This class will be about a single result:

Theorem (Atiyah-Singer). If X is a closed smooth manifold and D is an elliptic operator, then the analytic index of D is equal to the topological index of D.

Plan for the course.

(1) Define the analytic and the topological indices.

(2) Special cases: Gauß-Bonnet, Riemann-Roch, Hirzebruch signature theorem, Dirac operator.

(3) Applications to geometry and number theorem

(4) Bordism proof, K-theory proof, heat equation proof (supersymmetry).

(5) Generalizations: Index theorem for manifolds with boundary, G-equivariant index theorem (equality of virtual representations of G), B-family index theorem (equality of classes in K(B)), equivariant family version, local index theorem.

Today we start with (1) and (2).

Suppose E and F are smooth vector bundles (complex) and D: C^{∞}(E) → C^{∞}(F) is a complex-linear elliptic operator. Since D is elliptic, it is Fredholm, which means that its kernel and cokernel are finitedimensional. The analytic index is the difference of their dimensions: a-ind(D) := dim ker(D)−dim coker(D).

Now ch(D) ∈ H^{∗}(X) is the result of the Thom isomorphism applied to the Chern character of the difference bundle associated to the symbol of D and Td(X) ∈ H^{4∗}(X) is the Todd class of T X ⊗ C. Using the fact that ch(D) and [X] are twisted by the same character, the orientation character of X, one defines the topological index as t-ind(D) := ⟨Td(X) ∪ ch(D), [X]⟩.

In most cases we use a Riemannian metric g on X and hermitian metrics on E and F to define D and then ch(D) and Td(X) are represented by closed differential forms c̃h(D) and T̃d(X). Then the topological index of D is ∫_{X} c̃h(D) ∧ T̃d(X) ∈ R. Thus the topological index is an integral of a local expression in X, whereas the analytic index depends on the global structure of X.

Example: Gauß-Bonnet on a Riemann surface (X, g). The topological index of a certain elliptic operator D is (2π)^{−1} ∫_{X} scalar curvature(g). The analytic index is the Euler characteristic of X, which was originally defined in terms of a triangulation of X. The Gauß-Bonnet theorem implies that this integer depends neither on the metric nor on the triangulation!

Project 1. Dig into the history of the index theorem.

Definition. D: C^{∞}(E) → C^{∞}(F) is a differential operator if it is locally given by P_{i,j} ∈ C^{∞}(U)[∂_{1}, …, ∂_{n}]. Here we identify C^{∞}(E|_{U}) = C^{∞}(U, R^{p}) and likewise for F. The order of D is the maximum degree of P_{i,j}. The individual components of the symbol Q_{i,j} are obtained from P_{i,j} by substituting iξ_{k} for ∂_{k} and taking the top order degree. Observation: Q_{i,j} fit together on X to give the symbol σ(D): π^{∗}E → π^{∗}F as a homomorphism of vector bundles on the total space of the cotangent bundle T^{∗}X. Here π: T^{∗}X → X and σ_{x}(D)(ξ) = (Q_{i,j}(ξ_{1}, …, ξ_{n})). The symbol depends polynomially on ξ.

Definition. D is elliptic if σ(D)(ξ) is invertible for all ξ ≠ 0.

Example. X = pt. D: E → F is a morphism of finite-dimensional vector spaces. The analytical index is equal to dim ker D − dim cokerD. The topological index is ⟨ch(D) ∪ Td(X), [X]⟩ = ch^{0}(D) = dim E − dim F. Hence the index of a morphism of vector spaces is invariant under deformations and this will stay true for arbitrary elliptic operators.

Example. X = S^{1}, E = F = X × C, D = −i∂. We have σ = ξ. There is a basis of eigenvectors: −i∂(exp(inx)) = n exp(inx). We have dim ker D = dim cokerD = 1, hence ind(D) = 0. We can compute the index more easily by a deformation: ind(D) = ind(D + λ) = 0, because D + λ is invertible for non-integer λ.

Example. (Elaboration of the previous example.) X = S^{1} = R/Z, E = F = C, D = −i∂: C^{∞}(S^{1}) → C^{∞}(S^{1}). Consider L^{2}(S^{1}): (f, g) = ∫_{S^{1}} fḡdx. Orthonormal basis: exp(inx) for n ∈ Z. There is an isomorphism L^{2}(S^{1}) → l^{2}(Z). The inner product on l^{2}(Z) is given by (a, b) = ∑_{n∈Z} a_{n}b̄_{n}. The isomorphism is given by the formulas a ↦ ∑_{n∈Z} a_{n} exp(inx) and f ↦ (n ↦ (f, exp(inz)) = ∫_{S^{1}} f exp(−inx)dx.

The space L^{2}(S^{1}) admits a filtration L^{2} ⊃ C^{0} ⊃ C^{1} ⊃ ⋯ C^{∞} = ∩_{k}C^{k} ⊃ Pol. Likewise l^{2}(Z) admits a (Sobolev) filtration l^{2}(Z) = W^{0} ⊃ W^{1} ⊃ W^{2} ⊃ ⋯ W^{∞} = ∩_{k}W^{k} ⊃ C[Z]. Here W^{k} is the completion of C[Z] with respect to (a, b)_{k} = ∑_{n∈Z} a_{n}b̄_{n}(1+n^{2})^{k}. Thus a ∈ l^{2} belongs to W^{k} if and only if (n ↦ a_{n}n^{r}) ∈ l^{2} for all 0 ≤ r ≤ k if and only if ∥a∥_{k}^{2} < ∞.

Lemma 1. C^{k} ⊂ W^{k}.

Proof. Suppose f ∈ C^{k}, then for any 0 ≤ r ≤ k we have D^{r}f ∈ L^{2}, hence n^{r}f̂ ∈ l^{2}, i.e., f̂ ∈ W^{k}.

Lemma 2. W^{k+1} ⊂ C^{k}.

Proof. k = 0: f ∈ W^{1}; f̂ = a. We have ∑_{n∈Z} |a_{n}|^{2}(1 + n^{2}) < ∞. Cauchy’s inequality: (∑_{n∈Z} |a_{n}|)^{2} ≤ (∑_{n} |a_{n}|^{2}(1 + n^{2})) (∑_{n}(1 + n^{2})^{−1}) < ∞. Now ∥f∥_{∞} = sup_{x∈S^{1}} |f(x)| ≤ ∑_{n} |a_{n}| < ∞, hence f is continuous.

Now let’s do the example more carefully. D sends C^{k} to C^{k−1}. Moreover, it sends W^{k} to W^{k−1}. Elliptic regularity: The operator D: W^{k} → W^{k−1} is Fredholm and the kernel and the cokernel of D: W^{k} → W^{k−1} do not depend on k and equal their partners in C^{∞}.

Remark. W^{k} is the completion of C^{k} with respect to the norm f ↦ ∑_{|r|≤k} ∥∂^{r}f∥.

Remark. C^{k} ⊂ W^{k} for any smooth manifold M. Moreover, W^{k+⌊d/2⌋} ⊂ C^{k}, where d = dim M.

One way to define Sobolev spaces on a compact manifold M is to choose a partition of unity ψ indexed by I together with embeddings ϕ_{i}: U_{i} → T^{d}. and complete C^{k} in the norm ∥f∥_{k} := ∑_{i∈I} ∥(ψ_{i}f) ∘ ϕ_{i}^{−1}∥. (We define Sobolev spaces on a torus T^{r} via its Pontrjagin dual Z^{r} as for r = 1.) The chain rule implies that the Sobolev norms obtained from different choices are equivalent, see homework.

Remark. Given a metric on M, the Laplacian picks out a canonical inner product on W^{k}.

Example. X = S^{1}, E = F = S^{1} × C, D: C^{∞}(S^{1}) → C^{∞}(S^{1}), D = ∑_{0≤k≤r} f_{k}(x)(−i∂)^{k} (arbitrary order r differential operator on the trivial bundle).

D is elliptic if σ(D)(x, ξ) = f_{r}(x)ξ^{r} is invertible for ξ ≠ 0, in other words f_{r}(x) ≠ 0 for all x ∈ S^{1}.

Example. Constant coefficient operators D = ∑_{0≤k≤r} f_{k}(−i∂)^{k}. D extends to W^{r}(S^{1}) → L^{2}(S^{1}) and then to l^{2}(Z) → l^{2}(Z). The last operator has the form a ↦ a · p (pointwise multiplication), where p(n) = ∑_{0≤k≤r} f_{k}n^{k}. Hence dim ker(D) is the number of integers n such that p(n) = 0. By the index theorem this is also the dimension of the cokernel of D. (We will later prove the lemma that if X is odd-dimensional, then the topological index of D is zero.)

K(X) := KU(X) is the group completion of the commutative monoid of isomorphism classes of finitedimensional complex vector bundles over X with the direct sum as the monoid structure. Thus K(X) consists of formal differences of vector bundles module certain equivalence relation: [E_{0}]−[E_{1}] = [F_{0}]−[F_{1}] if and only if there is G such that E_{0} ⊕ F_{1} ⊕ G = E_{1} ⊕ F_{0} ⊕ G. Remark: The universal property of the group completion allows us to define pullbacks in K-theory: If we have a map f: X → Y, then the pullback f^{∗}: K(Y) → K(X) is given by extending pullbacks of vector bundles to their group completions. Homotopic maps induce identical maps on K-theory. Example: K(pt) = Z. Also K(S^{1}) = Z because every complex bundle on S^{1} is trivial. Likewise n-dimensional vector bundles on S^{k} are in bijection with π_{k−1}(GL_{n}(C)). We don’t know how to compute these groups, however, for K-theory we only need to compute π_{k−1}(GL_{∞}(C)). Bott periodicity states that these groups alternate between 0 and Z for k odd respectively even. Thus K(S^{k}) = Z for k odd and K(S^{k}) = Z ⊕ Z for k even.

Definition. The reduced K-theory of X is K̃(X) := coker(K(pt) → K(X)).

Lemma 1: K̃(X)⊕Z = K(X). Lemma 2: K̃(X) can be obtained by identifying E and F if E⊕m = F ⊕n for some trivial bundles m and n.

For a closed subset Y ⊂ X of a compact Hausdorff space, we define K(X, Y) as follows: Representatives are triples (E_{0}, E_{1}, α), where E_{i} → X and α: E_{0}/Y → E_{1}/Y is an isomorphism. The equivalence relation is the same as before: (E_{0}, E_{1}, α) ∼ (F_{0}, F_{1}, β) if there is (G, G, γ) such that E_{0} ⊕ F_{1} ⊕ G = E_{1} ⊕ F_{0} ⊕ G. Note: K(X, ∅) = K(X). The sequence K(X, Y) → K(X) → K(Y) is exact.

