Size: px
Start display at page:

Download ""

Transcription

1

2 DIFFERENTIAL GEOMETRY

3 PROCEEDINGS OF THE THIRD SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY Held at the University of Arizona Tucson, Arizona February 18-19, 1960 With the Support of the NATIONAL SCIENCE FOUNDATION CARL B. ALLENDOERFER EDITOR

4 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME III D I F F E R E N T I A L G E O M E T R Y AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1961

5 Library of Congress Catalog Card Number Prepared by the American Mathematical Society under Grant No. NSF-G10809 with the National Science Foundation Copyright 1961 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. Otherwise, this book, or parts thereof, may not be reproduced in any form without permission of the publishers.

6 CONTENTS INTRODUCTION PAGE A Report on the Unitary Group 1 By RAOUL BOTT Vector Bundles and Homogeneous Spaces 7 By M. F. ATIYAH and F. HIRZEBRUCH A Procedure for Killing Homotopy Groups of Differentiable Manifolds. 39 By JOHN MILNOR Some Remarks on Homological Analysis and Structures By D. C. SPENCER Vector Form Methods and Deformations of Complex Structures By ALBERT NIJENHUIS Almost-Product Structures 94 By A. G. WALKER Homology of Principal Bundles 101 By ELD ON DYER and R. K. LASH OF Alexander-Pontrjagin Duality in Function Spaces 109 By JAMES EELLS, JR. The Cohomology of Lie Rings 130 By RICHARD S. PALAIS On the Theory of Solvmanifolds and Generalization with Applications to Differential Geometry 138 By Louis AUSLANDER Homogeneous Complex Contact Manifolds 144 By WILLIAM M. BOOTHBY On Compact, Riemannian Manifolds with Constant Curvature. I By EUGENIO CALABI Elementary Remarks on Surfaces with Curvature of Fixed Sign By L. NlRENBERG Canonical Forms on Frame Bundles of Higher Order Contact 186 By SHOSHICHI KOBAYASHI On Immersion of Manifolds 194 By HANS SAMELSON Index. 197 vii

7 INTRODUCTION This Symposium on Differential Geometry was organized as a focal point for the discussion of new trends in research. As can be seen from a quick glance at the papers in this volume, modern differential geometry to a large degree has become differential topology, and the methods employed are a far cry from the tensor analysis of the differential geometry of the loso's. This development, however, has not been as abrupt as might be imagined from a reading of these papers. It has its roots in the movement toward differential geometry in the large to which mathematicians such as Hopf and Rinow, Cohn-Vossen, de Rham, Hodge, and Myers gave impetus. The objectives of their work were to derive relationships between the topology of a manifold and its local differential geometry. Other sources of inspiration were E. Cartan (whose fundamental contributions were recognized by many only after his death) and M. Morse and his calculus of variations in the large. One of the major new ideas was that of a fiber bundle which gave a global structure to a differentiable manifold more general than that included in the older theories. Methods and results of differential geometry were applied with outstanding success to the theories of complex manifolds and algebraic varieties and these in turn have stimulated differential geometry. The discovery by Milnor of invariants of the differential structure of a manifold which are not topological invariants established differential topology as a discipline of major importance. GAEL B. ALLENDOERFER University of Washington, Seattle, Washington Vll

