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The Atiyah-Singer Index Formula Part 3 The Atiyah-Singer Index Formula “Inperhapsmostcaseswhenwefailto findtheanswertoaquestion,thefailure iscausedbyunsolvedorinsufficiently solvedsimplerandeasierproblems.Thus alldependsonfindingtheeasierproblem andsolvingitwithtoolsthatareasperfect aspossibleandwithnotionsthatarecapable ofgeneralization.” (D.Hilbert,1900) CHAPTER 14 Introduction to Algebraic Topology (K-Theory) It is the goal of this part to develop a larger portion of algebraic topology by means of a theorem of Raoul Bott concerning the topology of the general linear group GL(N,C) on the basis of linear algebra, rather than the theory of “simplicial complexes” and their “homology” and “cohomology”. There are several reasons for doing so. First of all, is of course a matter of taste and familiarity as to which ap- proach “codifying qualitative information in algebraic form” (Atiyah) one prefers. In addition, there are objective criteria such as simplicity, accessibility and trans- parency, which speak for this path to algebraic topology. Finally, it turns out that this part of topology is most relevant for the investigation of the index problem. Before developing the necessary machinery, it seems advisable to explain some basic facts on winding numbers and the topology of the general linear group GL(N,C). Note that the group GL(N,C) moved to fore in Part 2 already in connection with the symbol of an elliptic operator, and that the group Z of integers was in a certain sense the topic of Part 1, the Fredholm theory. In the following, Part 3, the concern is (roughly) the deeper connection between the previous parts. Thereby we will be guided by the search for the “correct” and “promising” generalizations of the theo- rem of Israil Gohberg and Mark Krein on the index of Wiener-Hopf operators. See Chapter 5, Theorem 5.4 (p.87), Exercise 5.9 (p.90), and Theorem 5.11 (p.91). 1. Winding Numbers “How can numerical invariants be extracted from the raw material of geometry and analysis?” (Hirsch). A good example is the concept of winding number, surely the best known item of algebraic topology: In his studies of celestial mechanics the French physicist and mathematician Henri Poincaré turned to stability questions of planetary orbits. Many of the related problems are not completely solved even today (e.g., the “three body problem” of describing all possible motions of three points which are interact via gravitation. However, this problem is solved for practical purposes, as is shown by the successful landing of the lunar module Luna 1.) As a tool for the qualitative investigation of nonlinear (ordinary) differential equations, Poincaré introduced in 1881 the notion of the “index” I(P0) of a “singular point” P0 for a system of two ordinary differential equations x˙ = F (x, y), y˙ = G(x, y). To do this, surround P0 by a closed curve C in the phase portrait (see the following examples) and measure on it the angle of the rotation 191 192 14. INTRODUCTION TO ALGEBRAIC TOPOLOGY (K-THEORY) performed by the vector field (F (x, y),G(x, y)) when (x, y) traverses C once counterclockwise. The angle is an integral multiple of 2π, and this integer is I(P0). In case of a magnetic fieldonecanactuallyseeI(P0) in the rotation of the needle, when a compass is moved along C. Among other things, one has the theorem (see [AM, p.75-76]): If the equilibrium position P0 is stable, then I (P0)=1. We will come back to this in Section 17.5. Poincaré returned to this topological argument in 1895, when he considered allclosedcurvesinanarbitrary“space”andclassified them according to their deformation properties.1 His simplest result can be expressed in today’s terminology (we follow [Ati67b, p.237-241]) follows: 1 Theorem 14.1. Let f : S C× := C 0 be a continuous mapping of the → − { } circle X to the punctured plane of non-zero complex numbers C×.Inotherwords, we have a closed path in the plane not passing through the origin. The following hold: (i) f possesses a “winding number” which states how many times path rounds the origin; we write W(f,0) or deg(f). (ii) This degree is invariant under continuous deformations. (iii) deg(f) is the only such invariant, i.e., f can be deformed to g,ifandonlyif deg(f)=deg(g). (iv) For each integer m, there is a mapping f with deg(f)=m. Arguments: Instead of a formal proof, we briefly assemble the different ways of defining or computing deg(f). 