. 85 3.5 Local type of compact Lie transformation groups . 91 3.6Reconstructionofactions.................102 v vi CONTENTS 4 Compact (1;m)-quadrangles 115 4.1 Miscellaneous tools for compact polygons . 115 4.2 Line-homogeneous quadrangles . 119 4.3 Local type of compact Lie transformation groups . 130 4.4Reconstructionofactions.................145 5 Compact (4;m)-quadrangles 151 5.1CompactLiegroups....................152 5.2Compactnon-Liegroups..................161 5.3Characterizationtheorems.................182 Bibliography 187 Notation 201 Index 203 List of Tables 2.1 The simple compact Lie algebras and their subalgebras ofmaximaldimension................... 50 2.2 Structure of semi-simple compact Lie groups acting on spacesofsmalldimension................. 60 3.1 The semi-simple Lie algebras of rank at most 5 . 78 3.2 The compact Lie algebras for effective actions on S 8 .. 94 3.3 Actions of large compact connected groups on n-spheres 114 5.1 Large compact Lie algebras of given rank . 153 vii viii Preface To study an object is to study its symmetry – this thought is the heart of Felix Klein’s influential Erlangen Programme. In the present thesis, it is applied to topological spheres and to two classes of topological Tits buildings. In each setting, the actions of sufficiently high-dimensional compact connected groups are determined. The results are most eas- ily formulated for spheres. The group SOn+1R acts naturally on the n-dimensional sphere S n. If a compact connected group whose dimen- sion is greater than that of SO2R × SOn−2R acts continuously and effectively on S n then the action is equivalent to the natural action of a subgroup of SOn+1R . The proof combines the theory of Lie groups with algebraic topology. The Tits buildings which we study are generalized quadrangles. These geometries consist of points and lines. Their characteristic prop- erty is that for every non-incident point-line pair, there is a unique incident pair such that the four elements form a chain. We assume that this chain depends continuously on its ends with respect to some finite-dimensional compact topology, and that the points which lie on any one line form sets of dimension either 1 or 4. Each effective action of a sufficiently high-dimensional compact connected group on such a generalized quadrangle is equivalent to an action on a Moufang quad- rangle, i.e. on one which is associated to a BN-pair in a simple Lie group; the group, the quadrangle, and the action are described ex- plicitly. In particular, we obtain sharp upper bounds for the group dimensions. The interest in generalized quadrangles is the reason for working in the category of continuous actions. In the absence of differentiability assumptions, we are referred to the theory of topological transforma- tion groups which flourished in the context of the solution of Hilbert’s ix x PREFACE Fifth Problem by Montgomery and Zippin. Their work, as well as contributions by Borel, Bredon, and others, is essential. The second main source of inspiration for this thesis, and in fact its original motivation, is the work of Salzmann on compact projective planes, which are the smallest non-trivial compact buildings. Salz- mann and his school developed the use of the group dimension as the principal measure of symmetry, and many of their methods are relevant in our context. More recently, the investigation of general topological buildings was begun by Burns, Spatzier, and Thorbergsson in differ- ential geometry, and by Forst, Grundh¨ofer, Joswig, Knarr, Kramer, Schroth, the Stroppels, and Van Maldeghem in Salzmann’s tradition. A more detailed description of the results will now be given in con- nection with an overview of the five chapters. In the first, we provide some background material. Section 1.2 introduces cohomology mani- folds. Taking only sheaf-theoretic
Load more

Recommended publications
  • Groups and Geometry 2018 Auckland University Projective Planes
    Groups and Geometry 2018 Auckland University Projective planes, Laguerre planes and generalized quadrangles that admit large groups of automorphisms G¨unterSteinke School of Mathematics and Statistics University of Canterbury 24 January 2018 G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 1 / 26 (School of Mathematics and Statistics University of Canterbury) Projective and affine planes Definition A projective plane P = (P; L) consists of a set P of points and a set L of lines (where lines are subsets of P) such that the following three axioms are satisfied: (J) Two distinct points can be joined by a unique line. (I) Two distinct lines intersect in precisely one point. (R) There are at least four points no three of which are on a line. Removing a line and all of its points from a projective plane yields an affine plane. Conversely, each affine plane extends to a unique projective plane by adjoining in each line with an `ideal' point and adding a new line of all ideal points. G¨unterSteinke Projective planes, Laguerre planes, generalized quadrangles GaG2018 2 / 26 (School of Mathematics and Statistics University of Canterbury) Models of projective planes Desarguesian projective planes are obtained from a 3-dimensional vector space V over a skewfield F. Points are the 1-dimensional vector subspaces of V and lines are the 2-dimensional vector subspaces of V . In case F is a field one obtains the Pappian projective plane over F. There are many non-Desarguesian projective planes. One of the earliest and very important class is obtain by the (generalized) Moulton planes.
