Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
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Realizing Cohomology Classes As Euler Classes 1
Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 . -
Fermionic Finite-Group Gauge Theories and Interacting Symmetric
Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms Meng Guo1;2;3, Kantaro Ohmori4, Pavel Putrov4;5, Zheyan Wan6, Juven Wang4;7 1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA 2Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada 3Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada 4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 5 ICTP, Trieste 34151, Italy 6School of Mathematical Sciences, USTC, Hefei 230026, China 7Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA Abstract We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of 't Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). -
K-Theoryand Characteristic Classes
MATH 6530: K-THEORY AND CHARACTERISTIC CLASSES Taught by Inna Zakharevich Notes by David Mehrle [email protected] Cornell University Fall 2017 Last updated November 8, 2018. The latest version is online here. Contents 1 Vector bundles................................ 3 1.1 Grassmannians ............................ 8 1.2 Classification of Vector bundles.................... 11 2 Cohomology and Characteristic Classes................. 15 2.1 Cohomology of Grassmannians................... 18 2.2 Characteristic Classes......................... 22 2.3 Axioms for Stiefel-Whitney classes................. 26 2.4 Some computations.......................... 29 3 Cobordism.................................. 35 3.1 Stiefel-Whitney Numbers ...................... 35 3.2 Cobordism Groups.......................... 37 3.3 Geometry of Thom Spaces...................... 38 3.4 L-equivalence and Transversality.................. 42 3.5 Characteristic Numbers and Boundaries.............. 47 4 K-Theory................................... 49 4.1 Bott Periodicity ............................ 49 4.2 The K-theory spectrum........................ 56 4.3 Some properties of K-theory..................... 58 4.4 An example: K-theory of S2 ..................... 59 4.5 Power Operations............................ 61 4.6 When is the Hopf Invariant one?.................. 64 4.7 The Splitting Principle........................ 66 5 Where do we go from here?........................ 68 5.1 The J-homomorphism ........................ 70 5.2 The Chern Character and e invariant............... -
Hodge Decomposition
Hodge Decomposition Daniel Lowengrub April 27, 2014 1 Introduction All of the content in these notes in contained in the book Differential Analysis on Complex Man- ifolds by Raymond Wells. The primary objective here is to highlight the steps needed to prove the Hodge decomposition theorems for real and complex manifolds, in addition to providing intuition as to how everything fits together. 1.1 The Decomposition Theorem On a given complex manifold X, there are two natural cohomologies to consider. One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology which uses the complex structure. We’ll quickly go over the definitions of these cohomologies in order to set notation but for a precise discussion I recommend Huybrechts or Wells. If X is a manifold, we can define the de Rahm Complex to be the chain complex 0 E(Ω0)(X) −d E(Ω1)(X) −d ::: Where Ω = T∗ denotes the cotangent vector bundle and Ωk is the alternating product Ωk = ΛkΩ. In general, we’ll use the notation! E(E) to! denote the sheaf! of sections associated to a vector bundle E. The boundary operator is the usual differentiation operator. The de Rham cohomology of the manifold X is defined to be the cohomology of the de Rham chain complex: d Ker(E(Ωn)(X) − E(Ωn+1)(X)) n (X R) = HdR , d Im(E(Ωn-1)(X) − E(Ωn)(X)) ! We can also consider the de Rham cohomology with complex coefficients by tensoring the de Rham complex with C in order to obtain the de Rham complex! with complex coefficients 0 d 1 d 0 E(ΩC)(X) − E(ΩC)(X) − ::: where ΩC = Ω ⊗ C and the differential d is linearly extended. -
Arxiv:1910.04634V1 [Math.DG] 10 Oct 2019 ˆ That E Sdnt by Denote Us Let N a En H Subbundle the Define Can One of Points Bundle