8 Ct-adic topology, 24 Affine connections, 94,186 Affine spaces, locally, 142 Alexander-Pontrjagin duality, 109 theorem, 124 Almost complex, 22 Almost-product structure, 94 Artin-Rees lemma, 24 Atlas Eulerian, 159 Lagrangian, 160 Axioms of a cohomology theory, 14 Bidifferentiable transformations, closed pseudogroups of, 61 Borsuk's Extension Theorem, 120 Bott isomorphism, 13 periodicity, 7 Bundles complex vector, 8 homology of principal, 101 fc-trivial, 49 of r-frames, 188 orien table, 115 ring of complex vector, 7 transverse, 115 C-space, 146 Ci-map, 20 Canonical form differential, 189 structure equation of, 191 Cartan, E. invariant forms' for a continuous pseudogroup of differentiate transformations, 85 structure equations of, 186 Category, model, 59 Characteristic class, 102 relative, 102 Chern character, 15, 29 x-equivalent, 40, 46, 49, 53, 55 Classification theorem, 29 Classifying spaces, 7, 28 Clifford-Klein spaces, 156 differentiable family of, 159 Closed pseudogroups of bidiffenertiable transformations, 61 INDEX 197 Cobordism, 40 Cochains, invariant, 135 Cohomology, 109 group of a Lie d-ring, 137 of Lie rings, 130 operations, 18 with values in the sheaves of Lie algebras of infinitesimal groups, 56 with values in sheaves of nonabelian groups, 56 Cohomology theory axioms of, 14 periodic, 7 Compact, Riemannian manifolds, 155 Complete germ, 72 Completed representation ring of a torus, 26 Completions of modules, 24 Complex, almost, 22 Complex analytic differentiable r-manifold, 64 family of complex structures, 92 Complex contact manifold, 144 manifold, homogeneous, 146 structure, 144 Complex structures complex analytic family of, 92 deformation of, 87 equivalence of, 89 family of, 91 obstructions to deformation of, stability of, 90 variations of, 89 Complex vector bundles, 8 ring of, 7 Connections affine, 94 linear (affine), 186 Constant curvature, Contact form, 145 Continuous pseudogroup of differentiable transformations, 81 invariant Cartan forms for, 85 Coordinate transformation, 118 Co-orienting, 111 Curvature, 186 constant, Gauss, 181

9 198 INDEX d-trivial Lie d-ring, 131 Deformations, homological analysis of, 69 Deformations of complex structures, 87 obstructions to, Deformation of the T-manifold, 70 germ of, 71 Derivative lie, 134 torsional, 99 Differentiability Graves-Hildebrandt, 115 Differentiable complex analytic or real analytic T-manifold, 64 family of T-manifolds, 70 Differentiable transformations continuous pseudogroups of, 81 invariant Cartan forms for a continuous pseudogroup of, 85 Differential form, 189 manifolds, 39 Distributions, 94 Double exterior forms, 166 Duality theorem, 110 Alexander-Pontrjagin, 124 Eilenberg and Steenrod axioms, 7 Equivalence of complex structures, 89 Eulerian atlas, 159 Existence in homological analysis, problem of, 75 Extension Theorem of Borsuk, 120 Exterior forms, double, 166 /-relatedness for vector forms, 90 Family of complex structures, 91 complex analytic, 92 Family of r-manifolds, differentiable, 70 Frames, r-, 188 bundle of, 188 Function spaces, 109 Fundamental class of the oriented pair, 113 r-manifolds deformation of, 70 differentiable family of, 70 differentiable, real analytic or complex analytic, 64 germ of deformation of, 71 T-structure, 64 T-vectorfield,67 Gauss curvature, 181 Germ, complete, 72 effective, 72 of deformation of the r-manifold, 71 stable, 75 Gradient mapping, 182 Grating, 112 Graves-Hildebrandt differentiability, 115 Groups killing homotopy, 39, 50 sheaf of, 65 unitarv, 1, 8 Weyl,23 Gysin homomorphism, 20, 114 Hildebrandt-Graves differentiability, 115 Homogeneous complex contact manifold, 146 spaces, 7, 31 Homological analysis of deformations, 69 Homology of principal bundles, 101 Homomorphism, Gysin, 20, 114 Homotopy complements, 113 killing classes, 43 killing groups, 39, 50 Hypersurfaces, 181 Immersion of manifolds, 194 Implicit function theorem, Infinitesimal pseudogroup, 66 Infinitesimally surjective, 75 Interior product, 133 Invariant Cartan forms for a continuous pseudogroup of differentiable transformations, 85 cochains, 135 cohomology group of a Lie coring, 137 Invariants, ring of, 27 Isomorphism Bott, 13 Theorem of Leray, 111 r. (/) source of, 187 target of, 187 Jacobi identities for vector forms, 88 Jet, r-, 187 A;-parallelizable manifold, 49 fc-trivial bundle, 49

