Introduction to Index Theory Elmar Schrohe Institut für Analysis
Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a linear operator between two Banach spaces E and F. It is desirable that A be invertible, but this is not always possible. The next best thing: Definition A L(E, F ) is a Fredholm operator, if dim ker A < and codim im A <. In that case, let ind A = dim ker A codim im A.
Basics Some Background (i) Erik Ivar Fredholm (1866-1927) Theory of Integral Equations (ii) Fredholm Alternative for BVP: Essentially says that certain operators are Fredholm operators of index zero. Only interesting, if E, F are infinite-dimensional! ind A = dim ker A codim im A = dim E dim F, is independent of A, if both dimensions are finite use that dim ker A + dim im A = dim E.
Basics Obvious A invertible ind A = 0. The converse is false: Any A L(C n ) has index zero. Why this Strange Definition? While the dimensions of kernel and cokernel (codim im A) are very unstable quantities, the index has many good properties. Here is the first: Theorem: Composition A L(E, F ), B L(D, E) Fredholm AB Fredholm, and ind AB = ind A + ind B.
Basics Theorem: Stability Let A L(E, F ) be Fredholm. (a) A + S is Fredholm, if S L(E,F ) is small; ind (A + S) = ind A (b) A + K is Fredholm, for compact K; ind (A + K) = ind A. Corollary (a) ind (I + K) = 0 for compact K. (b) Homotopies through Fredholm operators preserve the index Theorem: Equivalent Characterization A is Fredholm B L(F, E) s.t. BA I and AB I are of finite rank B L(F, E) s.t. BA I and AB I are compact. i.e. A Fredholm A invertible modulo compacts.
Example 1: Abstract Fredholm Operators Consider the Hilbert space H = l 2 (N) = {x = (x 1, x 2,...) : x j C, x j 2 < } and the operators (left and right shift operators) S l (x 1, x 2,...) = (x 2, x 3,...) S r (x 1, x 2,...) = (0, x 1, x 2,...). Lemma Both are Fredholm with ind S l = 1 and ind S r = 1. Corollary Fredholm operators of all indices exist.
Example 2: Toeplitz Operators We consider the space L 2 (S 1 ) of L 2 -functions on S 1 C. Each such function has an expansion into a Fourier series u(z) = j Z a j z j with a j C and a j 2 <. Indeed, u 2 L 2 = a j 2. The space L 2 (S 1 ) has a closed subspace, namely the Hardy space H = {u L 2 : a j = 0 for j < 0} i.e. those u L 2 that extend holomorphically to the disk B(0, 1). According to basic Hiilbert space theory, there exists an orthogonal projection P : L 2 (S 1 ) H.
Example 2: Toeplitz Operators For f C(S 1 ) define the Toeplitz operator T f with symbol f by Theorem T f u = P(fu), u H. (i) (ii) T f +g = T f + T g T fg T f T g is compact (iii) T f = f sup. Theorem T f is a Fredholm operator, if and only if f (z) 0 z S 1. Then ind T f = wind(f ), the negative winding number of f. Note: T z is the right shift operator on H: 0 a jz j 0 a jz j+1
Example 3: The Gauß-Bonnet Theorem Let M be a closed surface. We consider the exterior derivative 0 Ω 0 (M) d 0 Ω 1 (M) d 1 Ω 2 (M) 0, where Ω j denotes the smooth j-forms. This is not a single operator, but a complex. One can define the index here as the sum ind d = dim(ker d 0 / im 0) dim(ker d 1 / im d 0 ) + dim(ker 0/ im d 1 ). This is just the alternating sum of the cohomology classes: ind d = dim H 0 dim H 1 + dim H 2 = χ(m), the Euler characteristic of M. Theorem (Gauß-Bonnet) χ(m) = ind d = 1 K dx 2π M where K is the curvature of a Riemannian metric on M.
Example 4: Differential Operators M closed manifold. A differential operator P of order m defines continuous maps P : C (M) C (M) and H m (M) L 2 (M) where H m is the Sobolev space of order m. In local coordinates P = p α (x)dx α α m Definition The principal symbol of P is the function σ P (x, ξ) = p α (x)ξ α α =m Although this seems to depend on coordinates, it is invariantly defined on the cotangent bundle T M.
Example 4: Differential Operators Observation Two differential operators of order m with the same principal symbol differ by an operator of order m 1, i.e. an operator which is compact as an operator H m (M) L 2 (M) (Rellich s Theorem). Definition P is elliptic, if the principal symbol is invertible on T M \ 0. By homogeneity: Suffices to require invertibilty on S M. Theorem P : H m (M) L 2 (M) Fredholm P is elliptic. Corollary The index of an elliptic differential operator depends only on the principal symbol.
