LECTURE 26: THE CHERN-WEIL THEORY
|
|
- Reginald Dickerson
- 5 years ago
- Views:
Transcription
1 LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if T v σ1,, v σk ) = T v 1,, v k ), σ S k. We will denote the space of all symmetric k-tensors on V by S k V. Let T S k V be any symmetric k-tensor. Then T induces a degree k homogeneous polynomial on V, P T v) := T v,, v). Conversely, it is not hard to see that T is completely determined by P T, since we have the following polarization formula k T v 1,, v k ) := 1 P T t 1 v t k v k ). k! t 1 t k Like wedge product, we can define a symmetric product : S k V S l V S k+l V via 1 T 1 T 2 v 1,, v k+l ) = T 1 v 1,, v k )T 2 v k+1,, v k+l ). k + l)! σ S k+l As usual we put S 0 V = R. Of course P T defined as above is not really a polynomial. At least one should need a conception of products of vectors on V before we can talk about polynomials on V.) However, we can associate to each T S k V a true polynomial as follows: We fix a basis {e 1,, e n } of V. Let R[x 1,, x n ] be the ring of polynomials in the variable x 1,, x n, and let R[x 1,, x n ] k be the subset of homogeneous polynomials of degree k. We define a map P : S k V R[x 1,, x n ] k by T p T x 1,, x n ) := P T x i e i ) = T x i e i,, x i e i ) Obviously P is a linear map and satisfies PT 1 T 2 ) = PT 1 )PT 2 ). Lemma 1.1. The map P is a linear isomorphism. In particular, dim S k V = n+k 1 n 1 Proof. By the polarization formula above we see P is injective. To show that P is surjective, we just notice that for k = 1, P maps T i S 1 V = V to the polynomial p Ti x 1,, x n ) = x i, where T i is the map T i x i e i ) := x i. For general k, by repeatedly using the rule PT 1 T 2 ) = PT 1 )PT 2 ) one immediately see that P is surjective. Remark. The fact PT 1 T 2 ) = PT 1 )PT 2 ) implies that P : S V R[x 1,, x n ] is in fact a ring isomorphism. ).) 1
2 2 LECTURE 26: THE CHERN-WEIL THEORY Now let V = g be the Lie algebra of a Lie group G. Then the adjoint action of G on g induces a G-action on S k g ) by g T )X 1,, X k ) := T Ad g 1X 1,, Ad g 1X k ), X i g, g G. Note that for the case that G is a linear Lie group, which we will always assume below, one has g T )X 1,, X k ) = T gx 1 g 1,, gx k g 1 ), where g is an invertible r r matrix while X i s are arbitrary r r matrices. Definition 1.2. T S k g ) is called invariant if g T = T, g G. The set of all G-invariant elements in S k g ) is denoted by I k G). So I G) = I k G) is a subring of S g ). Note that by definition and the polarization formula, T is invariant if and only if P T is invariant. Example. Consider G = GLr, R). For any positive integer k, we let p k/2 denote the degree k homogeneous polynomial which is the coefficient of λ r k for the following polynomial in λ det λi 1 ) r 2π A = p k/2 A,, A)λ r k, A gln, R). k=0 Obviously p k/2 is invariant. So p k/2 I k GLr, R)). They are called the k/2-th Pontrjagin polynomials. Note that for G = Or) GLr, R), the Lie algebra or) consists of skew-symmetric matrices, and hence det λi 1 ) 2π A = det λi + 1 ) 2π A, A or). This implies that p k/2 = 0 for k odd. So for On) one only need to study p k I 2k On)). Proposition 1.3. For any T I k G) and any X, X 1,, X k g, we have T [X, X 1 ], X 2,, X k ) + + T X 1,, X k 1, [X, X k ]) = 0. Proof. This follows from d dt T e tx X 1 e tx,, e tx X k e tx ) = 0. t=0 For T I k G) one can also define T F 1,, F k ) for F i Ω U) g, by extending linearly the relation T ω 1 X 1,, ω k X k ) := ω 1 ω k )T X 1,, X k ). Note that if ω is a 2-forms, or more generally is any even-form, then for any η one has ω η = η ω. As a consequence we see