10 INDEX 199 Killing homotopy classes, 43 groups, 39, 50 Klien-Clifford spaces, 156 differentiable family of, 159 < -ring, 132 Lagrangian atlas, 160 Leray Isomorphism Theorem, 111 Lie d-ring cohomology group of, 137 d-trivial, 131 invariant cohomology group of, 137 cc-module over, 132 over R, 131 Lie derivatives, 134 Lie group, compact, 25 connected, 23, 29, 36 representation ring of, 25 Lie rings, cohomology of, 130 Linear (affine) connection, 186 Locally afiine spaces, 142 stable, 74 trivial, 74 Manifold pair, orientation sheet of, 111 Manifolds, 39 compact, Riemannian, 155 complex contact, 144 deformation of the T-, 70 differentiable family of T-, 70 differentiable, real analytic or complex analytic T-, 64 germ of deformation of the T-, 71 homogeneous complex contact, 146 immersion of, 194 fc-parallelizable, 49 Mapping gradient, 182 monotone, 182 spherical image, 181 Model category, 59 Module over a Lie d-ring <, 132 basic, 132 cohomology of < with coefficients in, 136 invariant cohomology of with coefficients in, 136 impairing of two, 132 Modules completions of, 24 Monotone mappings, 182 Morse theory, 2 Multifoliate, 81 Multiplication, Pontrjagin, 125 Nilmanifold, 138 Noetherian ring, 24 Normal degree, Obstructions to deformation of a complex structure, Orient, 111, 115 Orientability, 111 Orientable bundle, 115 pair, 111 Orientation sheet of the manifold pair (X, F), 111 Oriented pair, fundamental class of, 113 Orienting, co-, 111 Parallel, 99 Periodic cohomology theory, 7 Periodicity, Bott, 7 7r-manifold, 46 Pontrjagin classes, 20 multiplication, 125 numbers, 41 Pontrjagin-Alexander duality, 109 theorem, 124 Primitive left, 167 right, 167 Principal bundles, homology of, 101 Product interior, 133 triple, 105 Projective space, 3 Projector, 94 Pseudogroup, 59 closed of bidifferentiable transformations, 61 infinitesimal, 66 of bidifferentiable, bianalytic or biholomorphic transformations, 64 resolution of the sheaf of vector fields associated with a continuous r (sheaf of T-vectorfields),85 Pseudogroup of differentiable transformations, continuous, 81 invariant Cartan forms for, 85

11 200 INDEX r-frames, 188 bundle of, 188 r-jet, 187 Real analytic, differentiable r-manifold, 64 Rees-Artin lemma, 24 Representation ring completed, 27 completed of a torus, 26 of a compact Lie group, 25 Resolution of the sheaf of vector fields associated with a continuous pseudogroup r (sheaf of T-vectorfields),85 Riemann-Roch theorem, 7, 20 Rigid, 78 Ring Noetherian, 24 of complex vector bundles, 7 of invariants, 27 Saddle surfaces, 182 Sequence, Wang, Sheaf of groups, 65 of vectorfieldsassociated with a continuous pseudogroup T (sheaf of r-vector fields), resolution of, 85 Solvmanifold, 138 Source of (/), 187 Spaces classifying, 7, 28 CUfford-Klein, 156 differentiable family of Clifford-Klein, 159 function, 109 homogeneous, 7, 31 locally affine, 142 projective, 3 structure on topological, 60 Spectral sequence, 7, 16 Spherical image mapping, 181 Spinor representation, 33 Stability of complex structures, 90 Stable germ, 75 locally, 74 Steenrod and Eilenberg axioms, 7 Stiefel-Whitney classes, 20 numbers, 41 Structure almost-product, 94 complex contact, 144 equation of the canonical form, 191 equations of E. Cartan, 186 T-,64 on a topological space, 60 (See Complex) Submanifold, closed relative, 113 Surfaces, 181 saddle, 182 Surgery, 39-42, 44, 46, 54 Surjective, infinitesimally, 75 Suspension, 9 Target of (/), 187 Tietze's Theorem, 119 Todd genus, 21 Topological space, structure on, 60 Topology, a-adic, 24 Torsion, 95, 186 for vector forms, 88 Torsional derivatives, 99 Torus, 26 completed representation ring of, 26 Transformation, coordinate, 118 Transregular, 116 Transverse bundle, 115 Triple product, 105 Trivial locally, 74 Unitary group, 1, 8 Universal Coefficient Theorem, 126 Variability, index of, 72 Variations of a complex structure, 89 Vector bundles, complex, 8 ring of, 7 Vector fields associated with a continuous pseudogroup r (sheaf of r-vector fields), resolution of the sheaf of, 85 r-,67 Vector forms, 87 /-relatedness for, 90 Jacobi identities, 88 torsion for, 88 types of, 88 vertical, 91 Wang sequence, Weyl group, 23 Whitney-Stiefel classes, 20 numbers, 41 BCDEFGHIJ-AMS