SPIN FRAME TRANSFORMATIONS AND DIRAC EQUATIONS R.NORIS(1)(2), L.FATIBENE(2)(3) (1) DISAT, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy (2)INFN Sezione di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy (3) Dipartimento di Matematica – University of Torino, via Carlo Alberto 10, I-10123 Torino, Italy Abstract. We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer re- quirements, transformations allowed by spin frames are more general than usual spin transformations and they usually do not preserve the induced metric structures. We study how these new transformations affect connections both on the spin bundle and on the frame bundle and how this reflects on the Dirac equations. 1. Introduction Dirac equations provide an important tool to study the geometric structure of manifolds, as well as to model the behaviour of a class of physical particles, namely fermions, which includes electrons. The aim of this paper is to generalise a key item needed to formulate Dirac equations, the spin structures, in order to extend the range of allowed transformations. Let us start by first reviewing the usual approach to Dirac equations. Let (M,g) be an orientable pseudo-Riemannian manifold with signature η = (r, s), such that r + s = m = dim(M). R arXiv:1910.04634v1 [math.DG] 10 Oct 2019 Let us denote by L(M) the (general) frame bundle of M, which is a GL(m, )-principal fibre bundle. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
The Atiyah-Singer Index Theorem*
CHAPTER 5 The Atiyah-Singer Index Theorem* Peter B. Gilkey Mathematics Department, University of Oregon, Eugene, OR 97403, USA E-mail: gilkey@ math. uo regon, edu Contents 0. Introduction ................................................... 711 1. Clifford algebras and spin structures ..................................... 711 2. Spectral theory ................................................. 718 3. The classical elliptic complexes ........................................ 725 4. Characteristic classes of vector bundles .................................... 730 5. Characteristic classes of principal bundles .................................. 737 6. The index theorem ............................................... 739 References ..................................................... 745 *Research partially supported by the NSF (USA) and MPIM (Germany). HANDBOOK OF DIFFERENTIAL GEOMETRY, VOL. I Edited by EJ.E. Dillen and L.C.A. Verstraelen 2000 Elsevier Science B.V. All fights reserved 709 The Atiyah-Singer index theorem 711 O. Introduction Here is a brief outline to the paper. In Section 1, we review some basic facts concerning Clifford algebras and spin structures. In Section 2, we discuss the spectral theory of self- adjoint elliptic partial differential operators and give the Hodge decomposition theorem. In Section 3, we define the classical elliptic complexes: de Rham, signature, spin, spin c, Yang-Mills, and Dolbeault; these elliptic complexes are all of Dirac type. In Section 4, we define the various characteristic classes for vector bundles that we shall need: Chern forms, Pontrjagin forms, Chern character, Euler form, Hirzebruch L polynomial, A genus, and Todd polynomial. In Section 5, we discuss the characteristic classes for principal bundles. In Section 6, we give the Atiyah-Singer index theorem; the Chern-Gauss-Bonnet formula, the Hirzebruch signature formula, and the Riemann-Roch formula are special cases of the index theorem. We also discuss the equivariant index theorem and the index theorem for manifolds with boundary. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
Introduction
Introduction Spin Geometry is the hidden facet of Riemannian Geometry. It arises from the representation theory of the special orthogonal group SOn, more precisely, from the spinor representation, a certain representation of the Lie algebra son which is not a representation of SOn. Spinors can always be constructed locally on a given Riemannian manifold, but globally there are topological obstructions for their exis- tence. Spin Geometry lies therefore at the cross-road of several subelds of modern Mathematics. Algebra, Geometry, Topology, and Analysis are subtly interwoven in the theory of spinors, both in their denition and in their applications. Spinors have also greatly inuenced Theoretical Physics, which is nowadays one of the main driv- ing forces fostering their formidable development. The Noncommutative Geometry of Alain Connes has at its core the Dirac operator on spinors. The same Dirac operator is at the heart of the Atiyah–Singer index formula for elliptic