3 LECTURE 26: THE CHERN-WEIL THEORY 3 Corollary 1.4. If F 1,, F k Ω even U) g, then for any T I k G) and A Ω U) g one has T [A, F 1 ], F 2,, F k ) + + T F 1,, F k 1, [A, F k ]) = Chern-Weil Theory Let G be a Lie group and for simplicity, assume G is a linear Lie group), with Lie algebra g. Let P be a principal G-bundle over M, which is defined by an open cover {U α ) and corresponding transition maps g αβ : U α U β G. Let A be a collection A α Ω 1 U α ) g which is a connection in P. Recall that the curvature F A of the connection A is given locally by the collection For any T I k G) we can define F α = da α [A α, A α ] Ω 2 U α ) g. p T F α ) := T F α,, F α ) Ω 2k U α ). Since T is Ad-invariant and since F β = g βα F α g 1 βα, we get p T F α ) = p T F β ) on U α U β. In other words, there is a globally-defined 2k-form p T F A ) Ω 2k M) so that p T F A ) = p T F α ) on U α. Note F A is not an element in Ω 2 M): it sits in Ω 2 M; AdP ).) It turns out that these p T F A ) s play a crucial role in modern math and physics. Theorem 2.1 Chern-Weil). Let P be a principal G-bundle over M. Then 1) For any T I k G) and any connection A in P, the form p T F A ) is closed. 2) The de Rham class [p T F A )] HdR 2k M) is independent of the choices of A. 3) The Chern-Weil map CW : I G), ) HdR M), ) that maps T to [p T F A )] is a ring homomorphism. Proof. 1) According to the Bianchi identity df α = [A α, F α ]. So by corollary 1.4, dp T F A ) = dt F α,, F α ) = T df α,, F α ) + + T F α,, df α ) = 0. This proves p T F A ) is closed. 2) To prove that the de Rham class [p T F A )] is independent of the choices of connection A, we let A 0 and A 1 be two connections defined by the collections A 0 α and A 1 α. By definition it is easy to check that the collection à α = 1 s)a 0 α + sa 1 α Ω 1 U α R) g defines a connection on a new principal bundle P R over M R This new principal bundle is defined over open sets {U α R} by using the same set of transition functions g αβ ). If we let ι 1 and ι 0 be the same as in the following lemma, then ι 0Ãα = A 0 α and ι 1Ãα = A 1 α. As a result, we see ι 0Fà = F A 0 and ι 1Fà = F A 1.
4 4 LECTURE 26: THE CHERN-WEIL THEORY So according to the next lemma, p T F A 0) p T F A 1) = ι 0p T FÃ) ι 1p T FÃ) = dqp T FÃ)) + Qdp T FÃ)). But we just proved that p T FÃ) is closed. So [p T F A 0)] = [p T F A 1)]. Lemma 2.2. Let ι 0, ι 1 : M M R be the inclusions ι 0 x) = x, 0), ι 1 x) = x, 1). Then there exists a collection of linear operators Q : Ω k M R) Ω k 1 M) so that ι 0ω ι 1ω = dqω) Qdω), ω Ω k M R). Proof. One just repeat the first four lines of the proof of theorem 2.6 in lecture 19, to get some linear map Q : Ω k M R) Ω k 1 M R) so that ω φ 1ω = d Qω + Qdω. It follows that ι 0ω ι 1ω = ι 0ω ι 0φ 1ω = ι 0d Qω + ι 0 Qdω = dι 0 Q)ω + ι 0 Q)dω. So the conclusion holds for Q = ι Q 0 : Ω k M R) Ω k 1 M). 3) It is straightforward to show that for T S k G) and S S l G), p T S F A ) = p T F A ) p S F A ). Details left as an exercise. Example. Consider the trivial principal G-bundle P = M G over M. In this case one can take U α = M, i.e. the open cover contains only one element. Then there is only one g αα, which equals e G at each point x M. For such a covering data one can take A α to be identically zero. It follows that F A = 0 and thus for any T I G),the class [p T F A )] = 0. Definition 2.3. For any T I G), the cohomology class [p T F A )]] is called the characteristic class for P corresponding to T. Remark. Obviously the above construction works if we replace the principal bundle P by a vector bundle E. So one can also talk about the charactersitic classes of vector bundle E over M associated with T I G), where G is the structural group of E. Example. Let E be any vector bundle over M. By choosing a Riemannian metric one can always reduce the structural group of E from GLr, R) to Or). Let p k be the Pontrjagin polynomial that we alluded above. Then p k E) := [p k F A )] H 4k drm) is called the kth Pontrjagin characteristic class of E. The Pontrjagin class p k M) of M is defined to be the Pontrjagin class of T M. They are important topological invariants to study manifolds. Example. Suppose r = 2p and consider G = SOr). For any A = a i j) sor) we let P fa) = 1 4π) r r/2)! σ S r 1) σ a σ1) σ2) aσ3) σ4) aσr 1) σr). One can check that P f is an Ad-invariant homogeneous polynomial of degree r/2. It is called the Pfaff polynomial.