1The “fundamental group” was introduced in Poincaré’s work Analysis Situs (Oeuvres 6, 193-288), whose theme is purely topological-geometric-algebraic: an “analysis situs in more than three dimensions.” Poincaré expected the abstract formalism to “do in certain cases the service usually expected of the figures of geometry.” He mentioned three areas of application: In addition to the Riemann-Picard problem of classifying algebraic curves and the Klein-Jordan problem of determining all subgroups of finite order in an arbitrary continuous group, he particularly stressed its relevance for analysis and physics: “one easily recognizes that the generalized analysis situs would allow treating the equations of higher order and specifically those of celestial mechanics (the same way as H. P. had done it before with simpler types of differential equations; B.B.). I also believe that I did not produce a useless work, when I wrote this treatise.” The complexity and limited understanding of the topological problems did not however permit Poincare to carry out his program completely: “Each time I tried to limit myself I slipped into darkness.” 1. WINDING NUMBERS 193 Geometrically: Replace f by g := f/ f . This is a mapping from S1 to S1. Approximate g by a differentiable map h, and| | count (algebraically, i.e. with a sign convention according to the derivative of the number of points in the preimage of a point which is in general position. This method can also be characterized as “counting of the intersection numbers”: Draw an arbitrary ray emanating from the origin which does not pass through a self-intersection point of the path. Now count the intersections of the path with the ray according to the “trafficoftherightof way” (H. Weyl) - thus with a plus sign if the path has the right of way, and a minus sign when the ray has the right of way. Combinatorially: We approximate with a piecewise linear path g,andthen use combinatorial methods; i.e., we permit deleting and adding of those edges of our polygonal path which are boundaries of 2-simplices (these are triangles whose interior is completely contained in C×, and thus do not contain the origin. 194 14. INTRODUCTION TO ALGEBRAIC TOPOLOGY (K-THEORY) Differential: We approximate by a differentiable g,andthensetdeg (f):= 1 dg . Here, we have regarded g as a map [0, 2π] C×, and then the integral 2πi g → 2π g0(τ) is deRfined as 0 g(τ) dτ. From the Cauchy Integral Formula, it follows that the integral is a multiple of 2πi, and hence deg (f) is an integer. R Algebraic: Approximate by a finite Fourier series k iνφ g (φ)= aν e ν= k X− 1 1 We regard g as a map S C× with S = z C : z =1 . Consider now the extension of g to the disk z→< 1, where it is a fi{nite∈ Laurent| | series.} We then obtain a meromorphic function h|,andset| deg (f):=N(h) P (h) where N(h) and P (h) denote the number of zeros and poles of h in z < 1.− | | Function Analytic:Setdeg (f):= index Tf ,whereTf is the Wiener-Hopf − 1 2 1 operator, assigned to f, on the space H0(S ) L (S ) spanned by the functions 0 1 2 1 ⊂ z ,z ,z ,.... Tf is defined (for u H0(S ))by ∈ ∞ fˆ(n k)ˆu (k) for n 0 T ˆu (n):= k=0 f 0− for n<≥ 0, ½ ¡ ¢ P where fˆ(m):= f,zm is the m-th Fourier coefficient of f. For the details of this, see Theoremh 5.4 abovei and Theorem 5.11 for the analogous representation deg (f) = index(I + Kφ) via the (continuous) Wiener-Hopf operator ∞ + 2 + (Kφu)(x):= φ(x y)u(y) dy; x R ,u L (R ), − ∈ ∈ Z0 1 ˆ 1 t i where φ L (R) with φ = f κ− and κt := t−+i is the Cayley transformation. ∈ ◦ 1 As a first example, one may recall the standard map a : S C× which is given → by a(z)=z, z C, z =1. Here the equivalence of the various definitionsisclear. We omit the proof∈ for| | complicated cases and refer to [Ah, p.151]. Compare also Theorem 14.2 (p. 196) and Theorem 14.17 (p. 208) below and in another context [Hir, p.120-131] or [BJ,p.161f]. In Theorem 14.1 all closed curves in the punctured plane are compared and the “essentially different” ones separated. The different definitions of winding number listed there reflect the main branches of topology with their different techniques, goals and connections. Accordingly, depending on the point of view chosen, very many generalizations of Theorem 14.1 to higher dimensions are possible (see also Exercise 14.22, p. 210): 1. WINDING NUMBERS 195 If one sticks with the classification of systems of ordinary differential equations (which was the point of departure for Poincaré’s topological papers) one would first try to distinguish the different possibilities of bending the real line into a closed curve in space or other higher dimensional spaces. In this fashion H. Poincaré (but also see the footnote above) conceived (among other things) the “fundamental group” π1(X, x0) of a space X which arises from the homotopic classification of 1 closed paths S X which pass through the point x0 X.
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