Definition. If Z is a locally compact, then K_{cs}(Z) := K(Z ∪ ∞, ∞). (Remark: If Z is Hausdorff, then Z is locally compact if and only if Z ∪ ∞ is Hausdorff.)

Classes in K_{cs}(Z) are represented by (E_{0}, E_{1}, ϕ), where ϕ: E_{0}/(Z \K) → E_{1}/(Z \K). Note: (Z ∪∞)\K are the open neighborhoods of ∞ in Z ∪ ∞.

Remark. Existence of partitions of unity implies that any vector bundle G embeds into a trivial bundle, moreover one can find a bundle H such that G ⊕ H is trivial using a hermitian metric (in this case, on the trivial bundle). Any bundle that pulls back from the universal bundle on the (infinite) Graßmannian BU(n) will come with an embedding into the trivial bundle, since the universal bundle does. So K(X) can only coincide with the homotopy theoretic definition [X, BU × Z] if we somehow require this property. For simplicity, we will work with compact Hausdorff spaces X in the following.

Serre-Swan theorem: The category of vector bundles over X is equivalent to the category of finitely generated projective C^{0}(X)-modules.

Definition. If D: C^{∞}(E) → C^{∞}(F) is an elliptic differential operator, then [σ(D)] ∈ K_{cs}(T^{∗}X) is represented by the triple (π^{∗}E, π^{∗}F, σ(D)) (the compact set off of which σ(D) is invertible is the image of the zero section, which is homeomorphic to X) Here π: T^{∗}X → X is the canonical projection.

To obtain the (cohomological description of the) topological index from the symbol, we apply the Chern character to [σ(D)], then we apply the (twisted) Thom isomorphism, then we multiply the result by Td(X) and evaluate it on the fundamental class.

There is also a K-theoretic description of the topological index of D which goes as follows: Pick an embedding f: X → R^{n}, then the composition of g: T^{∗}X ≅ T X and T f: T X → T R^{n} is proper. Moreover, there is always a complex structure on the normal bundle of this map, even if X is not oriented. For proper maps with complex normal bundle we will define a pushforward in K-theory: [σ(D)] ∈ T f_{!}: K_{cs}(T^{∗}X) → K_{cs}(R^{2n}) = K̃(S^{2n}) → H̃^{2∗}(S^{2n}, Q) = Q. By Bott periodicity, we can also identify K̃(S^{2n}) = Z directly and this gives us a direct proof of the integrality of the topological index.

Books on K-theory: Atiyah, Karoubi, Rosenberg, …

Fredholm operators on Fréchet spaces. A Fréchet space is a metrizable complete locally convex Hausdorff topological vector space. Alternatively, it is a complete topological vector space whose topology is defined by an increasing sequence of seminorms. Examples: Banach spaces, C^{∞}(X) for compact manifolds X (take Sobolev norms), C^{0}(X) if X is σ-compact (norms are suprema on compact subsets), C^{∞}(X) for non-compact manifolds X (norms are Sobolev norms on compact submanifolds).

Fact. The dual space V′ has many topologies. One of them (the strong topology) agrees with the norm topology if V is Banach. But V′ in this topology is Fréchet if and only if V is Banach. So we will be very careful when working with topologies on mapping spaces of Fréchet spaces.

The open mapping theorem. If V and W are Fréchet spaces and a continuous linear map A: V → W is surjective, then A is open.

Definition. An operator A: V → W is Fredholm if its kernel and cokernel are finite-dimensional.

Lemma. The image of a Fredholm operator is closed.

Proof. We can assume that A: V → W is injective. Pick a complement Z for the image of A in W. Since A is Fredholm, Z is finite-dimensional and Hausdorff, in particular it is Fréchet. The map A⊕id_{Z}: V ⊕Z → W is continuous, linear, bijective, and therefore a homeomorphism (by the open mapping theorem). Since V is closed in V ⊕ Z, its image is closed in W.

Lemma. A: V → W is Fredholm if and only if it is invertible up to finite-rank operators if and only if it is invertible up to compact operators.

Proof. The first equivalence follows from the above lemma. For the second equivalence we observe that the 1-eigenspace (i.e., the invariant subspace) of a compact operator is finite-dimensional. Moreover, if K is compact then id + K always has a finite-dimensional cokernel.

Remark. The property of ellipticity is important because the index of a non-elliptic operator may vary if we compute it in C^{∞} versus in L^{2}. Think of the Fredholm operator C^{∞}(S^{1}) → C^{∞}(S^{1}) given by the multiplication by z − 1. Its extension to L^{2} is not Fredholm because its image is not closed. In fact, the image is dense in L^{2} but the operator cannot be invertible since zero lies in the spectrum, which is the image of the map z − 1: S^{1} → C. This is related to the fact that the evaluation map C^{∞}(S^{1}) → C at 1 ∈ S^{1} does not extend to L^{2}. You should contrast that with the elliptic operator of differentiation that we studied above. There the cokernel is detected by the integration map C^{∞}(S^{1}) → C, which does extend to L^{2}. In this sense the unbounded differentiation operator is simpler than the bounded multiplication operator. One just has to be precise about its domain, which we can take to be the first Sobolev space W^{1}(S^{1}).

Chern-Gauß-Bonnet theorem. If X is a closed smooth n-manifold then χ(X) = ⟨e(T X), [X]⟩. If X comes with a cell structure, then the Euler characteristic χ(X) is ∑_{0≤k≤n}(−1)^{k}n_{k}, where n_{k} is the number of k-dimensional cells. We can also compute it as the alternating sum ∑_{k}(−1)^{k}b_{k} of Betti numbers (for any ordinary cohomology theory). Now ∑_{k}(−1)^{k} dim H^{k}(X, R) is also the analytic index of the de Rham differential d_{•}: → Ω^{k−1}X → Ω^{k}X → Ω^{k+1}X →, thought of as an elliptic complex (see below).

The right hand side of the Chern-Gauß-Bonnet theorem ⟨e(T X), [X]⟩ is the topological index of d_{•}, where e(T X) is the Euler class of the tangent bundle of X. By Chern-Weil theory ⟨e(T X), [X]⟩ = ∫_{X} ẽ(T X, g)dvol_{g} for any metric g on X.

Definition. D_{•}: 0 → C^{∞}(E_{0}) → C^{∞}(E_{1}) → ⋯ → C^{∞}(E_{k}) → 0 is an elliptic complex if it is a complex of differential operators and the associated complex of symbols is exact.

An elliptic complex with one differential operator is an elliptic differential operator (if a sequence 0 → A → B → 0 is exact, then the map A → B is an isomorphism).

Index theorem generalizes to elliptic complexes: The analytic index of an elliptic complex equals its topological index. The analytic index is ∑_{k}(−1)^{k} dim H^{i}(D_{•}). The topological index comes from the Ktheory class [σ(D_{•})] ∈ K_{cs}(T^{∗}X) represented by the corresponding complex of symbols (see Problem 4d in the homework).

To translate elliptic complexes back into elliptic operators, we choose a Riemannian metric on X and hermitian metrics on E_{i}. If D: C^{∞}(E) → C^{∞}(F) is a differential operator, then there is a unique differential operator D^{∗}: C^{∞}(F) → C^{∞}(E) such that ⟨D(ϕ), ψ⟩_{X} = ⟨ϕ, D^{∗}ψ⟩_{X}. This operator can be constructed locally via integration by parts (using compactly supported sections for the defining property) and is then glued together to an operator on X via partition of unity (and uniqueness). We have σ(D^{∗}) = σ(D)^{∗} so D is elliptic if and only if D^{∗} is elliptic. Note that D^{∗∗} = D.

It follows from elliptic regularity (see below) that the natural map ker D^{∗} → cokerD is an isomorphism. Note that this is easy to see on Banach spaces but in general not true on Fréchet spaces: The multiplication operator m_{(z−1)} on C^{∞}(S^{1}) has adjoint m_{(z̄−1)} and both have trivial kernel and 1-dimensional cokernel. As we have discussed before, this operator is not elliptic and the problem doesn’t arise by elliptic regularity. It follows that a-ind(D^{∗}) = − a-ind(D).

To pass from an elliptic complex D_{•} to an elliptic differential operator, set E_{even} = ⊕_{i even}E_{i} and E_{odd} = ⊕_{i odd}E_{i} and D: C^{∞}(E_{even}) → C^{∞}(E_{odd}) is given by D = ∑_{i even}(D_{i} + D_{i+1}^{∗}).

Lemma. We have a-ind(D_{•}) = a-ind(D) and t-ind(D_{•}) = t-ind(D).

Proof. If we have an elliptic complex 0 → V_{0} →^{D_{0}} V_{1} →^{D_{1}} V_{2} → ⋯, we consider its adjoint 0 ← V_{0} ←^{D_{0}^{∗}} V_{1} ←^{D_{1}^{∗}} V_{2} ← · · ·. Since the inner product is positive definite, it follows that ker D_{i}^{∗}D_{i} = ker D_{i} and from D_{i}D_{i−1} = 0 we see that im D_{i} ∩ker D_{i}^{∗} = 0 and im D_{i}^{∗} ∩ker D_{i} = 0. Moreover, the fact that ker D_{i}^{∗} = cokerD_{i} implies the decomposition V_{i} = im D_{i−1} ⊕ im D_{i}^{∗} ⊕ (ker D_{1}^{∗} ∩ ker D_{2}). It follows that the operators D_{i} and D_{i}^{∗} restrict to isomorphisms im D_{i}^{∗} → im D_{i} and im D_{i} → im D_{i}^{∗} and that they are zero on the other summands in V_{i}. This easily implies the claim.

In the homework we saw how to associate a K-theory class to a complex of vector bundles and in the case of the symbol complex of an elliptic complex D_{•} it follows from that construction that [σ(D_{•})] = [σ(D)] ∈ K(T^{∗}X). We have thus shown that the index theorem for elliptic operators is equivalent to the following result:

Index Theorem for elliptic complexes. If D_{•}: 0 → C^{∞}(E_{0}) → C^{∞}(E_{1}) → ⋯ → C^{∞}(E_{k}) → 0 is an elliptic complex, then a-ind(D_{•}) = t-ind(D_{•}).

Hodge Theory. Given an elliptic operator D: C^{∞}(E) → C^{∞}(F), one can construct the self-adjoint elliptic operator D̃ = \binom{0 D^{∗}}{D 0}: C^{∞}(E ⊕ F) → C^{∞}(E ⊕ F). From the self-adjointness it follows that its index is zero but we have ker D̃ = ker D ⊕ ker D^{∗}. If we define a Z/2-grading on E ⊕ F by declaring E to be even and F to be odd, then D̃ is an odd operator and sdim ker(D̃) = a-ind(D). Here sdim V := dim V^{even} − dim V^{odd} if V = V^{even} ⊕ V^{odd}. One also associates to a Z/2-graded vector bundle E an element in K-theory via [E] := E^{even} − E^{odd} ∈ K(X).