5 LECTURE 26: THE CHERN-WEIL THEORY 5 Now suppose E is an oriented vector bundle over M of rank r. Then the structural group of E can be reduced to SOr). Thus we get a characteristic class ee) := [P ff A )] H r drm) which is called the Euler characterstic class of E. Again the Euler class em) of a smooth manifold M is defined to be the Euler class of T M. It generalizes the classical notion of Euler characteristic. Example. All discusstions above are over R, but they can be generalized to objects over C. For example, one can talk about complex vector bundle E over smooth manifold M: They are vector bundles over M whose fibers are C r and whose structural group is GLr, C). For any A glr, C) = the set of all r r complex matrices), one can define c k to be such that det λi r 1 ) 2πi A = c k A)λ n k. Again c k is a homogeneous polynomial of degree k and is Ad-invariant. As a result, for any complex vector bundle E of complex rank r, one gets a complex-valued de Rham cohomology class c k E) := c k F A ) HdRM; 2k C) which is called the kth Chern class of E. If one fix an Hermitian metric on E i.e. fix a Hermitian inner product on each fiber of E which vary smoothly with respect to base points this is always possible by using partition of unity), then one can reduce the structural group to Un). But for A un) = the set of all r r complex matrices with A + A T = 0), one has det λi r 1 ) 2πi A = det λi r 1 ) 2πi A. It follows that each c k is a real-valued polynomial and thus each c k E) H 2k dr M), i.e. c ke) is a real-valued) de Rham cohomology class. It is of fundamental importance in algebraic topology, differential geometry, algebraic geometry and mathematical physics. The End
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 informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationE 0 0 F [E] + [F ] = 3. Chern-Weil Theory How can you tell if idempotents over X are similar?
. Characteristic Classes from the viewpoint of Operator Theory. Introduction Overarching Question: How can you tell if two vector bundles over a manifold are isomorphic? Let X be a compact Hausdorff space.
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationMath 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 informationHolomorphic line bundles
Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationLECTURE: 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 informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationMATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY
MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract
More informationLecture 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 informationHyperkähler 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 informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationWe have the following immediate corollary. 1
1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
More informationLECTURE 5: SURFACES IN PROJECTIVE SPACE. 1. Projective space
LECTURE 5: SURFACES IN PROJECTIVE SPACE. Projective space Definition: The n-dimensional projective space P n is the set of lines through the origin in the vector space R n+. P n may be thought of as the
More informationEilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )
II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
More informationA Bridge between Algebra and Topology: Swan s Theorem
A Bridge between Algebra and Topology: Swan s Theorem Daniel Hudson Contents 1 Vector Bundles 1 2 Sections of Vector Bundles 3 3 Projective Modules 4 4 Swan s Theorem 5 Introduction Swan s Theorem is a
More informationLecture III: Neighbourhoods
Lecture III: Neighbourhoods Jonathan Evans 7th October 2010 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 1 / 18 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 2 / 18 In
More informationCHERN-WEIL THEORY ADEL RAHMAN
CHERN-WEIL THEORY ADEL RAHMAN Abstract. We give an introduction to the Chern-Weil construction of characteristic classes of complex vector bundles. We then relate this to the more classical notion of Chern
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationCHAPTER 3 SUBMANIFOLDS
CHAPTER 3 SUBMANIFOLDS One basic notion that was not addressed in Chapter 1 is the concept of containment: When one manifold is a subset on a second manifold. This notion is subtle in a manner the reader
More informationA little taste of symplectic geometry
A little taste of symplectic geometry Timothy Goldberg Thursday, October 4, 2007 Olivetti Club Talk Cornell University 1 2 What is symplectic geometry? Symplectic geometry is the study of the geometry
More informationV = 1 2 (g ijχ i h j ) (2.4)
4 VASILY PESTUN 2. Lecture: Localization 2.. Euler class of vector bundle, Mathai-Quillen form and Poincare-Hopf theorem. We will present the Euler class of a vector bundle can be presented in the form
More informationPractice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.
Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts
More informationSubgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.
Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this
More information(1) Let π Ui : U i R k U i be the natural projection. Then π π 1 (U i ) = π i τ i. In other words, we have the following commutative diagram: U i R k
1. Vector Bundles Convention: All manifolds here are Hausdorff and paracompact. To make our life easier, we will assume that all topological spaces are homeomorphic to CW complexes unless stated otherwise.
More informationMetrics and Holonomy
Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it
More informationLECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL TANGENT VECTORS.
LECTURE 2. (TEXED): IN CLASS: PROBABLY LECTURE 3. MANIFOLDS 1. FALL 2006. TANGENT VECTORS. Overview: Tangent vectors, spaces and bundles. First: to an embedded manifold of Euclidean space. Then to one
More informationThe Atiyah bundle and connections on a principal bundle
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata
More informationA MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS
A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of
More informationMath 225B: Differential Geometry, Final
Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of
More informationQualifying Exams I, 2014 Spring
Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationDirac 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 informationREPRESENTATION THEORY WEEK 7
REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable
More informationFibre Bundles and Chern-Weil Theory. Johan Dupont
Fibre Bundles and Chern-Weil Theory Johan Dupont Aarhus Universitet August 2003 Contents 1 Introduction 3 2 Vector Bundles and Frame Bundles 8 3 Principal G-bundles 18 4 Extension and reduction of principal
More informationCharacteristic classes of vector bundles
Characteristic classes of vector bunles Yoshinori Hashimoto 1 Introuction Let be a smooth, connecte, compact manifol of imension n without bounary, an p : E be a real or complex vector bunle of rank k
More informationSOME 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 informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationChern forms and the Fredholm determinant
CHAPTER 10 Chern forms and the Fredholm determinant Lecture 10: 20 October, 2005 I showed in the lecture before last that the topological group G = G (Y ;E) for any compact manifold of positive dimension,
More informationAtiyah-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 informationEQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE FOUR: LOCALIZATION 1
EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE FOUR: LOCALIZATION 1 WILLIAM FULTON NOTES BY DAVE ANDERSON Recall that for G = GL n (C), H G Pn 1 = Λ G [ζ]/(ζ n + c 1 ζ n 1 + + c n ), 1 where ζ =
More informationU N I V E R S I T Y OF A A R H U S D E P A R T M E N T OF M A T H E M A T I C S ISSN: X FIBRE BUNDLES AND CHERN-WEIL THEORY.
U N I V E R S I T Y OF A A R H U S D E P A R T M E N T OF M A T H E M A T I C S ISSN: 0065-017X FIBRE BUNDLES AND CHERN-WEIL THEORY By Johan Dupont Lecture Notes Series No.: 69 August 2003 Ny Munkegade,
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationTransparent connections
The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection
More informationNONNEGATIVE 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 informationTHE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES
THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES Denis Bell 1 Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu This
More informationCHAPTER 8. Smoothing operators
CHAPTER 8 Smoothing operators Lecture 8: 13 October, 2005 Now I am heading towards the Atiyah-Singer index theorem. Most of the results proved in the process untimately reduce to properties of smoothing
More informationHARMONIC COHOMOLOGY OF SYMPLECTIC FIBER BUNDLES
HARMONIC COHOMOLOGY OF SYMPLECTIC FIBER BUNDLES OLIVER EBNER AND STEFAN HALLER Abstract. We show that every de Rham cohomology class on the total space of a symplectic fiber bundle with closed Lefschetz
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationNOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES
NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES SHILIN U ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of
More informationSurvey on exterior algebra and differential forms
Survey on exterior algebra and differential forms Daniel Grieser 16. Mai 2013 Inhaltsverzeichnis 1 Exterior algebra for a vector space 1 1.1 Alternating forms, wedge and interior product.....................
More informationLECTURE 10: THE PARALLEL TRANSPORT
LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationVector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle
Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,
More informationTHE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES
Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 -EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main
More informationCohomology jump loci of local systems
Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to
More informationTopics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map
Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map Most of the groups we will be considering this semester will be matrix groups, i.e. subgroups of G = Aut(V ), the group
More informationElliptic Curves Spring 2015 Lecture #7 02/26/2015
18.783 Elliptic Curves Spring 2015 Lecture #7 02/26/2015 7 Endomorphism rings 7.1 The n-torsion subgroup E[n] Now that we know the degree of the multiplication-by-n map, we can determine the structure
More informationGeneralized complex geometry and topological sigma-models
Generalized complex geometry and topological sigma-models Anton Kapustin California Institute of Technology Generalized complex geometry and topological sigma-models p. 1/3 Outline Review of N = 2 sigma-models
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationLECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory
More information1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces.