12

Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem

Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents

More information

Modern Geometric Structures and Fields

Modern Geometric Structures and Fields Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

More information

CHARACTERISTIC CLASSES

CHARACTERISTIC CLASSES 1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

More information

Celebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013

Celebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013 Celebrating One Hundred Fifty Years of Topology John Milnor Institute for Mathematical Sciences Stony Brook University (www.math.sunysb.edu) ARBEITSTAGUNG Bonn, May 22, 2013 Algebra & Number Theory 3 4

More information

Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

More information

Exercises on characteristic classes

Exercises on characteristic classes Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives

More information

Characteristic classes and Invariants of Spin Geometry

Characteristic classes and Invariants of Spin Geometry Characteristic classes and Invariants of Spin Geometry Haibao Duan Institue of Mathematics, CAS 2018 Workshop on Algebraic and Geometric Topology, Southwest Jiaotong University July 29, 2018 Haibao Duan

More information

Submanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition.

Submanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition. le 27 AIPEI CHENNAI TAIPEI - Series in Pure Mathematics Volume 27 Total Mean Curvature and Submanifolds of Finite Type Second Edition Bang-Yen Chen Michigan State University, USA World Scientific NEW JERSEY

More information

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

More information

Patrick Iglesias-Zemmour

Patrick Iglesias-Zemmour Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

More information

Contents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39.

Contents. Chapter 3. Local Rings and Varieties Rings of Germs of Holomorphic Functions Hilbert s Basis Theorem 39. Preface xiii Chapter 1. Selected Problems in One Complex Variable 1 1.1. Preliminaries 2 1.2. A Simple Problem 2 1.3. Partitions of Unity 4 1.4. The Cauchy-Riemann Equations 7 1.5. The Proof of Proposition

More information

Fundamentals of Differential Geometry

Fundamentals of Differential Geometry - Serge Lang Fundamentals of Differential Geometry With 22 luustrations Contents Foreword Acknowledgments v xi PARTI General Differential Theory 1 CHAPTERI Differential Calculus 3 1. Categories 4 2. Topological

More information

Syllabuses for Honor Courses. Algebra I & II

Syllabuses for Honor Courses. Algebra I & II Syllabuses for Honor Courses Algebra I & II Algebra is a fundamental part of the language of mathematics. Algebraic methods are used in all areas of mathematics. We will fully develop all the key concepts.

More information

SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY

SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY M. F. ATIYAH, V. K. PATODI AND I. M. SINGER 1 Main Theorems If A is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then

More information

The Riemann Legacy. Riemannian Ideas in Mathematics and Physics KLUWER ACADEMIC PUBLISHERS. Krzysztof Maurin

The Riemann Legacy. Riemannian Ideas in Mathematics and Physics KLUWER ACADEMIC PUBLISHERS. Krzysztof Maurin The Riemann Legacy Riemannian Ideas in Mathematics and Physics by Krzysztof Maurin Division of Mathematical Methods in Physics, University of Warsaw, Warsaw, Poland KLUWER ACADEMIC PUBLISHERS DORDRECHT

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

Math 550 / David Dumas / Fall Problems

Math 550 / David Dumas / Fall Problems Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;

More information

Quaternionic Complexes

Quaternionic Complexes Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

More information

Math 231b Lecture 16. G. Quick

Math 231b Lecture 16. G. Quick Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector

More information

Lecture Notes in Mathematics

Lecture Notes in Mathematics Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 766 Tammo tom Dieck Transformation Groups and Representation Theory Springer-Verlag Berlin Heidelberg New York 1979 Author T. tom Dieck Mathematisches

More information

Exotic spheres. Overview and lecture-by-lecture summary. Martin Palmer / 22 July 2017

Exotic spheres. Overview and lecture-by-lecture summary. Martin Palmer / 22 July 2017 Exotic spheres Overview and lecture-by-lecture summary Martin Palmer / 22 July 2017 Abstract This is a brief overview and a slightly less brief lecture-by-lecture summary of the topics covered in the course