    [Show full text]
  • Generalisations of the Fundamental Theorem of Projective
    GENERALIZATIONS OF THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOMETRY A thesis submitted for the degree of Doctor of Philosophy By Rupert McCallum Supervisor: Professor Michael Cowling School of Mathematics, The University of New South Wales. March 2009 Acknowledgements I would particularly like to thank my supervisor, Professor Michael Cowling, for conceiving of the research project and providing much valuable feedback and guidance, and providing help with writing the Extended Abstract. I would like to thank the University of New South Wales and the School of Mathematics and Statistics for their financial support. I would like to thank Dr Adam Harris for giving me helpful feedback on drafts of the thesis, and particularly my uncle Professor William McCallum for providing me with detailed comments on many preliminary drafts. I would like to thank Professor Michael Eastwood for making an important contribution to the research project and discussing some of the issues with me. I would like to thank Dr Jason Jeffers and Professor Alan Beardon for helping me with research about the history of the topic. I would like to thank Dr Henri Jimbo for reading over an early draft of the thesis and providing useful suggestions for taking the research further. I would like to thank Dr David Ullrich for drawing my attention to the theorem that if A is a subset of a locally compact Abelian group of positive Haar measure, then A+A has nonempty interior, which was crucial for the results of Chapter 8. I am very grateful to my parents for all the support and encouragement they gave me during the writing of this thesis.
    [Show full text]
  • Arxiv:1009.4249V12 [Math.MG] 17 Oct 2012 H Dniyof Identity the a K Ulig Fsmsml Leri Ruswihetbihsti.W O P Now We This
    A LOCAL-TO-GLOBAL RESULT FOR TOPOLOGICAL SPHERICAL BUILDINGS RUPERT McCALLUM Abstract. Suppose that ∆, ∆′ are two buildings each arising from a semisimple algebraic group over a field, a topological field in the former case, and that for both the buildings the Coxeter diagram has no isolated nodes. We give conditions under which a partially defined injective chamber map, whose domain is the subcomplex of ∆ generated by a nonempty open set of chambers, and whose codomain is ∆′, is guaranteed to extend to a unique injective chamber map. Related to this result is a local version of the Borel-Tits theorem on abstract homomorphisms of simple algebraic groups. Keywords: topological geometry, local homomorphism, topological building, Borel-Tits the- orem Mathematicsl Subject Classification 2000: 51H10 1. Introduction Throughout the history of Lie theory there has been a notion of “local isomorphism”. Indeed, the original notion of Lie groups considered by Sophus Lie in [9] was an essentially local one. Suppose that G is an algebraic group defined over a Hausdorff topological field k. One may consider the notion of a local k-isogeny from the group G to a group G′ defined over k of the same dimension. This is a mapping defined on a nonempty open neighbourhood of the identity of G(k) in the strong k-topology, (see Definition 1.10), which “locally” acts as a k-isogeny, in the sense that it is a local k-homomorphism and its range is Zariski dense. As far as I know this notion has not been investigated systematically. In this paper we shall produce a local version of the Borel-Tits theorem which hints at the possibility that it may be fruitful to investigate this notion.