1. Tangent Vectors to R n ; Vector Fields and One Forms All the vector spaces in this note are all real vector spaces. The set of n-tuples of real numbers is denoted by R n. Suppose that a is a real number
More informationEQUIVARIANT 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 informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationAn introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109
An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups
More informationLecture 4: Harmonic forms
Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15
More informationA LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS
A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions
More informationLECTURE 2: SYMPLECTIC VECTOR BUNDLES
LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over
More informationIntegration and Manifolds
Integration and Manifolds Course No. 100 311 Fall 2007 Michael Stoll Contents 1. Manifolds 2 2. Differentiable Maps and Tangent Spaces 8 3. Vector Bundles and the Tangent Bundle 13 4. Orientation and Orientability
More informationLecture VI: Projective varieties
Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still
More informationDirac Operator. Texas A&M University College Station, Texas, USA. Paul Baum Penn State. March 31, 2014
Dirac Operator Paul Baum Penn State Texas A&M University College Station, Texas, USA March 31, 2014 Miniseries of five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. Beyond
More informationLecture 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 informationTopic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016
Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces
More informationTransverse Euler Classes of Foliations on Non-atomic Foliation Cycles Steven HURDER & Yoshihiko MITSUMATSU. 1 Introduction
1 Transverse Euler Classes of Foliations on Non-atomic Foliation Cycles Steven HURDER & Yoshihiko MITSUMATSU 1 Introduction The main purpose of this article is to give a geometric proof of the following
More informationIntersection of stable and unstable manifolds for invariant Morse functions
Intersection of stable and unstable manifolds for invariant Morse functions Hitoshi Yamanaka (Osaka City University) March 14, 2011 Hitoshi Yamanaka (Osaka City University) ()Intersection of stable and
More informationNOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY
NOTES ON DIFFERENTIAL FORMS. PART 5: DE RHAM COHOMOLOGY 1. Closed and exact forms Let X be a n-manifold (not necessarily oriented), and let α be a k-form on X. We say that α is closed if dα = 0 and say
More informationCup product and intersection
Cup product and intersection Michael Hutchings March 28, 2005 Abstract This is a handout for my algebraic topology course. The goal is to explain a geometric interpretation of the cup product. Namely,
More informationMath Topology II: Smooth Manifolds. Spring Homework 2 Solution Submit solutions to the following problems:
Math 132 - Topology II: Smooth Manifolds. Spring 2017. Homework 2 Solution Submit solutions to the following problems: 1. Let H = {a + bi + cj + dk (a, b, c, d) R 4 }, where i 2 = j 2 = k 2 = 1, ij = k,
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationLECTURE 1: LINEAR SYMPLECTIC GEOMETRY
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 *********************************************************************************
More informationk=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 informationRepresentations. 1 Basic definitions
Representations 1 Basic definitions If V is a k-vector space, we denote by Aut V the group of k-linear isomorphisms F : V V and by End V the k-vector space of k-linear maps F : V V. Thus, if V = k n, then
More informationMath 752 Week s 1 1
Math 752 Week 13 1 Homotopy Groups Definition 1. For n 0 and X a topological space with x 0 X, define π n (X) = {f : (I n, I n ) (X, x 0 )}/ where is the usual homotopy of maps. Then we have the following
More informationL 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 informationRIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan
RIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan On multiframings of 3 manifolds Tatsuro Shimizu 1
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationLecture 11: Hirzebruch s signature theorem
Lecture 11: Hirzebruch s signature theorem In this lecture we define the signature of a closed oriented n-manifold for n divisible by four. It is a bordism invariant Sign: Ω SO n Z. (Recall that we defined
More informationDIFFERENTIAL FORMS AND COHOMOLOGY
DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the
More informationIntroduction to the Baum-Connes conjecture
Introduction to the Baum-Connes conjecture Nigel Higson, John Roe PSU NCGOA07 Nigel Higson, John Roe (PSU) Introduction to the Baum-Connes conjecture NCGOA07 1 / 15 History of the BC conjecture Lecture
More informationSection 18 Rings and fields
Section 18 Rings and fields Instructor: Yifan Yang Spring 2007 Motivation Many sets in mathematics have two binary operations (and thus two algebraic structures) For example, the sets Z, Q, R, M n (R)
More informationLECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
More informationLECTURE 5: COMPLEX AND KÄHLER MANIFOLDS
LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationTHE QUANTUM CONNECTION
THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,
More information