More information

L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S

L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S L A U R E N T I U M A X I M U N I V E R S I T Y O F W I S C O N S I N - M A D I S O N L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S i Contents 1 Basics of Homotopy

More information

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

More information

THE DIAGONAL COHOMOLOGY CLASS OF VERTICAL BUNDLES

THE DIAGONAL COHOMOLOGY CLASS OF VERTICAL BUNDLES THE DIAGONAL COHOMOLOGY CLASS OF VERTICAL BUNDLES SPUR FINAL PAPER, SUMMER 2017 JUAN CARLOS ORTIZ MENTOR: JACKSON HANCE Abstract Given a manifold M, Milnor and Stasheff studied in [1] the diagonal cohomology

More information

Chern Classes and the Chern Character

Chern Classes and the Chern Character Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the

More information

SOME ASPECTS OF STABLE HOMOTOPY THEORY

SOME ASPECTS OF STABLE HOMOTOPY THEORY SOME ASPECTS OF STABLE HOMOTOPY THEORY By GEORGE W. WHITEHEAD 1. The suspension category Many of the phenomena of homotopy theory become simpler in the "suspension range". This fact led Spanier and J.

More information

Cobordant differentiable manifolds

Cobordant differentiable manifolds Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated

More information

Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 8: More characteristic classes and the Thom isomorphism Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable

More information

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula 20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

More information

Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

More information

CHARACTERISTIC CLASSES

CHARACTERISTIC CLASSES CHARACTERISTIC CLASSES ARUN DEBRAY CONTENTS 1. Four approaches to characteristic classes 1 2. Stiefel-Whitney classes 6 3. Stable cohomology operations and the Wu formula 10 4. Chern, Pontrjagin, and Euler

More information

Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017

Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017 Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory

More information

An Introduction to Spectral Sequences

An Introduction to Spectral Sequences An Introduction to Spectral Sequences Matt Booth December 4, 2016 This is the second half of a joint talk with Tim Weelinck. Tim introduced the concept of spectral sequences, and did some informal computations,

More information

Osaka Journal of Mathematics. 37(2) P.1-P.4

Osaka Journal of Mathematics. 37(2) P.1-P.4 Title Katsuo Kawakubo (1942 1999) Author(s) Citation Osaka Journal of Mathematics. 37(2) P.1-P.4 Issue Date 2000 Text Version publisher URL https://doi.org/10.18910/4128 DOI 10.18910/4128 rights KATSUO

More information

Real affine varieties and obstruction theories

Real affine varieties and obstruction theories Real affine varieties and obstruction theories Satya Mandal and Albert Sheu University of Kansas, Lawrence, Kansas AMS Meeting no.1047, March 27-29, 2009, at U. of Illinois, Urbana, IL Abstract Let X =

More information

The Riemann-Roch Theorem

The Riemann-Roch Theorem The Riemann-Roch Theorem TIFR Mumbai, India Paul Baum Penn State 7 August, 2015 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond ellipticity 5. The Riemann-Roch

More information

Math 797W Homework 4

Math 797W Homework 4 Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition

More information

A (Brief) History of Homotopy Theory

A (Brief) History of Homotopy Theory April 26, 2013 Motivation Why I m giving this talk: Dealing with ideas in the form they were first discovered often shines a light on the primal motivation for them (...) Why did anyone dream up the notion

More information

NONNEGATIVE CURVATURE AND COBORDISM TYPE. 1. Introduction

NONNEGATIVE CURVATURE AND COBORDISM TYPE. 1. Introduction NONNEGATIVE CURVATURE AND COBORDISM TYPE ANAND DESSAI AND WILDERICH TUSCHMANN Abstract. We show that in each dimension n = 4k, k 2, there exist infinite sequences of closed simply connected Riemannian

More information

Atiyah-Singer Revisited

Atiyah-Singer Revisited Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator

More information

Differential Geometry, Lie Groups, and Symmetric Spaces

Differential Geometry, Lie Groups, and Symmetric Spaces Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island PREFACE PREFACE TO THE

More information

something on spin structures sven-s. porst

something on spin structures sven-s. porst something on spin structures sven-s. porst spring 2001 abstract This will give a brief introduction to spin structures on vector bundles to pave the way for the definition and introduction of Dirac operators.