Now starting with an elliptic complex D_{•}, we can construct the associated operators D and D̃. It is not hard to see that D̃ = ∑_{i}(D_{i}+D_{i}^{∗})^{2}: C^{∞}(⊕_{i}E_{i}) → C^{∞}(⊕E_{i}). In particular, for the de Rham complex D_{•} = d we obtain the Hodge Laplacian and for the Dolbeault complex D_{•} = ¯∂ we get the Laplace-Beltrami operator. It follows from the above considerations that ker D̃ ≅ ⊕_{i}H^{i}(D_{•}) and sections in ker D̃ are called harmonic. Note that sdim ker D̃ = a-ind(D) = a-ind(D•).

Thom isomorphism. Let E → X be a real n-dimensional vector bundle over a paracompact space X with metric and orientation. Then there is a class u(E) ∈ H^{n}(DE, SE) = H̃^{n}(DE/SE) such that the map H^{k}(X) → H^{k+n}(DE, SE) (a ↦ u(E) ∪ π^{∗}(a)) is an isomorphism. Moreover, u(E) is uniquely determined by the requirement that its restriction to all fibres gives the orientation class of E.

Remark. If X is compact, then E ∪ ∞ ≈ DE/SE and hence H_{cs}^{∗}(E) := H^{∗}(E ∪ ∞, ∞) ≅ H^{∗}(DE, SE).

Example. For the Hopf bundle H over CP^{n} we have homeomorphisms

Th(H) = DH/SH ≈ DH/S^{2n+1} ≈ Dν(CP^{n}, CP^{n+1})/S^{2n+1} ≈ Dν(CP^{n}, CP^{n+1}) ∪_{S^{2n+1}} D^{2n} ≈ CP^{n+1}.

It follows that the map H^{2k}(CP^{n}) → H^{2k+2}(CP^{n+1}) (a ↦ π^{∗}(a) ∪ u) is an isomorphism. This allows us to compute the cohomology ring of CP^{n} inductively: H^{∗}(CP^{n}) = Z[x]/(x^{n+1}).

Definition. (a) e(E) := i_{0}^{∗}u(E) ∈ H^{n}(X) (the Euler class of E). (b) If L is complex r-dimensional bundle, then c_{r}(L) := e(L_{R}) ∈ H^{2r}(X) (rth Chern class).

Proof of Thom isomorphism. Use induction on the number of trivializing charts (works for compact spaces). For one chart (the trivial bundle) we have U × R^{n} → U. Now (DE, SE) = (U × D^{n}, U × S^{n−1}) = U × (D^{n}, S^{n−1}). Künneth isomorphism: a ∈ H^{k}(U) ↦ a ⊗ ±u_{0} ∈ H^{k}(U) ⊗ H^{n}(D^{n}, S^{n−1}) ↦ a × u_{0} ∈ H^{k}(U × (D^{n}, S^{n−1})). To fix the sign of u_{0} we use the orientation H^{n}(DE, SE) → H^{n}(DE_{x}, SE_{x}) ≅ Z. The result follows from a Mayer-Vietoris argument (and the 5-lemma) that uses the existence and uniquess of u inductively.

Definition. The orientation sheaf o(E) is a locally constant sheaf on X with stalks H_{n}(E_{x}, E_{x} \ 0).

With this definition we extend Thom isomorphism to non-oriented bundles: We get a unique (twisted) Thom-class u(E) ∈ H^{n}(DE, SE, o(E)), which restricts on each fibre to the identity map in the twisted cohomology groups H^{n}(E_{x}, E_{x} \ 0; H_{n}(E_{x}, E_{x} \ 0)) ≅ Hom(H_{n}(E_{x}, E_{x} \ 0), H_{n}(E_{x}, E_{x} \ 0)). The Thom isomorphism H^{k}(X; o(E)) ≅ H^{k+n}(DE, SE) is still given by a ↦ u(E) ∪ π^{∗}(a).

Remark. As a consequence, the (twisted) Euler class lives in H^{n}(X, o(E)).

A quick trip through characteristic classes.

The uniqueness of the Thom class readily implies the following multiplicative properties:

(0) u(E, −θ) = −u(E, θ) and e(E, −θ) = −e(E, θ), where θ is an orientation on E.

(1) u(E_{1} × E_{2}, θ_{1} × θ_{2}) = u(E_{1}, θ_{1}) × u(E_{2}, θ_{2}).

(2) e(E_{1} × E_{2}) = e(E_{1}) × e(E_{2}) with orientations suppressed.

(3) e(E_{1} ⊕ E_{2}) = e(E_{1}) ∪ e(E_{2}).

Remark. Since all sections of a vector bundle are homotopic one can get the Euler class via pullback from the Thom class using an arbitrary section (rather then the zero section as in the definition). If a vector bundle E admits a non-vanishing section it follows that e(E) = 0 because the Thom class is relative to the complement of the zero-section. It follows from this and property (3) above that e is not stable: e(E⊕R) = 0.

Corollary to the Thom isomorphism. (Gysin sequence.) If E → X is an oriented n-dimensional vector bundle, then the sequence

⋯ → H^{i+n−1}(E^{0}) → H^{i+n}(E, E^{0}) → H^{i+n}(E) → H^{i+n}(E^{0}) → H^{i+n+1}(E, E^{0}) → ⋯

is exact and it is isomorphic to

⋯ → H^{i+n−1}(E^{0}) → H^{i}(X) → H^{i+n}(X) → H^{i+n}(E^{0}) → H^{i+1}(X) → ⋯

The map H^{i}(X) → H^{i+n}(E, E^{0}) is given by cup product with u(E) and the map H^{i+n}(X) → H^{i+n}(E) is π^{∗}. Hence the map H^{i}(X) → H^{i+n}(X) is given by cup product with e(E).

Definition. Suppose E → X is a complex n-dimensional vector bundle. Define c_{n}(E) := e(E_{R}, θ_{C}) ∈ H^{2n}(X). Here E_{R} is the underlying oriented real bundle of E. By induction define c_{i}(E) ∈ H^{2i}(X) for i < n via the Gysin sequence for E_{1} → SE, where the fiber of E_{1} over v ∈ SE_{x} is the orthogonal complement v^{⊥} ⊂ E_{x}. For k ≤ n − 1 the image of c_{k}(E) ∈ H^{2k}(X) under π^{∗} is c_{k}(E_{1}) ∈ H^{2k}(SE).

Remark. Chern classes are natural and for the Hopf bundle one can normalize c_{1}(H) ∈ H^{2}(CP^{1}) as the generator that evaluates as −1 on the complex orientation of CP^{1}.

Theorem. Let c(E) := 1 + c_{1}(E) + c_{2}(E) + ⋯ be the total Chern class. Then c(E_{1} ⊕ E_{2}) = c(E_{1}) ∪ c(E_{2}). Together with naturality and normalization properties this characterizes Chern classes.

Consider universal bundles γ_{n} over the n-dimensional Graßmannian Gr_{n} := ∪_{m≥0}Gr_{n}(C^{m+n}).

Theorem. H^{∗}(Gr_{n}) = Z[c_{1}, …, c_{n}] where c_{i} := c_{i}(γ_{n}) are the universal Chern classes.

Proof. Induction on n. The case n = 1 has been done above via the Thom isomorphism. Lemma: Consider the map p: S(γ_{n}) → Gr_{n−1} ((V, v) ↦ v^{⊥}). This map is a fiber bundle with the fiber over W ⊂ C^{∞} being equal to S(W^{⊥}), which is contractible (since it is an infinite dimensional sphere). Now we use the Gysin sequence, which shows that the map H^{i}(Gr_{n}) → H^{i+2n}(Gr_{n}) given by cup product with c_{n} is injective and by induction on n we conclude that its cokernel is isomorphic to H^{i+2n}(Gr_{n−1}). Another induction on i finishes the proof.

Lemma. p_{m,n} := c(γ_{m} × γ_{n}) = (1 + c_{1} + ⋯ + c_{m})(1 + c′_{1} + ⋯ + c′_{n}) =: q_{m,n}.

Proof. Induction on m and n. Restricting to Gr_{m} × Gr_{n−1} respectively Gr_{m−1} × Gr_{n} shows by induction p_{m,n} ≡ q_{m,n} (mod c_{m}c′_{n}). Then the result follows from property (3) above, which says that for any two vector bundles E_{i} of dimensions m and n one has c_{m+n}(E_{1} ⊕ E_{2}) = c_{m}(E_{1}) ∪ c_{n}(E_{2}).

Theorem 1. If X is paracompact, then [X, Gr_{n}] ≈ Vect_{n}(X), via f ↦ f^{∗}(γ_{n}). In other words, Gr_{n} is the classifying space for n-dimensional complex vector bundles.

Theorem 2. There are unique Chern classes c = 1 + c_{1} + ⋯ + c_{n}: Vect_{n}(X) → H^{2∗}(X) characterized by (i) naturality; (ii) normalization: ⟨c_{1}(H → CP^{1}), [CP^{1}]⟩ = −1; (iii) Whitney sum formula: c(E_{1} ⊕ E_{2}) = c(E_{1}) ∪ c(E_{2}).

Proof of Theorem 2. We already constructed Chern classes satisfying (i) and (ii). The sum formula follows from the Lemma above and the fact that E_{1} × E_{2} is the pullback of γ_{n_{1}} × γ_{n_{2}} via f_{1} × f_{2}. Uniqueness follows from Theorem 1. Outline of the proof of Theorem 1. (Assuming X is compact.) The map α is surjective: Any n-dimensional bundle E → X embeds into some trivial bundle X × C^{N} → X and therefore is a pullback of γ_{n} for the appropriate map f: X → Gr_{n}(C^{N}) → Gr_{n}. The map α is injective: If f_{1}^{∗}(γ_{n}) ≅ f_{0}^{∗}(γ_{n}), one uses this isomorphism to construct a bundle E over X ×[0, 1] whose pullbacks along the two boundary embeddings are f_{i}^{∗}(γ_{n}). The bundle E gives us a homotopy between f_{i} just as in the surjectivity part of the argument, if we are careful to construct the embedding into the trivial bundle relative to the given embeddings on X ×{0, 1}.