Example 4: Differential Operators Observation Let P 1 and P 2 be two differential operators whose principal symbols are homotopic. Then ind P 1 = ind P 2. Vector Bundles In general, elliptic operators will not act on scalar functions, but on sections of vector bundles (e.g. differential forms, spinors, etc.): P : H m (M, E 1 ) L 2 (M, E 2 ). The principal symbol then is a homomorphism the pull-back of E 1/2 to T M. σ P : π E 1 π E 2,
Example 4: Differential Operators There is one more invariance of the index Stability Let F be another vector bundle over M. Instead of P consider ( ) P 0 P = : H m (M, E 0 Λ 1 F ) L 2 (M, E 2 F ), where Λ : H m (M, F ) L 2 (M, F ) is a fixed isomorphism with symbol I F. Then ind P = ind P. Combining this with the observation on homotopy invariance: Corollary The index only depends on the stable homotopy class of the principal symbol. So it must be possible to compute it from that.
K-theory Atiyah and Singer solved the index problem using K-theory. Short Introduction to K-theory Let X be a compact manifold (space), V a vector bundle over it. By [V ] denote the isomorphism class of V, i.e. all vector bundles isomorphic to V. There is an addition on these objects by [V ] + [W ] = [V W ]. Grothendieck s group construction: We consider formal differences [V ] [W ] and identify [V ] [W ] and [V ] [W ] if [V ] + [W ] = [V ] + [W ]. Over compact spaces, K-theory classes are simply formal differences of (isomorphism classes of) vector bundles. Write K(X ) for the K-classes over X.
K-theory Noncompact Manifolds Does not work for noncompact spaces X. Instead define: K-class = triple (V, W, φ), V and W are vector bundles over X, φ : V W is a map which is an isomorphism outside a cpt set. Might only be defined outside compact set. Write K c (X ). In Our Case The symbol σ P of P : H m (M, E 1 ) L 2 (M, E 2 ) defines the class [σ P ] = (π E 1, π E 2, σ P ) K c (T M). Question What have we gained??
The Topological Index Map Answer There is a map χ : K c (T M) Z, the topological index map. So we have two ways of associating an integer to P: The Fredholm index of P The topological index of the symbol: χ([σ P ]). Theorem (Atiyah and Singer) Both maps coincide: ind P = χ([σ P ]).
The Topological Index Map Definition of the Topological Index Map Embed M into R N for some large N Induces an embedding of T M into R 2n Thom homomorphism induces map K c (T M) K c (R 2N ) Bott periodicity induces a map K c (R 2N ) K(pt) = Z. The topological index map is the composition of these.
C*-algebras Definition A C*-algebra is a Banach algebra A with a sesquilinear involution, such that x x = x 2, x A. It is called unital, if it has a unit. Example Any closed symmetric subalgebra of L(H), H a Hilbert space. C 0 (X ), X locally compact Hausdorff space. Unital, iff X cpt. Theorem (Gelfand-Neimark-Segal) Every C*-algebra is isomorphic to a closed subalgebra of L(H). Every commutative C*-algebra is isomorphic to C 0 (X ), X locally compact Hausdorff.
K-theory for C*-algebras Definition A projection is a p A such that p = p 2 = p. A partial isometry is a v A such that v v is a projection. For projections in a unital A, one has three notions of equivalence. equivalent: p q, if there exists a partial isometry v such that vv = p and v v = q unitarily equivalent: p u q, if there exists a unitary u in A such that p = u qu homotopic: p h q, provided there is a norm continuous path of projections π(t), 0 t 1 such that π(0) = p and π(1) = q. Lemma: p h q p u p p q
K-theory for C*-algebras Observation If A is a (unital) C*-algebra, then so is M n (A), the algebra of n n matrices with entries in A. Moreover, we have an embedding M n (A) M n+1 (A) by a 11... a 1n 0 a 11... a 1n.. a n1... a nn.. a n1... a nn 0 0... 0 0.. By M (A) we denote the inductive limit with respect to this identification. Lemma In M (A), all three notions of equivalence coincide.
K-theory for C*-algebras Let A be a unital C*-algebra. Observation: Addition on equivalence classes of projections [p] + [q] = [p q]. Definition We denote by K 0 (A) the set of all formal differences This is an abelian group. [p] [q], p, q projections in A. Relation to classical K-theory (Swan s Theorem) If X is compact and p Mat n (C(X )) is a projection, then the ranges of p(x), x X, define a vector bundle over X. All vector bundles are obtained this way K(X ) = K 0 (C(X )).
K-theory for C*-algebras Let A be a unital C*-algebra. Observation We may consider Gl n (A) as a subset ( of ) Gl n+m (A), m N, by x 0 identifying x with the element. By Gl 0 1 (A) we denote m the inductive limit with respect to these embeddings. Definition K 1 (A) = Gl (A)/ Gl (A) 0, where Gl (A) 0 denotes the connected component of the identity in Gl (A + ). K 1 (A) becomes an abelian group with the multiplication [( )] x 0 [x][y] = [xy] =. 0 y
K-theory for C*-algebras The Main Tool to Compute K-theory Let 0 A B C 0 be a short exact sequence of C*-algebras. Theorem: Six Term Exact Sequence There exist maps ind and exp such that K 0 (A) ind α K0 (B) β K0 (C) exp (1) K 1 (C) β K1 (B) α K1 (A) is an exact sequence of abelian groups. In general the maps exp and ind are hard to determine.