More information

A users guide to K-theory

A users guide to K-theory A users guide to K-theory K-theory Alexander Kahle alexander.kahle@rub.de Mathematics Department, Ruhr-Universtät Bochum Bonn-Cologne Intensive Week: Tools of Topology for Quantum Matter, July 2014 Outline

More information

Thinking of Iaοsi, my hometown

Thinking of Iaοsi, my hometown LECTURES ON THE GEOMETRY OF MANIFOLDS Liviu I. Nicolaescu Thinking of Iaοsi, my hometown i Introduction Shape is a fascinating and intriguing subject which has stimulated the imagination of many people.

More information

On the Van Est homomorphism for Lie groupoids

On the Van Est homomorphism for Lie groupoids Fields Institute, December 13, 2013 Overview Let G M be a Lie groupoid, with Lie algebroid A = Lie(G) Weinstein-Xu (1991) constructed a cochain map VE: C (G) C (A) from smooth groupoid cochains to the

More information

Lecture on Equivariant Cohomology

Lecture on Equivariant Cohomology Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove

More information

Introduction to surgery theory

Introduction to surgery theory Introduction to surgery theory Wolfgang Lück Bonn Germany email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/ Bonn, 17. & 19. April 2018 Wolfgang Lück (MI, Bonn) Introduction to surgery theory

More information

Foundation Modules MSc Mathematics. Winter Term 2018/19

Foundation Modules MSc Mathematics. Winter Term 2018/19 F4A1-V3A2 Algebra II Prof. Dr. Catharina Stroppel The first part of the course will start from linear group actions and study some invariant theory questions with several applications. We will learn basic

More information

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1) Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

More information

NOTES ON FIBER BUNDLES

NOTES ON FIBER BUNDLES NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement

More information

J-holomorphic curves in symplectic geometry

J-holomorphic curves in symplectic geometry J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely

More information

Introduction (Lecture 1)

Introduction (Lecture 1) Introduction (Lecture 1) February 2, 2011 In this course, we will be concerned with variations on the following: Question 1. Let X be a CW complex. When does there exist a homotopy equivalence X M, where

More information

From Algebraic Geometry to Homological Algebra

From Algebraic Geometry to Homological Algebra From Algebraic Geometry to Homological Algebra Sepehr Jafari Università Degli Studi di Genova Dipartimento Di Matematica November 8, 2016 November 8, 2016 1 / 24 Outline 1 Historical Events Algebraic Geometry

More information

Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions

Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions by Shizuo Kaji Department of Mathematics Kyoto University Kyoto 606-8502, JAPAN e-mail: kaji@math.kyoto-u.ac.jp Abstract

More information

Introductory Lectures on Manifold Topology: Signposts

Introductory Lectures on Manifold Topology: Signposts Surveys of Modern Mathematics Volume VII Introductory Lectures on Manifold Topology: Signposts Thomas Farrell Department of Mathematical Sciences Binghamton University Yang Su Academy of Mathematics and

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

KO -theory of complex Stiefel manifolds

KO -theory of complex Stiefel manifolds KO -theory of complex Stiefel manifolds Daisuke KISHIMOTO, Akira KONO and Akihiro OHSHITA 1 Introduction The purpose of this paper is to determine the KO -groups of complex Stiefel manifolds V n,q which

More information

Remarks on the Milnor number

Remarks on the Milnor number José 1 1 Instituto de Matemáticas, Universidad Nacional Autónoma de México. Liverpool, U. K. March, 2016 In honour of Victor!! 1 The Milnor number Consider a holomorphic map-germ f : (C n+1, 0) (C, 0)

More information

WHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014

WHAT IS K-HOMOLOGY? Paul Baum Penn State. Texas A&M University College Station, Texas, USA. April 2, 2014 WHAT IS K-HOMOLOGY? Paul Baum Penn State Texas A&M University College Station, Texas, USA April 2, 2014 Paul Baum (Penn State) WHAT IS K-HOMOLOGY? April 2, 2014 1 / 56 Let X be a compact C manifold without

More information

Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

More information

Lecture 6: Classifying spaces

Lecture 6: Classifying spaces Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

More information

THE HOMOTOPY TYPES OF SU(5)-GAUGE GROUPS. Osaka Journal of Mathematics. 52(1) P.15-P.29