Corollary: The Splitting Principle. Consider the map i: Gr_{1}^{n} → Gr_{n} given by taking direct sums of subspaces, or given by the classifying map of the sum of line bundles π_{i}^{∗}(γ_{1}). The induced map in cohomology is i^{∗}: H^{∗}(Gr_{n}) = Z[c_{1}, …, c_{n}] → H^{∗}(Gr_{1}^{n}) = Z[x_{1}, …, x_{n}]. We have x_{i} = π_{i}^{∗}(c_{1}(γ_{1})). The class c_{k} = c_{k}(γ_{n}) maps to σ_{k}(x_{i}) = c_{k} (⨁ π_{i}^{∗}(γ_{1})), which by the Whitney sum formula is the kth elementary symmetric polynomial. It follows that the map i^{∗} is injective and its image consists of symmetric polynomials, i.e., polynomials that are invariant under the action of the symmetric group S_{n}.

Generalization. Vect_{n}(X) = Vect_{n}^{h}(X) = Bun_{U(n)}(X), where Vect^{h} denotes (isomorphism classes of) vector bundles with hermitian metric and Bun_{G} denotes (isomorphism classes of) G-principal bundles on X. Recall that BG is the classifying space of G, i.e., the base space of the universal principal G-bundle EG → BG, which is characterized up to homotopy by the fact that EG is contractible. For example, BU(n) is Gr_{n} and EU(n) is the bundle of orthonormal frames over Gr_{n}. Characteristic classes for principal G-bundles are elements of H^{∗}(BG).

Theorem (Borel). If G is a compact Lie group and T ⊂ G is a maximal torus with Weyl group W := N(T)/T, then H^{∗}(BG, Q) ≅ H^{∗}(BT, Q)^{W} = Q[x_{1}, …, x_{dim T}]^{W}. For example, for U(n) the maximal torus is U(1)^{n} and the Weyl group is S_{n}.

Sometimes, one can also make similar statements for other coefficients. For example, H^{∗}(BO(n), Z/2) = Z/2[w_{1}, …, w_{n}], where w_{i} = w_{i}(γ_{n}) ∈ H^{i}(BO(n), Z/2) is the Stiefel-Whitney class. These can be defined in exactly the same way we defined Chern classes, starting with w_{n}(E) := e(E) modulo 2 for a real ndimensional vector bundle E.

For example, H^{∗}(BO(2n), Q) = H^{∗}(BT^{n}, Q)^{W} = Q[y_{1}, …, y_{n}]^{W} = Q[σ_{k}(y_{i}^{2})] because the Weyl group of O(2n) consists of reflections and permutations: W = (Z/2)^{n} ˜×S_{n}. For the oriented case we have in addition the Euler class e(γ_{2n}^{R}) = y_{1} ⋯ y_{n} ∈ H^{∗}(BSO(2n), Q) = Q[y_{1}, …, y_{n}]^{W} = Q[σ_{k}(y_{i}^{2}), y_{1} ⋯ y_{n}] = Q[p_{k}, e]. This follows from the fact that only pairs of reflections lie in the Weyl group of SO(2n) and hence the product of all y_{i} is invariant.

Pontrjagin classes. If E → X is a real vector bundle, define p_{i}(E) := (−1)^{i}c_{2i}(E ⊗ C) ∈ H^{4i}(X). It is not hard to see that the universal Pontrjagin class p_{k}(γ_{2}^{R}n) = σ_{k}(y_{i}^{2}) in rational cohomology.

Lemma. (a) c_{k}(Ē) = (−1)^{k}c_{k}(E); (b) c_{1}(E) = c_{1}(det E); (c) c_{1}(L_{1} ⊗ L_{2}) = c_{1}(L_{1}) + c_{1}(L_{2}).

Proof. By the above splitting principle it suffices to check (a) and (b) on sums of line bundles. (a) Since c_{k} = σ_{k} it suffices to check c_{1}(L̄) = −c_{1}(L). Since the endomorphism bundle End(L) has a non-vanishing (identity) section, it follows that

0 = c_{1}(End(L)) = c_{1}(L^{∗} ⊗ L) = c_{1}(L^{∗}) + c_{1}(L) = c_{1}(L̄) + c_{1}(L).

(b) By part (c) we have c_{1}(L_{1} ⊕⋯⊕L_{n}) = c_{1}(L_{1})+⋯+c_{1}(L_{n}) = c_{1}(L_{1} ⊗⋯⊗L_{n}) = c_{1}(det(L_{1} ⊕⋯⊕L_{n})).

(c) It suffices to check this on Vect_{2}(CP^{1}) = [S^{1}, U(2)] where it follows from the isomorphism for the Hopf bundle H: (H ⊗ H) ⊕ C ≅ H ⊕ H.

Hirzebruch genera. We have the isomorphism H^{∗}(Gr_{n}) = Z[c_{1}, …, c_{n}] = Z[x_{1}, …, x_{n}]^{S_{n}} = H^{∗}(Gr_{1}^{n})^{S_{n}} and similarly, the completions Ĥ^{∗} := ∏_{i} H^{i} are formal power series rings in variables c_{k} respectively x_{k}. Start with a formal power series g in one variable and define g_{n}^{a} := ∑_{1≤i≤n} g(x_{i}) ∈ Λ_{n}^{S_{n}} and g_{n}^{m} := ∏_{1≤i≤n} g(x_{i}) ∈ Λ_{n}^{S_{n}}. Here Λ_{n} = Z[[x_{1}, …, x_{n}]].

Lemma. There are characteristic classes ĝ^{a}, ĝ^{m}: Vect(X) → Ĥ^{2∗}(X) such that (i) ĝ^{a}(L) = g(c_{1}(L)) = ĝ^{m}(L); (ii) ĝ^{a}(E_{1} ⊕ E_{2}) = ĝ^{a}(E_{1}) + ĝ^{a}(E_{2}) and ĝ^{m}(E_{1} ⊕ E_{2}) = ĝ^{m}(E_{1}) ∪ ĝ^{m}(E_{2}). Moreover, (i) and (ii) determine ĝ^{a} and ĝ^{b} uniquely.

Proof. By the splitting principle we have ĝ^{m}(E) := ĝ^{m}(f^{∗}(γ_{n})) = f^{∗}(ĝ^{m}(γ_{n})) = f^{∗}(g_{n}^{m}). Thus (i) can be proved as follows: ĝ^{m}(L) = f^{∗}(g_{1}^{m}) = f^{∗}(g(x)) = g(f^{∗}(x)) = g(c_{1}(L)). Case (ii) works similarly.

Remark. (i) ĝ^{a} extends uniquely to a group homomorphism ĝ^{a}: K(X) → Ĥ^{∗}(X). (ii) ĝ^{m} is stable, i.e., ĝ^{m}(E⊕C) = ĝ^{m}(E), if and only if g(0) = 1. If g(0) = 1, then ĝ^{m} extends uniquely to a group homomorphism ĝ^{m}: K̃(X) → Ĥ^{2∗}(X)^{×}. Here Ĥ^{2∗}(X)^{×} is the group (under cup product) of classes 1 + a_{1} + · · ·.

Example. Take g(x) = exp(x) = 1 + x + x^{2}/2 + ⋯ ∈ Q[[x]].

Lemma. The Chern character ch := ĝ^{a}: K(X) → Ĥ^{2∗}(X, Q) is a ring homomorphism.

Proof. Need to check ch(E_{1} ⊗ E_{2}) = ch(E_{1}) ∪ ch(E_{2}) for vector bundles E_{i} over X. It suffices to check this statement for direct sums of line bundles. In fact, by additivity, it is enough to check it on line bundles. Now ch(L_{1} ⊗ L_{2}) = ch(L_{1}) ∪ ch(L_{2}) is implied by exp(c_{1}(L_{1} ⊗ L_{2})) = exp(c_{1}(L_{1}) + c_{1}(L_{2})) = exp(c_{1}(L_{1})) ∪ exp(c_{1}(L_{2})).

If X is a finite-dimensional CW-complex, then the map ch: K(X, Q) → H^{2∗}(X, Q) is an isomorphism.

Definition. The Todd character Td := ĝ^{m}: K̃(X) → Ĥ^{2∗}(X, Q)^{×} is induced by g(x) = x/(1 − exp(−x)) = 1+x/2+x^{2}/12+· · ·. The Todd class of a vector bundle E is Td([E]) and the Todd class of a real manifold X is Td(X) := Td(T X ⊗_{R} C). The Todd genus of X is ⟨Td(X), [X]⟩ ∈ Q.

Now the topological index is fully defined because the Chern character for relative K-theory can be defined using the isomorphism K(X, Y) = K̃(X/Y).

Lemma. We have e(T X) ∪ Th^{−1}(ch(σ(D))) = ch(E) − ch(F) ∈ H^{∗}(X, Q) for any elliptic differential operator D: E → F, or more generally, for any symbol class.

Proof. We have a commutative diagram

K(T^{∗}X, T^{∗}X \ 0) ⟶ K(T^{∗}X)
↓^{ch}  ↓^{ch}
H^{∗}(T^{∗}X, T^{∗}X \ 0) ⟶ H^{∗}(T^{∗}X)
∪u(T^{∗}X)↑≅  _{π^{∗}}↑≅
H^{∗−d}(X, Q)  ⟶ H^{∗}(X, Q)

This lemma is most useful when cup product with the Euler class is injective. Therefore, we will study elliptic operators whose symbols come from representation theory.

Definition. Let G be a Lie group. A G-structure on a manifold X consists of a principal G-bundle P over X, a real representation V of G, and an isomorphism P ×_{G} V ≅ T X over X. G-structures form a category, whose isomorphism classes we can usually compute.

Examples. (0) Every manifold has a canonical GL(R^{d})-structure. (1) An O(d)-structure is a Riemannian metric. Here V = R^{d}. (2) An SO(d)-structure is a Riemannian metric with an orientation. (3) A GL(C^{d/2})- structure (d is even) is an almost complex structure. (4) A Spin(d)-structure is a Spin-structure.

The topological index t-ind: K_{cs}(T^{∗}X) → Q, where σ ↦ ⟨Td(X) ∪ Th^{−1}(ch(σ)), [X]⟩ can be computed most easily for those σ that come (via G-representations) from a G-structure on X. Let [σ] ∈ K_{cs}(T^{∗}X) be represented by (E, F, π^{∗}E/(T^{∗}X \ 0) →^{≅} π^{∗}F/(T^{∗}X \ 0)). Assume there are complex representations M and N of G such that E = P ×_{G} M and F = P ×_{G} N. If f: X → BG classifies P, then we form vector bundles on BG: Ṽ := EG ×_{G} V, M̃ := EG ×_{G} M and Ñ := EG ×_{G} N.

Running assumptions. There is an isomorphism π̃^{∗}(M̃)/(Ṽ \ 0) →^{≅} π̃^{∗}(Ñ)/(Ṽ \ 0) and the Euler class is nontrivial: 0 ≠ e(V^{∗}) ∈ H^{∗}(BG, Q). This is what we mean when we say that the symbol comes from G-representations.