    [Show full text]
  • Geometrie Galois Actions 1
    London Mathematical Society Lecture Note Series. 242 Geometrie Galois Actions 1. Around Grothendieck's Esquisse d'un Programme Edited by Leila Schneps CNRS and Pierre Locha!< CNRS gCAMBRIDGE V UNIVERSITY PRESS 32 Alexandre Grothendieck Esquisse d'un Programme 33 par le diviseur à croisements normaux de Mumford-Deligne. Une autre dif­ de voisinage tubulaire d'un sous-topos - et il est étonnant d'ailleurs que ce ficulté, moins sérieuse sans doute, c'est que le soi-disant "espace" modulaire travail (pour autant que je sache) n'ait toujours pas été fait, c'est-à-di:e est en fait une multiplicité - techniquement, cela s'exprime surtout par la né­ que personne (depuis plus de vingt ans qu'il existe un contexte de t?polo~le cessité de remplacer l'ensemble d'indices 1 pour les strates par une catégorie étale) ne semble en avoir eu besoin; un signe sûrement que la compre.hensl~n (essentiellement finie) cl 'indices, en l'occurrence celle d~s "graphes MDli 1 qui de la structure topologique des schémas n'a pas tellement progresse depms uparamètrent" les "structures combinatoires" possibles d'une courbe stable le travail d'Artin-Mazur... de type (g, v). Ceci dit, je puis affirmer que la théorie de dévissage générale, Une fois accompli le double travail de dégrossissage (plus ou moins heuris­ spécialement développée SO~S la pression du besoin de cette cause, s'est tique) autour de la notion de dévissage d'un espace ou d'un topos stratifié, révélée en effet un guide précieux, conduisant à une compréhension progres­ qui a été une étape cruciale dans ma compréhension des multiplicités modu­ sive, d'une cohérence sans failles, de certains aspects essentiels de la tour laires il est d'ailleurs apparu que pour les besoins de ces dernières, on peut de Teichmüller (c'est à dire, essentiellement de la "structure à l'infini" des sans doute court-circuiter au moins une bonne partie de cette théorie par groupes de Teichmüller ordinaires).
    [Show full text]
  • Arxiv:Math/0603537V3 [Math.GR] 16 Jul 2008
    Final Group Topologies, Kac-Moody Groups and Pontryagin Duality Helge Gl¨ockner, Ralf Gramlich and Tobias Hartnick Abstract. We study final group topologies and their relations to compactness properties. In particular, we are interested in situations where a colimit or direct limit is locally compact, a kω-space, or locally kω. As a first application, we show that unitary forms of complex Kac-Moody groups can be described as the colimit of an amalgam of subgroups (in the category of Hausdorff topological groups, and the category of kω-groups). Our second application concerns Pontryagin duality theory for the classes of almost metrizable topological abelian groups, resp., locally kω topological abelian groups, which are dual to each other. In particular, we explore the relations between countable projective limits of almost metrizable abelian groups and countable direct limits of locally kω abelian groups. Introduction Given a group G and a family (fi)i∈I of maps fi : Xi → G from certain topological spaces to G, there exists a finest group topology on G making all of the maps fi continuous, the so-called final group topology with respect to the family (fi)i∈I . Such topologies arise naturally in connection with colimits of topological groups (notably, direct limits), which carry the final group topology with respect to the family of limit maps. Although a final group topology O always exists, it may be quite elusive in the sense that it may not be clear at all how one could check whether a given subset U ⊆ G belongs to O. For example, consider an ascending sequence G1 ⊆ G2 ⊆ ··· of topological groups such that all inclusion maps are continuous homomorphisms.
    [Show full text]
  • Topological Classification of Morse–Smale Diffeomorphisms Without Heteroclinic Curves on 3-Manifolds Christian Bonatti, V
    Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds Christian Bonatti, V. Z. Grines, Francois Laudenbach, Olga Pochinka To cite this version: Christian Bonatti, V. Z. Grines, Francois Laudenbach, Olga Pochinka. Topological classifica- tion of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds. Ergodic The- ory and Dynamical Systems, Cambridge University Press (CUP), 2019, 39 (9), pp.2403-2432. 10.1017/etds.2017.129. hal-01467144v2 HAL Id: hal-01467144 https://hal.archives-ouvertes.fr/hal-01467144v2 Submitted on 22 Sep 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Topological classification of Morse-Smale diffeomorphisms without heteroclinic curves on 3-manifolds Ch. Bonatti V. Grines F. Laudenbach O. Pochinka ∗ Contents 1 Introduction and formulation of the result 2 2 Dynamics of diffeomorphisms in the class G(M) 4 3 Compatible foliations 7 4 Proof of the classification theorem 12 4.1 Necessity . 13 4.2 Sufficiency . 13 5 Topological background 27 Abstract We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on a 3-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a charac- teristic space.