THE HOMOTOPY TYPES OF SU(5)-GAUGE GROUPS. Osaka Journal of Mathematics. 52(1) P.15-P.29 Title THE HOMOTOPY TYPES OF SU(5)-GAUGE GROUPS Author(s) Theriault, Stephen Citation Osaka Journal of Mathematics. 52(1) P.15-P.29 Issue Date 2015-01 Text Version publisher URL https://doi.org/10.18910/57660

More information

BERNOULLI NUMBERS, HOMOTOPY GROUPS, AND A THEOREM OF ROHLIN

BERNOULLI NUMBERS, HOMOTOPY GROUPS, AND A THEOREM OF ROHLIN 454 BERNOULLI NUMBERS, HOMOTOPY GROUPS, AND A THEOREM OF ROHLIN By JOHN W. MILNOR AND MICHEL A. KERVAIRE A homomorphism J: 7T k _ 1 (SO w ) -> n m+k _ 1 (S m ) from the homotopy groups of rotation groups

More information

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds

A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn

More information

Foliations, Fractals and Cohomology

Foliations, Fractals and Cohomology Foliations, Fractals and Cohomology Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder Colloquium, University of Leicester, 19 February 2009 Steven Hurder (UIC) Foliations, fractals,

More information

The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

More information

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms

EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be

More information

INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY

INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY YOUNG-HOON KIEM 1. Definitions and Basic Properties 1.1. Lie group. Let G be a Lie group (i.e. a manifold equipped with differentiable group operations mult

More information

CYCLES, SUBMANIFOLDS, AND STRUCTURES ON NORMAL BUNDLES

CYCLES, SUBMANIFOLDS, AND STRUCTURES ON NORMAL BUNDLES CYCLES, SUBMANIFOLDS, AND STRUCTURES ON NORMAL BUNDLES C. BOHR, B. HANKE, AND D. KOTSCHICK Abstract. We give explicit examples of degree 3 cohomology classes not Poincaré dual to submanifolds, and discuss

More information

The Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick

The Based Loop Group of SU(2) Lisa Jeffrey. Department of Mathematics University of Toronto. Joint work with Megumi Harada and Paul Selick The Based Loop Group of SU(2) Lisa Jeffrey Department of Mathematics University of Toronto Joint work with Megumi Harada and Paul Selick I. The based loop group ΩG Let G = SU(2) and let T be its maximal

More information

arxiv: v3 [math.kt] 14 Nov 2014

arxiv: v3 [math.kt] 14 Nov 2014 Twisted K-homology, Geometric cycles and T-duality Bei Liu Mathematisches Institut, Georg-August-Universitä Göettingen arxiv:1411.1575v3 [math.kt] 14 Nov 2014 16th January 2018 Abstract Twisted K-homology

More information

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1 Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

More information

Algebraic geometry over quaternions

Algebraic geometry over quaternions Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

More information

Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,

Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix, Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n

More information

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical

More information

Isometric Immersions without Positive Ricci Curvature

Isometric Immersions without Positive Ricci Curvature Contemporary Mathematics Isometric Immersions without Positive Ricci Curvature Luis Guijarro Abstract. In this note we study isometric immersions of Riemannian manifolds with positive Ricci curvature into

More information

Topology in the 20th century: a view from the inside

Topology in the 20th century: a view from the inside Russian Math. Surveys 59:5 803 829 Uspekhi Mat. Nauk 59:5 3 28 c 2004 RAS(DoM) and LMS DOI 10.1070/RM2004v059n05ABEH000770 Topology in the 20th century: a view from the inside S. P. Novikov Foreword The

More information

Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism

Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized

More information

Characteristic classes in the Chow ring

Characteristic classes in the Chow ring arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

More information

SOME EXERCISES IN CHARACTERISTIC CLASSES

SOME EXERCISES IN CHARACTERISTIC CLASSES SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined

More information

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE

LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds

More information

Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

More information

Atiyah classes and homotopy algebras

Atiyah classes and homotopy algebras Atiyah classes and homotopy algebras Mathieu Stiénon Workshop on Lie groupoids and Lie algebroids Kolkata, December 2012 Atiyah (1957): obstruction to existence of holomorphic connections Rozansky-Witten

More information

An Introduction to Complex K-Theory

An Introduction to Complex K-Theory An Introduction to Complex K-Theory May 23, 2010 Jesse Wolfson Abstract Complex K-Theory is an extraordinary cohomology theory defined from the complex vector bundles on a space. This essay aims to provide

More information

Classics in Mathematics

Classics in Mathematics Classics in Mathematics Friedrich Hirzebruch Topological Methods in Algebraic Geometry Friedrich Hirzebruch Topological Methods in Algebraic Geometry Reprint of the 1978 Edition Springer Friedrich Hirzebruch

More information

The Riemann-Roch Theorem

The Riemann-Roch Theorem The Riemann-Roch Theorem Paul Baum Penn State Texas A&M University College Station, Texas, USA April 4, 2014 Minicourse of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology?