operators on compact manifolds, linking in a spectacular way the topology of a manifold to the space of solutions to elliptic equations. Signicantly, the classical Riemann–Roch formula and its generalization by Hirzebruch; the Gauß–Bonnet formula and its extension by Chern; and nally Hirzebruch’s topological signature theorem, provide most of the examples in the index formula, but it was the Dirac operator acting on spinors which turned out to be the keystone of the index formula, both in its formulation and in its subsequent developments. The Dirac operator appears to be the primordial example of an elliptic operator, while spinors, although younger than differential forms or tensors, illustrate once again that aux âmes bien nées, la valeur n’attend point le nombre des années. -
An Introduction to Pseudo-Differential Operators
An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality). -
A Combinatorial Computation of the First Pontryagin Class of the Complex Projective Plane
LAZAR MILIN A COMBINATORIAL COMPUTATION OF THE FIRST PONTRYAGIN CLASS OF THE COMPLEX PROJECTIVE PLANE ABSTRACT. This paper carries out an explicit computation of the combinatorial formula of Gabrielov, Gel'fand, and Losik for the first Pontryagin class of the complex projective plane with the 9-vertex triangulation discovered by Wolfgang Kfihnel. The conditions of the original formula must be modified since the 8-vertex triangulation of the 3-sphere link of each vertex cannot be realized as the complex of faces of a convex polytope in 4-space, but it can be so realized by a star-shaped polytope, and the space of all such realizations is not connected. 0. INTRODUCTION In a series of papers [24]-[26] in the 1940s, L. S. Pontryagin defined and studied 'characteristic cycles' on smooth manifolds. In the introduction to the first paper of the series, he stressed the importance of having 'a definition of characteristic cycles which would be applicable to combinatorial mani- folds' since it would provide an algorithm for their calculation from 'the combinatorial structure of the manifolds'. In the late 1950s, Rdne Thom [33], and independently Rohlin and Sarc [28], proved that the rational Pontryagin classes are combinatorial invariants, and in the 1960s. S. P. Novikov [23] proved that these classes are topological invariants. No combinatorial formula for Pontryagin classes was proved until the mid-1970s, when Gabrielov et al. [6], [7] established a formula for the first Pontryagin class pl(X) of a combinatorial manifold X. The formula ex- presses Pl(X) in terms of the simplicial structure of X and some additional structure imposed on X, so in this sense the formula is not purely combinatorial. -
Vector Bundles. Characteristic Classes. Cobordism. Applications
Math 754 Chapter IV: Vector Bundles. Characteristic classes. Cobordism. Applications Laurenţiu Maxim Department of Mathematics University of Wisconsin [email protected] May 3, 2018 Contents 1 Chern classes of complex vector bundles 2 2 Chern classes of complex vector bundles 2 3 Stiefel-Whitney classes of real vector bundles 5 4 Stiefel-Whitney classes of manifolds and applications 5 4.1 The embedding problem . .6 4.2 Boundary Problem. .9 5 Pontrjagin classes 11 5.1 Applications to the embedding problem . 14 6 Oriented cobordism and Pontrjagin numbers 15 7 Signature as an oriented cobordism invariant 17 8 Exotic 7-spheres 19 9 Exercises 20 1 1 Chern classes of complex vector bundles 2 Chern classes of complex vector bundles We begin with the following Proposition 2.1. ∗ ∼ H (BU(n); Z) = Z [c1; ··· ; cn] ; with deg ci = 2i ∗ Proof. Recall that H (U(n); Z) is a free Z-algebra on odd degree generators x1; ··· ; x2n−1, with deg(xi) = i, i.e., ∗ ∼ H (U(n); Z) = ΛZ[x1; ··· ; x2n−1]: Then using the Leray-Serre spectral sequence for the universal U(n)-bundle, and using the fact that EU(n) is contractible, yields the desired result. Alternatively, the functoriality of the universal bundle construction yields that for any subgroup H < G of a topological group G, there is a fibration G=H ,! BH ! BG. In our A 0 case, consider U(n − 1) as a subgroup of U(n) via the identification A 7! . Hence, 0 1 there exists fibration U(n)=U(n − 1) ∼= S2n−1 ,! BU(n − 1) ! BU(n): Then the Leray-Serre spectral sequence and induction on n gives the desired result, where 1 ∗ 1 ∼ we use the fact that BU(1) ' CP and H (CP ; Z) = Z[c] with deg c = 2.