    [Show full text]
  • Projective Planes and Their Look-Alikes
    j. differential geometry 64 (2003) 1-55 PROJECTIVE PLANES AND THEIR LOOK-ALIKES LINUS KRAMER Abstract We classify all closed topological manifolds which have the same integral homology as a projective plane. In this paper we classify manifolds which look like projective planes. More precisely,we consider 1-connected closed topological manifolds M with integral homology ∼ 3 H•(M) = Z . A straight-forward application of Poincar´e duality shows that for such a manifold there exists a number m ≥ 2 such that Hk(M)=Z,for k =0,m,2m; in particular,dim( M)=2m is even. It follows from Adams’ Theorem on the Hopf invariant that m divides 8. We construct a family of topological 2m-manifolds M(ξ) which are Thom spaces of certain topological Rm-bundles (open disk bundles) ξ over the sphere Sm,for m =2, 4, 8,and which we call models. This idea seems to go back to Thom and was exploited further by Shimada [53] and Eells-Kuiper [14]. A particular case is worked out in some detail in Milnor-Stasheff [46,Ch. 20]. However,these authors used vector bundles instead of Rm-bundles. We will see that the non-linearity of Rm-bundles yields many more manifolds than the construction by Eells- Kuiper. In [14,p. 182],the authors expressed the hope that “the given combinatorial examples form a complete set [...] for n =4”. Our results show that in dimension n =2m = 16,their construction missed 27 /28 of the (infinitely many) combinatorial and topological solutions,while Supported by a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft.
    [Show full text]
  • Hints and Solutions to Selected Exercises
    Hints and Solutions to Selected Exercises Chapter 1 1.1. Using the formula of Remark 1.4, we obtain II[~ ~]II = J2IUI2+1~J4IUI2+1, II [~ ~] II = max{lul, Ivl}, C?S t - sin t] II = 1 cosh t sinh t] II = {t -t} II [smt cost ' II [sinh t cosh t max e ,e . 1.2. (a) From the definition of the matrix norm, -1 { IBAB-Ixl }{ IAxl } IIBAB II = max Ixl :x t 0 = max IBxl: x t 0 = max {I~~I :x to} = IIAII. (b) The inequality IICAC- I II ~ IICIlIIAIIIIC- I II always holds. 1.3. (a) We have Ixl 2 = x·x by definition, so Since the unit sphere {x E en : Ixl = 1} is closed and bounded, the continuous 2 function x I---> IAxl defined on it attains its supremum, hence 2 (b) The matrix A*A is hermitian and for any x E en, x*A*Ax = IAxI , which is real and non-negative; it also vanishes precisely when Ax = O. By the general theory of hermitian matrices, there is an orthonormal basis {UI, ... ,Un} consisting of eigenvalues AI, ... ,An which are all real. For each j, 303 304 Matrix Groups: An Introduction to Lie Group Theory since ujUj = 1; hence Aj ~ O. If A i= 0 there must be at least one non-zero eigenvalue of A*A since otherwise IAxl2 would always vanish; so in this case we have a largest positive eigenvalue A which we might as well assume is A= AI. Now if Ixl = 1 we can 2 2 write x = tIU1 + ..
    [Show full text]
  • 1 Introduction
    The Octonions John C. Baez Department of Mathematics University of California Riverside CA 92521 email: [email protected] May 16, 2001 Abstract The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathe- matics. Here we describe them and their relation to Clifford algebras and spinors, Bott period- icity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry. 1 Introduction There are exactly four normed division algebras: the real numbers (R), complex numbers (C), quaternions (H), and octonions (O). The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative. Most mathematicians have heard the story of how Hamilton invented the quaternions. In 1835, at the age of 30, he had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between C and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843.