More information

On algebraic index theorems. Ryszard Nest. Introduction. The index theorem. Deformation quantization and Gelfand Fuks. Lie algebra theorem

On algebraic index theorems. Ryszard Nest. Introduction. The index theorem. Deformation quantization and Gelfand Fuks. Lie algebra theorem s The s s The The term s is usually used to describe the equality of, on one hand, analytic invariants of certain operators on smooth manifolds and, on the other hand, topological/geometric invariants

More information

1.1 Definition of group cohomology

1.1 Definition of group cohomology 1 Group Cohomology This chapter gives the topological and algebraic definitions of group cohomology. We also define equivariant cohomology. Although we give the basic definitions, a beginner may have to

More information

HODGE GENERA OF ALGEBRAIC VARIETIES, II.

HODGE GENERA OF ALGEBRAIC VARIETIES, II. HODGE GENERA OF ALGEBRAIC VARIETIES, II. SYLVAIN E. CAPPELL, ANATOLY LIBGOBER, LAURENTIU MAXIM, AND JULIUS L. SHANESON Abstract. We study the behavior of Hodge-theoretic genera under morphisms of complex

More information

Lectures on the Orbit Method

Lectures on the Orbit Method Lectures on the Orbit Method A. A. Kirillov Graduate Studies in Mathematics Volume 64 American Mathematical Society Providence, Rhode Island Preface Introduction xv xvii Chapter 1. Geometry of Coadjoint

More information

INERTIA GROUPS AND SMOOTH STRUCTURES OF (n - 1)- CONNECTED 2n-MANIFOLDS. Osaka Journal of Mathematics. 53(2) P.309-P.319

INERTIA GROUPS AND SMOOTH STRUCTURES OF (n - 1)- CONNECTED 2n-MANIFOLDS. Osaka Journal of Mathematics. 53(2) P.309-P.319 Title Author(s) INERTIA GROUPS AND SMOOTH STRUCTURES OF (n - 1)- CONNECTED 2n-MANIFOLDS Ramesh, Kaslingam Citation Osaka Journal of Mathematics. 53(2) P.309-P.319 Issue Date 2016-04 Text Version publisher

More information

Algebraic geometry versus Kähler geometry

Algebraic geometry versus Kähler geometry Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2

More information

110:615 algebraic topology I

110:615 algebraic topology I 110:615 algebraic topology I Topology is the newest branch of mathematics. It originated around the turn of the twentieth century in response to Cantor, though its roots go back to Euler; it stands between

More information

Hodge Theory of Maps

Hodge Theory of Maps Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

More information

Polynomial Hopf algebras in Algebra & Topology

Polynomial Hopf algebras in Algebra & Topology Andrew Baker University of Glasgow/MSRI UC Santa Cruz Colloquium 6th May 2014 last updated 07/05/2014 Graded modules Given a commutative ring k, a graded k-module M = M or M = M means sequence of k-modules

More information

EXACT BRAIDS AND OCTAGONS

EXACT BRAIDS AND OCTAGONS 1 EXACT BRAIDS AND OCTAGONS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/ aar Lewisfest, Dublin, 23 July 2009 Organized by Eva Bayer-Fluckiger, David Lewis and Andrew Ranicki. 2 3 Exact braids

More information

Oxford 13 March Surgery on manifolds: the early days, Or: What excited me in the 1960s. C.T.C.Wall

Oxford 13 March Surgery on manifolds: the early days, Or: What excited me in the 1960s. C.T.C.Wall Oxford 13 March 2017 Surgery on manifolds: the early days, Or: What excited me in the 1960s. C.T.C.Wall In 1956 Milnor amazed the world by giving examples of smooth manifolds homeomorphic but not diffeomorphic

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information