Remark. H^{∗}(BG, Q) = H^{∗}(BT, Q)^{W} has no zero divisors and hence one can uniquely divide by any nonzero class. In particular, the above lemma shows that under our assumptions the class Th^{−1}(ch(σ)) only depends on the vector bundles in question (and not on the symbol class σ):

Lemma. Th^{−1}(ch(σ)) = (ch(M̃) − ch(Ñ))e(Ṽ^{∗})^{−1} ∈ Ĥ^{∗}(BG, Q).

Example: Chern-Gauß-Bonnet. Let G = SO(2n), V = R^{2n}, M = Λ^{even}V^{∗} ⊗ C, N = Λ^{odd}V^{∗} ⊗ C. We will show that t-ind(X, P, G, V, M, N) = ⟨e(T X), [X]⟩. For any elliptic operator D: C^{∞}(E) → C^{∞}(F) whose symbol comes from G-representations we therefore have a-ind(D) = ⟨e(T X), [X]⟩. For example, we can take D = (d + d^{∗})/(Λ^{even}V ⊗ C) and a-ind(D) = χ(X).

Example: Hirzebruch-Riemann-Roch. Let G = U(C^{m}), V = C^{m} (X is almost Hermitian), M = Λ^{even}V, N = Λ^{odd}V. We will show that in this case t-ind(X) = ⟨Td(T X), [X]⟩. Note that T X is a complex bundle, hence we don’t need to tensor it with C as we usually do. For example, we can take the Dolbeault operator (¯∂+ ¯∂^{∗})/Λ^{even}V whenever X is a complex manifold (integrability is needed to define the Dolbeault operator). Thus its analytic index is the holomorphic Euler characteristic χ_{h}(X), also known as arithmetic genus.

Claim. [Td(Ṽ ⊗ C) ∪ (ch(M̃) − ch(Ñ))e(Ṽ^{∗})^{−1}]^{deg=2n} = (−1)^{n}e(Ṽ) ∈ H^{2n}(BSO(2n), Q).

Proof. Suppose M is a complex representation of a compact Lie group G. Denote by T the maximal torus of G. Now M as a representation of T splits as a direct sum of M_{r}, where each M_{r} corresponds to some weight w_{r} of T. For a weight w_{r} an element t ∈ T acts on m ∈ M_{r} by t(m) = exp(2πiw_{r}(t))m. Now Hom(T, S^{1}) = H^{2}(BT). To be continued.

A distraction: Rigid and integrable G-structures. An example of a G-structure is the flat G-structure on a vector space V, which is given by P := V × G → V and the isomorphism is given by P ×_{G} V = (V × G) ×_{G} V ≅ V × V ≅ T V via translation in V.

Definition. A G-structure on X is integrable if it is locally flat.

Examples. (i) An integrable GL_{n}(C)-structure (V = C^{n}) is a complex structure: There are complex charts. (ii) For G = Sp(2n) and V = R^{2n} integrable structures are symplectic structures by Darboux’s theorem. (iii) For G = O(n) and V = R^{n} integrable structures are flat metrics. (iv) For G = U(n) and V = C^{n} integrable structures are flat Kähler structures.

Example. The total space of the cotangent bundle T^{∗}X is always symplectic: ω = dα. A metric on X induces a metric on T^{∗}X. This metric is integrable if and only if the original metric is flat. It turns out that T^{∗}X is always Kähler but with respect to a different metric. See http://mathoverflow.net/questions/26776/.

Chart versions of integrable G-structures. Choose a covering collection of charts on a manifold X with codomain being an open subset of a vector space V. We can now say that an integrable G-structure is a lift of the derivatives of transition functions along the map ρ: G → GL(V) that satisfies the usual cocycle condition.

Definition. A G-structure on X is rigid if the transition functions are restrictions of the action of the semidirect product of G and the additive group of V on V. For G = O(n) and V = R^{n} we get the notion of a rigid Euclidean manifold. For general rigid geometry we should generalize the model space from a vector space to an arbitrary (homogenous) G-space.

Hirzebruch-Riemann-Roch theorem. We have a-ind(¯∂_{•}) = χ_{hol}(X) := ∑_{0≤i≤n}(−1)^{i} dim_{C} H^{0,i}(X) = ⟨Td(T X), [X]⟩ =: t-ind(¯∂_{•}).

Observation. If this holds for X = CP^{n} for n ≥ 0, then the formal power series that defines Td(X) is uniquely determined.

Proof. We have χ_{hol}(CP^{n}) = 1 because CP^{n} is Kähler with the Kähler form representing a non-trivial class in H^{1,1}(CP^{n}). Lemma: T CP^{n} ⊕ C = ⊕_{1≤i≤n+1}H^{∗}. Observe that C^{n+1} = H ⊕ H^{⊥} and T CP^{n} = Hom(H, H^{⊥}). Now the total Chern class of T CP^{n} is c(⊕^{n+1}H^{∗}) = ∏^{n+1} c(H^{∗}) = (1 + a)^{n+1}, where a = c_{1}(H^{∗}) = −c_{1}(H) is a “complex” generator of H^{2}(CP^{n}). Recall that the Todd genus is given by the power series x/(1 − exp(−x)). Hence T̃d^{m}(T CP^{n}) = ∏^{n+1} Td(H^{∗}) = Td(c_{1}(H^{∗}))^{n+1} = Td(a)^{n+1} ∈ H^{∗}(CP^{n}). Thus ⟨Td(T CP^{n}), [CP^{n}]⟩ is the coefficient of a^{n} in Td(a)^{n+1}. By Cauchy formula this coefficient equals (2πi)^{−1} ∫_{S^{1}} z^{−n−1}(z/(1 − exp(−z)))^{n+1}dz = 1.

Suppose X is a compact 2d-dimensional oriented manifold and D: C^{∞}(E) → C^{∞}(E′) is an elliptic operator. Then t-ind(D) := (−1)^{d}⟨Td(X) ∪ Th^{−1}(ch(σ(D))), [X]⟩. We have so far ignored the additional sign. Recall from homework that the topological index always vanishes on odd-dimensional manifolds.

Theorem. If X has a G-structure and σ(D) comes from G-representations, then

t-ind(D) = (−1)^{d} ⟨ ∏_{1≤k≤d}(Td(η_{k})Td(−η_{k})/η_{k}) ∑_{1≤k≤m}(exp(w_{k}) − exp(w′_{k})), a⟩

for any a such that f_{∗}([X]) = i_{∗}(a). Here H_{2n}(BT, Q) →^{i_{∗}} H_{2d}(BG, Q) ←^{f_{∗}} H_{2d}(X).

The assumption about representations means the following: T X ≅ P ×_{G} V^{2d}, E ≅ P ×_{G} M, E′ ≅ P ×_{G} M′ for some complex G-modules M and M′. The w_{k} respectively w′_{k} are the complex weights of M respectively M′ and the η_{k} are the real weights of V. All of these live in H^{2}(BT). Our running assumption is that

P ×_{G} (V^{∗} × M) ≅ π^{∗}E ⟶^{σ(D)} π^{∗}E′ = P ×_{G} (V^{∗} × M′)
↓  ↓  ↓
P ×_{G} V^{∗}  ≅ T^{∗}X = T^{∗}X

comes from a G-equivariant map σ̃: V^{∗} × M → M′ and that 0 ≠ η_{k} ∈ H^{2}(BT, Q).

Definition. (a) Suppose G is a compact connected group and T ≤ G is a maximal torus. If M is a complex G-module, then M|_{T} ≅ M_{1} ⊕ ⋯ ⊕ M_{m} and for m ∈ M_{k} we have t(m) = exp(2πiw_{k}(t))m where w_{k} ∈ Hom(T, S^{1}) = H^{2}(BT) are “complex weights of M”. (b) If V is a real oriented 2d-dimensional Gmodule, i.e., we have a map ρ: G → SO(V), then its real weights η_{k} are defined by V |_{T} = V_{1} ⊕ ⋯ ⊕ V_{d}, where dim V_{k} = 2. Here η_{k} ∈ Hom(T, S^{1}) = H^{2}(BT).

Recall that we have an isomorphism i^{∗}: H^{∗}(BG, Q) → H^{∗}(BT, Q)^{W} ⊂ Q[[z_{1}, …, z_{r}]]. In particular, the cohomology ring does not have zero divisors, hence the formula for t-ind(D) makes sense because the product η_{1} ⋯ η_{d} is non-zero.

Remark about notation: For ρ: G → GL(M) the character is defined by cha(ρ)(t) := tr_{M}(ρ(t)) = ∑_{1≤k≤m} exp(2πiw_{k}(t)). Compare this to Ĥ^{2∗}(BT) ∋ i^{∗}ch(M) := i^{∗}ch(EG ×_{G} M) = ∑_{1≤k≤m} exp(w_{k}).

Lemma. (a) If M has complex weights w_{k} then M̄ has complex weights −w_{k}. (b) If V has real weights η_{1}, …, η_{d}, then V ⊗_{R} C has complex weights ±η_{1}, …, ±η_{d}. (c) If M has complex weights w_{1}, …, w_{m}, then the underlying real representation of M has real weights w_{1}, …, w_{m}.

Proof of Theorem. ⟨(T X ⊗ C) ∪ Th^{−1}(ch(σ(D))), [X]⟩ = ⟨i^{∗}(Td(V ⊗ C) ∪ (ch(M) − ch(M′))e(V^{∗})^{−1}), a⟩. Here Ṽ = EG ×_{G} V. We have the following commutative diagram:

V  ∈  R(G)  ⟶  R(T)  =  Z[T̂]
↓  ↓  ↓
EG ×_{G} V = Ṽ ∈ K(BG) ⟶ K(BT)
↓^{ch}  ↓^{ch}
Ĥ^{∗}(BG, Q) ⟶^{i^{∗}}  Ĥ(BT, Q) ≅ Q[[y_{1}, …, y_{k}]]

Remark: Atiyah-Segal completion theorem. Consider the augmentation ideal I := ker(dim: R(G) → Z). Define R(G)Î:= lim_{n} R(G)/I^{n}. The map R(G)Î → K(BG) = lim_{n} K((BG)^{(n)}) is an isomorphism.