    [Show full text]
  • [Math.DG] 22 Feb 2007
    (SEMI-)RIEMANNIAN GEOMETRY OF (PARA-)OCTONIONIC PROJECTIVE PLANES ROWENA HELD, IVA STAVROV, BRIAN VANKOTEN Abstract. We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octo- nionic projective plane; we discuss these manifolds in the later sections of the paper. 1. Introduction I. Porteous [11] and H. Aslaksen [2] have coordinatized the octonionic (Cayley) projective plane, OP 2, using the reduced homogeneous coordinates. Points in this model of OP 2 take the form [u,v,w], with at least one of the octonions u,v,w equal to 1. For the appropriate coordinatization of lines in OP 2 the reader is referred to [2], where it is also shown that this model of projective plane geometry is non-Desarguesian. D. Allcock [1] provided an identification between the model of OP 2 involving reduced homogeneous coordinates and H. Freudenthal’s [3] classic approach via Jordan algebras. Classically, the octonionic projective plane can be seen as a 16- dimensional quotient manifold F4/Spin(9). This manifold can be equipped with a Riemannian metric with respect to which OP 2 is a 2-point homogenous space. In this paper we use the reduced homogeneous coordinates of Porteous to study the Riemannian geometry of OP 2. We explicitly write down the metric in terms of the reduced homogeneous coordinates, verify the homogeneity, and compute its curvature. We also provide an elementary approach to the other manifolds tightly related to the OP 2: the idefinite octonionic projective plane OP (1,1), the octonionic hyperbolic plane OH2 and the para-octonionic projective plane O′P 2.
    [Show full text]
  • The Role of Intuition and Formal Thinking in Kant, Riemann, Husserl, Poincare, Weyl, and in Current Mathematics and Physics
    Kairos. Journal of Philosophy & Science 22, 2019 Center for the Philosophy of Sciences of Lisbon University The Role of Intuition and Formal Thinking in Kant, Riemann, Husserl, Poincare, Weyl, and in Current Mathematics and Physics “Mathematics is a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty”. (Hermann Weyl, 1949) “Many people have an impression that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules for the manipulation of numbers, symbols and equations, rather like the preparation of a complicated income tax return. Good mathematics is quite opposite to this. Mathematics is an art of human understanding… […] A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie — or best of all, an integrated combination of several different representations”. (William Thurston, 2009) Luciano Boi Ecole des Hautes Etudes en Sciences Sociales, Centre de Mathématiques [email protected] DOI 10.2478/kjps-2019–0007 Abstract. According to Kant, the axioms of intuition, i.e. space and time, must pro- vide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjecti- vity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless intuition comes into play in a fun- damentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition.
    [Show full text]
  • Topology from Art and Math to Contemporary Cities
    AUSART Journal for Research in Art. 1 (2013), 1, pp. 255-264 ISSN: 2340-8510 www.ehu.es/ojs/index.php/ausart ISSNe: 2340-9134 UPV/EHU Topology from Art and Math to contemporary cities Michele Emmer Univ. Roma “La Sapienza”, “Castelnuvo”, Dipto Mat. Abstract Mathematics is not a mere tool of kitchen recipes, but has contributed to, if not determined, our way of conceiving space on Earth and even in the Universe. A lack of awareness of mathematics as an essential tool of our culture has led to a major delay in the use of instruments that mathematicians have used for decades. A case in point is topology, the science of transformations, the science of invariants. Its relatively recent discovery by architects has led to a remarkable transformation of the art of concei- ving and building space. A notorious cultural leap has led to construction using techniques and material that allow realizing transformation, rendering it almost continuous, a sort of contradiction between the finished construction and its deformation. The so called fluid topological architecture is the final result of a set of new ideas of space. New topological ideas are used in art, and has been materialized in recent buildings in towns all around the world, such as the Guggenheim Museum in Bilbao, the National Library in Astana, the Moebius Bridge in Bristol, the MAXXI museum in Rome, the Max Reinhardt Haus of Eise- mann, the Moebius House of Van Berkel. I will illustrate this story using various examples from the topo- logical gold jewels in pre-Colombian culture in South America to the new development in modern cities.
    [Show full text]