Theorem. Suppose X is a 2d-manifold with G-structure (P, V, α) and the elliptic complex (C^{∞}(E), D) = 0 → C^{∞}(E_{0}) ⟶^{D_{0}} C^{∞}(E_{1}) ⟶^{D_{1}} ⋯ → C^{∞}(E_{n}) → 0 comes from G-representations, i.e., E_{i} ≅ P ×_{G} M_{i} for some complex G-representation M_{i} satisfying our running assumptions: 0 ≠ e(V) ∈ H^{2d}(BG) and there is a G-equivariant (linear for each v ∈ V^{∗}) sequence of maps σ̃_{i}: V^{∗} × M_{i} → M_{i+1} such that σ(D) = id_{P} ×_{G} σ̃_{•}. Then t-ind(D) = (−1)^{d}⟨Td(V ⊗ C)ch(M)e(V^{∗})^{−1}, f_{∗}^{P} [X]⟩ = (−1)^{d}⟨∏_{1≤k≤d} Td(η_{k})Td(−η_{k})η_{k}^{−1}i^{∗}ch(M), a⟩ for any a ∈ H_{2d}(BT, Q) such that i_{∗}(a) = f_{∗}^{P} [X]. Here f^{P}: X → BG classifies P, η_{k} are R-weights of V, and ch(M) := ∑_{0≤i≤n}(−1)^{i}ch(M_{i}) ∈ Ĥ^{2∗}(BG, Q) pulls back via i to ∑_{0≤i≤n}(−1)^{i} ∑_{1≤k≤m_{i}} exp(w_{i}^{k}), where w_{i}^{k} are the C-weights of M_{i}.

Example: Chern-Gauß-Bonnet theorem. We have G = SL(R^{2d}), V = R^{2d}, M = Λ(V^{∗})⊗C (complexified exterior algebra). The R-weights of V are y_{k} ∈ H^{∗}(BSO(2d), Q). Now we compute H^{∗}(BSO(n), Q) ≅ H^{∗}(BT, Q)^{W} = Q[y_{1}, …, y_{⌊n/2⌋}]^{W}, where W is the semidirect product of (Z/2)^{⌊n−1/2⌋} with S_{⌊n/2⌋}. We have the following polynomial generators:

n = 1  \begin{matrix}2 \\ y_{1} \end{matrix}  \begin{matrix}3 \\ y_{1}^{2} \end{matrix}  \begin{matrix}4 \\ y_{1}^{2} + y_{2}^{2} \\ y_{1}y_{2} \end{matrix}  \begin{matrix}5 \\ y_{1}^{2} + y_{2}^{2} \\ y_{1}^{2}y_{2}^{2} \end{matrix}  \begin{matrix}6 \\ y_{1}^{2} + y_{2}^{2} + y_{3}^{2} \\ y_{1}^{2}y_{2}^{2} + y_{2}^{2}y_{3}^{2} + y_{1}^{2}y_{3}^{2} \\ y_{1}y_{2}y_{3} \end{matrix}

It is easy to check from our definition that i^{∗}e(V) = y_{1} ⋯ y_{d} and i^{∗}p_{i}(V) = σ_{i}(y_{k}). Therefore we have isomorphisms H^{∗}(BSO(2k), Q) ≅ Q[p_{1}, …, p_{k−1}, e] with p_{k} = e^{2} and H^{∗}(BSO(2k + 1), Q) ≅ Q[p_{1}, …, p_{k}]. In fact, these isomorphisms also hold for any coefficient ring that is an integral domain containing 1/2.

Lemma. If M has complex weights w_{k}, then i^{∗}ch(Λ^{∗}M) = ∏_{1≤k≤m}(1 − exp(w_{k})) ∈ Ĥ^{2∗}(BT, Q).

Proof. M|_{T} = ⨁_{1≤k≤m} M_{k}, where M_{k} are 1-dimensional with weights w_{k}. Then i^{∗}ch(ΛM) = ch(Λ^{∗}(M_{1} ⊕ ⋯⊕M_{m})) = ch(Λ^{∗}(M_{1}))⊗⋯⊗ch(Λ^{∗}(M_{m})) = ∏_{1≤k≤m}(1−exp(w_{k})) because ch(Λ^{∗}(M_{k})) = ch(1)−ch(M_{k}).

Proof of last steps in Chern-Gauß-Bonnet theorem. The R-weights of V are y_{k} ∈ H^{∗}(BT, Q)^{W}, hence the C-weights of V ⊗ C are ±y_{k}, where 1 ≤ k ≤ d = dim T. By Lemma ch(Λ^{∗}(V^{∗}) ⊗ C) = ∏_{1≤i≤d}(1 − exp(y_{k}))(1−exp(−y_{k})). Thus t-ind(D) = (−1)^{d}⟨∏_{1≤k≤d} Td(y_{k})Td(−y_{k})(1−exp(y_{k}))(1−exp(−y_{k}))y_{k}^{−1}, a⟩. We have Td(y_{k}) = y_{k}/(1 − exp(−y_{k})) and therefore t-ind(D) = (−1)^{d}⟨∏_{1≤k≤d}(−y_{k}), a⟩ = ⟨∏_{1≤k≤d} y_{k}, a⟩ = ⟨e(V), f_{∗}^{P} [X]⟩ = ⟨e(T X), [X]⟩.

Example: Hirzebruch-Riemann-Roch. G = GL(C^{m}), V = C^{m}, M^{k} = Λ^{k}(V̄^{∗}), i.e., X is almost complex. Complex weights of V are x_{k} ∈ H^{2}(BT), hence real weights of V_{R} are also x_{k}. We have ch(M) = ∏_{1≤k≤m}(1−exp(x_{k})) by Lemma. Note that the Dolbeault operator D exists only for complex manifolds. We have t-ind(D) = (−1)^{m}⟨∏_{1≤k≤m} Td(x_{k})Td(−x_{k})(1 − exp(x_{k}))x_{k}^{−1}, a⟩ = ⟨∏_{1≤k≤m} x_{k}/(1 − exp(−x_{k})), a⟩ = ⟨∏_{1≤k≤m} Td(x_{k}), a⟩ = ⟨i^{∗}Td(V), a⟩ = ⟨Td(V), f^{P} [X]⟩ = ⟨Td(T X), [X]⟩.

Twisted Dolbeault operator: G = GL(C^{m}) × GL(C^{n}) and M = Λ(V) ⊗ W. We have ch(M) = ∏_{1≤k≤m}(1 − exp(x_{k})) × ch(W) by Lemma. Thus the factor ch(W) appears in all formulas above and we obtain t-ind(D) = ⟨Td(T X)∪ch(P ×_{GL(C^{n})}W), [X]⟩. Here P ×_{GL(C^{n})}W is the twisting holomorphic bundle.

Summary: Examples of Index Theorem.

Chern-Gauß-Bonnet  Hirzebruch-Riemann-Roch  Hirzebruch Signature Theorem
operator  d  ¯∂  (d + d^{∗})|_{Ω^{∗}(X,C)^{+}}
G  SL_{2d}(R)  GL_{d}(C)  SO_{2d}
M^{∗}  Λ^{∗}(V^{∗}) ⊗ C  Λ^{∗}(V̄^{∗})  (Λ^{∗}(V^{∗}) ⊗ C)^{±}
t-ind  ⟨e(T X), [X]⟩  ⟨Td(T X), [X]⟩  ⟨L(T X), [X]⟩
a-ind  χ(X)  χ_{hol}(X)  σ(X)

Recall that the symbol of the de Rham differential evaluated at a point ξ ∈ T_{x}^{∗} is the exterior multiplication by ξ from the left: σ(d_{i}): T_{x}^{∗} × Λ^{i}(T_{x}^{∗}) → Λ^{i+1}(T_{x}^{∗}) is hence given by (ξ, w) ↦ ξ ∧ w for all i. Since this map is SL(V)-equivariant and linear in w, we see that our running assumption is satisfied: The symbol of the de Rham operator comes from SL(V)-representations. Moreover, the resulting sequence of operators is exact whenever ξ ≠ 0, thus the resulting complex is elliptic. A geometric proof can be immediately obtained from the geometric interpretation of the exterior algebra. This finally completes our derivation of the Chern-Gauß-Bonnet theorem from the Atiyah-Singer index theorem.

For the Dolbeault operator ¯∂ its symbol evaluated at ξ ∈ \overline{T_{x}^{∗}X} = Λ_{x}^{0,1}(X) is the exterior multiplication by ξ followed by the multiplication by the imaginary unit. Hence again our running assumptions for computing t-ind(¯∂) via representations are satisfied and the Dolbeault complex is elliptic. Hence the Hirzebruch-Riemann-Roch theorem follows from the Atiyah-Singer index theorem.

We recall the “wrapping” operation of elliptic complexes: After choosing a Riemannian metric on the manifold X and hermitian inner products on the vector bundles E_{i}, we can pass from an elliptic complex

D_{•}: 0 → C^{∞}(E_{0}) ⟶^{D_{0}} C^{∞}(E_{1}) ⟶^{D_{1}} C^{∞}(E_{2}) ⋯ → C^{∞}(E_{n}) → 0 to an elliptic operator D := ∑_{i} D_{2i} + D_{2i−1}^{∗}

going from the even part of C^{∞} (⨁_{i} E_{i}) to its odd part. The analytic index is preserved under this operation.

If we apply wrapping to the de Rham complex, we obtain an operator d + d^{∗}: Ω^{even}(X) → Ω^{odd}(X). The analytic index of d + d^{∗} is χ(X). However, we can also think of d + d^{∗} as an operator Ω^{+}(X) → Ω^{−}(X), where Ω^{±}(X) are the ±1-eigenspaces of the Hodge star on Ω^{∗}(X, C) (adjusted in such a way that it is its own inverse). The Hodge star operator depends on an orientation of X. The analytic index of (d + d^{∗})|_{Ω^{+}(X)}, the so-called signature operator, is the signature of X, which is defined as the signature of the non-degenerate symmetric bilinear form H_{dR}^{2d}(X) ⊗ H_{dR}^{2d}(X) → R, where dim X = 4d (otherwise the signature is zero).

To compute the topological index of the signature operator recall that the L-genus is given by the power series L(x) = x/ tanh(x) = 1 + x^{2}/3 − x^{4}/45 + · · ·. Note that only even powers are present. Recall that we have an isomorphism i^{∗}: H^{∗}(BSO(2d), Q) = Q[p_{1}, …, p_{d−1}, e] → H^{∗}(BT^{d}, Q)^{W} = Q[y_{1}, …, y_{d}]^{W}, where i^{∗}p_{k} = σ_{k}(y_{i}^{2}). Thus ∏_{1≤i≤d} L(y_{i}) = 1 + L_{1}(p_{1}) + L_{2}(p_{1}, p_{2}) + ⋯ and for a real vector bundle E → X we have L(E) = 1 + L_{1}(p_{1}(E)) + L_{2}(p_{1}(E), p_{2}(E)) + · · ·.

Example. We have c(T CP^{n})c(\overline{T CP^{n}}) = c(T CP^{n} ⊗ C), i.e., (1 + a)^{n+1}(1 − a)^{n+1} = (1 − a^{2})^{n+1}. Here a ∈ H^{2}(CP^{n}) is the “complex” generator. Thus p(T CP^{n}) = (1 + a^{2})^{n+1} since p_{k}(E) = (−1)^{k}c_{2k}(E ⊗_{R} C). Hence ⟨L(T CP^{n}), [CP^{n}]⟩ = ⟨(a/ tanh(a))^{n+1}, [CP^{n}]⟩. This is the coefficient of a^{−1} in 1/ tanh(a)^{n+1}, which can be computed via the Cauchy formula to be equal to 1 if n is even and 0 if n is odd. Thus the signature theorem is true for all complex projective spaces (and this fact determines the formula for the L-genus). Hirzebruch showed that this implies the signature theorem for all manifolds using Thom’s computation of the rational bordism ring of oriented manifolds. This bordism proof was later generalized to all elliptic operators by Atiyah-Singer, that’s why we’ll explain Thom’s results next.

Signature theorem (Hirzebruch, Thom). If X is a smooth oriented closed 4d-manifold, then a-ind(D) = σ(X) = ⟨L(T X), [X]⟩ = t-ind(D), where D = (d + d^{∗})|_{Ω^{∗}(X,C)^{+}} and the L-genus is given by L(x) = x/ tanh(x) = 1 + x^{2}/3 − x^{4}/45 + · · ·.

Corollary (Milnor). (a) There is a smooth structure on S^{7} not diffeomorphic to the standard smooth structure. (b) There is a closed PL-manifold T^{8} without a smooth structure.

Lemma 1. For any k ≡ 2 (mod 4) there is a 4-dimensional vector bundle E_{k} → S^{4} such that e(E_{k}) = u, where u is the generator of H^{4}(S^{4}) and p_{1}(E_{k}) = ku ∈ H^{4}(S^{4}).

Proof of Lemma 1. For the tangent bundle we have e(T S^{4}) = 2u, p_{1}(T S^{4}) = 0. Consider now the 4dimensional tautological bundle γ_{H} → HP^{1}, where HP^{1} = H ∪ ∞ = S^{4} is the quaternion projective line. The sphere bundle γ_{H} → HP^{1} is isomorphic to S^{4+3} → S^{4}. By Gysin sequence e(γ_{H}) = u and p_{1}(γ_{H}) = −c_{2}(γ_{H} ⊗ C) = −c_{2}(γ_{H} ⊕ γ̄_{H}) = −c_{2}(γ_{H} ⊕ γ_{H}) = −2c_{2}(γ_{H}) = −2u. We have Vect_{R}^{4}(S^{4}) ≅ π_{4}(BO(4)) = π_{3}(O(4)) = π_{3}(Spin(4)) = π_{3}(SU(2) × SU(2)) = Z × Z. We observe that Vect_{R}^{4}(S^{4}) is a group and (p_{1}, e) is a homomorphism from this group to Z × Z.

Remark. The condition k ≡ 2 (mod 4) is necessary because p_{1} ≡ p(w_{2}) + 2e (mod 4), where the cohomology operation p: H^{2}(X, Z/2) → H^{4}(X, Z/4) is known as the Pontrjagin square.

Lemma 2. For any k ≡ 2 (mod 4) we have S(E_{k}) ≃ S^{7}.

Proof. We have π_{i}(S(E_{k})) = 0 for 0 ≤ i ≤ 2. By Gysin sequence H^{∗}(S(E_{k})) ≅ H^{∗}(S^{7}), hence H_{∗}(S(E_{8})) = H_{∗}(S^{7}). By Hurewicz π_{i}(S(E_{k})) = π_{i}(S^{7}) for all i. Thus by Whitehead theorem the obvious degree one map S(E^{k}) → S^{7} is a homotopy equivalence. Proof of Corollary. By Smale’s h-cobordism theorem S(E_{k}) is PL-equivalent to S^{7}. This also follows from the existence of smooth function f: S(E_{k}) → R with exactly two critical points, which Milnor wrote down in his short Fields medal paper. If we cut out two 7-balls, the rest is diffeomorphic to S^{6} × [0, 1]. We can then glue one ball to obtain D^{7} and the manifold S(E_{k}) is diffeomorphic to the union of two smooth 7-balls, glued along a diffeomorphism of S^{6}. Any such diffeomorphism extends to a PL-homeomorphism to D^{7} (via coning) but not necessarily to a diffeomorphism. That’s why S(E_{k}) is PL-homeomorphic to S^{7} but not necessarily diffeomorphic.

Recall that L(x)/ tanh(x) = 1+x^{2}/3−x^{4}/45+· · ·. Thus L(V) = 1+L_{1}(p_{1}(V))+L_{2}(p_{1}(V), p_{2}(V))+⋯ = 1 + p_{1}(V)/3 + p_{2}(V)/9 − (p_{1}(V)^{2} − 2p_{2}(V))/45 + · · ·. Let T_{k}^{8} := D(E_{k}) ∪_{S(E_{k})≅S^{7}} D^{8}. If T_{k}^{8} had a smooth structure (e.g., if S(E_{k}) was diffeomorphic to S^{7}) then we would get the following contradiction for k = 6: T_{6} does not satisfy the signature theorem: 1 = σ(T_{k}) ≠ ⟨L(T (T_{k})), [T_{k}]⟩ = ⟨7p_{2}(T (T_{k})) − p_{1}(T (T_{k}))^{2}, [T_{k}]⟩/45. If σ = ⟨L, [T]⟩ then 45 = 7⟨p_{2}, [T]⟩ − k^{2} would imply that k ≡ ±2 (mod 7).

Original (bordism) proof of signature theorem uses results of Serre (1951): Stable homotopy groups of spheres are finite (except π_{0}) and Thom (1952), who computed the rational oriented bordism ring. Steps in the proof of signature theorem: (1) It is true for CP^{n} for all n; (2) Both sides have the following properties:

(a) They are additive under coproduct of manifolds; (b) They are multiplicative under product of manifolds;

(c) They are invariant under oriented bordism: End of proof: Denote by Ω_{∗} the oriented bordism ring (Ω_{n} consists of closed oriented smooth n-manifolds modulo compact oriented smooth (n + 1)-cobordisms). We just proved that σ and L are ring homomorphisms Ω_{∗} → Q.

Theorem (Thom). Ω_{∗} ⊗ Q = Q[CP^{2}, CP^{4}, CP^{6}, …].

Proof of (c): We need to show that σ and L vanish on ∂W^{n+1}. Observe that T W |_{M} ≅ T M ⊕ R. Hence ⟨L(T M ⊕ R), [M]⟩ = ⟨L(T W), i_{∗}[M] = 0⟩ = 0. Remark: If H_{n}(M^{2n}, Q) contains a Lagrangian (for the intersection form), then σ(M) = 0. Lemma: L := ker(H_{n}(M, Q) → H_{n}(W, Q)) is a Lagrangian. Here dim M = 2n and dim W = 2n + 1. Proof: (a) The intersection form vanishes on L because the intersection points of two n-cycles in M bound intersection arcs of the bounding (n + 1)-cochains in W.

(b) 2 dim_{Q} L = dim_{Q} H_{n}(M, Q) by Lefschetz duality. Proof of Thom’s theorem. The first step is the Pontrjagin-Thom construction: Consider an embedding of manifolds M → S^{d+n} (here dim M = d and n is large). Take the normal bundle νM of this embedding and construct the collapse map S^{d+n} → Th(νM) (map a tubular neighborhood of M diffeomorphically to the normal bundle νM and map everything outside the tubular neighborhood to the basepoint). The embedding gives a classifying map M → BO(n, n + d) ⊂ BO(n) for the normal bundle νM. It is the pullback of the universal bundle γ^{n} via this map. Thus we get a map Th(ν): Th(νM) → Th(γ^{n}), which we compose with the collapse map to arrive at a map t_{M}: S^{d+n} → Th(γ^{n}), associated to our manifold M.

Theorem. (a) For the unoriented bordism groups we have Ω_{d}^{un} ≅ π_{d+n}(Th(γ_{O}^{n})) as vector spaces over Z/2. The forward map is given by sending a manifold M to the map t_{M}. The backward map is given by sending a map S^{d+n} → Th(γ_{O}^{n}) to the preimage of the zero section. (First we deform the map to make it smooth and transversal.) (b) Ω_{d} ≅ π_{d+n}(Th(γ_{SO}^{n})) and more generally: (c) Ω_{d}^{ξ} ≅ π_{d+n}(Th(ξ_{n})) for stable normal structures ξ: B → BO like Spin, complex, symplectic. Here we define spaces B_{n} as pullbacks of a fibration ξ: B → BO under the inclusions BO(n) → BO and hence they come equipped with an ndimensional bundle ξ_{n}. In particular, for framed bordism we can use the contractible spaces B_{n} = EO(n) and hence Th(ξ_{n} → B_{n}) ≃ Th(R^{n} → pt) ≃ S^{n}. We thus obtain Ω_{d}^{fr} ≅ π_{d+n} Th(ξ_{n} → EO(n)) ≅ π_{d+n}(S^{n}) for large n. These are the stable homotopy groups of spheres; Serre’s theorem states that these groups π_{d}^{s} are finite for d > 0.

Remark. The spaces Th(ξ_{n}) for n ≥ 0 form a spectrum Mξ, i.e., these are pointed spaces together with maps Th(ξ_{n})∧S^{1} → Th(ξ_{n+1}), arising from the fact that S^{1} = Th(R). In the oriented case ξ: BSO → BO we have the following sequence of isomorphisms: Ω_{d} ⊗ Q → π_{d}(MSO) ⊗ Q → H_{d}(MSO, Q) = H_{d}(BSO, Q) = Hom(H^{d}(BSO, Q), Q). This composition is given by Pontrjagin numbers, i.e., evaluation of products of Pontrjagin classes on the fundamental class of a closed oriented d-manifold. The rational Hurewicz homomorphism, used here for the spectrum MSO, is an isomorphism for any spectrum because of Serre’s finiteness theorem. If one looks at all degrees together, then the above maps are actually ring isomorphisms if one uses the H-space structure on BSO given by direct sum of vector bundles. It induces the structure of a ring spectrum on MSO. The last step in Thom’s rational computation of the bordism ring is to show that the CP^{2n}, n ≥ 0 do not satisfy any polynomial identities and hence can be used as polynomial generators. This can be seen by computing their Pontrjagin numbers as in Milnor and Stasheff.

The Pontrjagin-Thom construction sends elements of π_{n+d}(S^{n}) to d-dimensional closed smooth submanifolds of S^{n+d} with a framing of the normal bundle modulo framed bordism in S^{n+d} × I. This can be discussed for arbitrary (and fixed) n, not just in the stable case where n → ∞ as in the previous lecture. For example, π_{3}(S^{2}) consists of framed 1-manifolds in S^{3} modulo bordism. Knotting and linking is not an issue because of the existence of Seifert surfaces. So one obtains normal framings on S^{1} modulo bordism, which are isomorphic to π_{1}(SO(2)) ≅ Z.

We have suspension morphism π_{3}(S^{2}) → π_{4}(S^{3}), the elements of the latter group correspond to framed 1-manifolds in S^{4} modulo bordism, which are isomorphic to π_{1}(SO(3)) = Z/2. The induced map Z → Z/2 is non-trivial because of the fibration SO(2) → SO(3) → S^{2}. By transversality, the sequence stabilizes starting from π_{4}(S^{3}).

Recall that Ω_{d}^{ξ} ≅ π_{n+d}(Th(ξ_{n})) = π_{d}(Mξ) for all n ≫ d.

Corollaries of Pontrjagin-Thom construction. (a) Ω_{d} ⊗ Q ≅ π_{d}(MSO) ⊗ Q ≅ H_{d}(MSO, Q) ≅ H_{d}(BSO, Q) ≅ Hom(H^{d}(BSO, Q), Q). Thus Pontrjagin numbers detect elements of bordism group. (b) For non-oriented bordism ring we have Z/2[x_{i}] = N_{∗} = Ω_{∗}^{O} → H_{∗}(BO, Z/2) via Stiefel-Whitney numbers. Here i ≠ 2^{k} − 1. The tool for computing π_{d}(E) → H_{d}(E, R) is Adams spectral sequence. (c) Unitary bordism: Z[a_{2n}] ≅ Ω_{∗}^{U} → H_{∗}(BU). Milnor did the computation first and later Quillen explained it via the relation to formal group laws. Rationally we have Q[CP^{n}] ≅ Ω_{∗}^{U} ⊗ Q. (d) For Spin we have K-theory Hurewicz map: Ω_{∗}^{Spin} → KO_{∗}(BSpin). Together with Stiefel-Whitney numbers, it detects spin bordism.

Spectra and (co)homology theories. Examples of spectra: (1) Thom spectra Mξ; (2) Suspension spectra Σ^{∞}X, for example S := Σ^{∞}(S^{0}). (3) Eilenberg-Mac Lane spectra HA, where A is an abelian group; (4) Ktheory spectra: KU, KO, KQ.

Theorem. Any spectrum E gives homology and cohomology theories as follows:

E_{d}(X) := [S^{d}, X ∧ E] = π_{d}(X ∧ E) = colim_{n} π_{n+d}(X ∧ E_{n})

and

E^{d}(X) := [X, E]_{−d} = colim_{n}[X ∧ S^{n}, E_{n+d}]

satisfying homotopy axiom, Mayer-Vietoris axiom, and the wedge axiom. For example, we have

Ω_{∗}^{Spin}  ≅ π_{∗}(MSpin) = MSpin_{∗}(S^{0})
↓  ↓
π_{∗}(KO ∧ MSpin) = KO_{∗}(MSpin) = MSpin_{∗}(KO)

Pushforwards in cohomology theories. We can assume that our cohomology theory comes from an Ω-spectrum, i.e., the map E_{n} → ΩE_{n+1} (the adjoint to E_{n} ∧ S^{1} → E_{n+1}) is a homeomorphism. For example, E = HA, E = K, E = KO. For E = K observe that Ω(BU × Z) = ΩBU ≅ U. If E is an Ω-spectrum, then E^{n}(X) = [X, E_{n}]. We have E^{n+k}(X ∧ S^{k}) = [X ∧ S^{k}, E_{n+k}] = [X, Ω^{k}E_{n+k} ≅ E_{n}]. Thus we can define E^{−k}(X) = E^{0}(X ∧ S^{k}), where k ≥ 0. For K-theory we have K^{0}(X) ≅ K^{2n}(X) as before and K^{1}(X) ≅ K^{2n+1}(X) = [X, U]. We have H^{0}(pt, A) = A and H^{k}(pt, A) = 0 for k ≠ 0, which characterizes ordinary cohomology theories. K-theory is extra-ordinary because we have K^{n}(pt) = K^{0}(pt) = Z for even n and K^{n}(pt) = K^{1}(pt) = 0 for odd n. As a consequence, K-theory is a 2-periodic cohomology theory.

If E is an Ω-spectrum, then a ring spectrum consists of maps E_{m} ∧ E_{n} → E_{m+n} and 1 ∈ E_{0} that are associative up to coherent higher homotopies. For example, we have a concrete model for (HA)_{n} = K(A, n) (points in S^{n} marked with elements of A). Multiplication is given by multiplying points and their labels.

Definition. Given a fibration ξ: B → BO, a ring spectrum E is called ξ-oriented if it is equipped with a natural multiplicative Thom isomorphism E^{n+k}(DV, SV) ≅ E^{k}(X) given by a ∈ E^{k}(X) ↦ π^{∗}(a) ∪ u_{E}(V) ∈ E^{n+k}(DV, SV) for some Thom classes u_{E}(V, c̃_{V}) ≅ E_{cs}^{n}(V). Here π: V → X is a vector bundle with classifying map c_{V}: X → BO(n) that is lifted to a ξ-structure c̃_{V}: X → B_{n}.

Remark. A ξ-orientation is the same structure as a ring spectrum map u: Mξ → E. In one direction, the maps u_{n}: Th(ξ_{n}) = (Mξ)_{n} → E_{n} are the universal Thom classes.

Examples of orientations. MU → MSO → HZ, MO → HZ/2, MSpin → KO, MU → MSpin^{c} → K.

Lemma. Suppose E is ξ-orientable. If M and N are smooth manifolds and f: M → N is a proper embedding with a ξ-structure on the normal bundle of f, then define f_{!}: E_{cs}^{k}(M) → E_{cs}^{k+n−m}(N) as the composition of maps E_{cs}^{k}(M) → E_{cs}^{k+n−m}(ν(f)) (Thom isomorphism, i.e., cup product with u_{E}(ν(f))) and E_{cs}^{k+n−m}(ν(f)) → E_{cs}^{k+n−m}(N) (extend by zero). Here we use the fact that E_{cs}^{n}(Z) is π_{0} of the space of maps Z → E_{n} that are equal to the basepoint of E_{n} outside of a compact set.

Lemma. Every morphism of manifolds f: M → N can be decomposed as a composition of a proper embedding and a projection: M → N × R^{s} → N, where i: M → R^{s} is a proper embedding of M into R^{s} for some large s. We now define f_{!} := (π_{1})_{!} ∘ (f × i)_{!}: E_{cs}^{k}(M) → E_{cs}^{k+s+n−m}(N × R^{n}) → E_{cs}^{k+n−m}(N). Here m = dim M, n = dim N, and (π_{1})_{!} is defined as the inverse of the Thom isomorphism (for the trivial bundle). This map is independent of the choice of i by the multiplicativity of universal Thom classes and the homotopy invariance of cohomology. It depends on a ξ-structure on the stable vector bundle −T M ⊕ f^{∗}T N. The dependence comes from the fact that a ξ-structure on ν(f × i) is the same thing as a stable ξ-structure on −T M ⊕ f^{∗}T N.

Example. If M → N is a fiber bundle, then the pushforward map is sometimes called “integration over the fibers” (literally the case for de Rham cohomology). The stable normal bundle is in this case the inverse to the tangent bundle along the fibres.

Theorem. K-theory is complex oriented, i.e., Thom classes exist for complex vector bundles.

Proof. For a complex n-dimensional vector bundle V → X we need a class u_{K}(V) ∈ K_{cs}^{2n}(V) and these classes must behave multipicatively (as for ordinary cohomology).

Theorem. K-theory is complex oriented, i.e., there are Thom isomorphisms K_{cs}(X) →^{≅} K_{cs}(V) for any complex vector bundle p: V → X given by the map a ↦ p^{∗}(a) ∪ u_{K}(V), where u_{K}(V) ∈ K_{cs}(V) is the Thom class. Here K_{cs}(Y) = K̃(Y^{∞}).

Proof. The Thom class u_{K}(v) is represented by 0 → Λ_{C}^{0}p^{∗}V → Λ_{C}^{1}p^{∗}V → ⋯ → Λ_{C}^{n}p^{∗}V → 0. All maps are exterior products with a base vector v ∈ V. Recall that this complex is exact on V \ 0. Bott periodicity implies that i_{x}^{∗}(u_{K}(v)) ∈ K_{cs}(V_{x}) ≅ Z is a generator. We have the following isomorphism given by the tensor product: ⨂^{n} K_{cs}(C) → K_{cs}(V_{x}), which is isomorphic to ⨂^{n} K̃(S^{2}) → K̃(S^{2n}), which is isomorphic to ⨂^{n} Z → Z. Here K̃(S^{2}) ≅ Z is generated by 1 − H, which is the Thom class u_{K}(C).

Definition of shriek/Gysin/pushforward/integration-over-the-fibers/wrong-way/Umkehr maps in K-theory. Let f: X → Y be a complex oriented morphism of smooth manifolds, i.e., the stable normal bundle ν(f) := (−T X) ⊕ f^{∗}(T Y) has a complex structure. Then we get a pushforward map f_{!}: K_{cs}(X) → K_{cs}(Y). Case 1: f equals p or i, where p: V → X is the projection map of a vector bundle and i is a section. The p_{!} and i_{!} come from Thom isomorphism. Note that p is not proper. Case 2: An arbitrary map f: X → Y can be decomposed as a section i: X → ν(f × j) followed by an open inclusion ν(f × j) → Y × C^{N} followed by a projection p_{1}: Y × C^{N} → Y. The pushforward map is given by the composition of individual pushforward maps. This map is independent of the choice of a proper embedding j: X → C^{N} because u_{K} is multiplicative and K is a homotopy functor.

Henceforth we denote K(X) := K_{cs}(X) for a locally compact space X. This functor is contravariant for proper continuous maps, covariant for smooth maps with complex normal bundle, and satisfies twisted Bott periodicity (Thom isomorphism): K(X) ≅ K(V) for complex vector bundles p: V → X, where a ↦ p^{∗}(a) ⊗ u_{K}(V).

Compare this to H_{cs}^{∗}(X). We have a pushforward map H_{cs}^{k}(X) → H_{cs}^{k+dim Y −dim X}(Y) for a smooth (or even continuous) map f: X → Y of oriented manifolds, which is given by taking the Poincaré dual of the pushforward in homology. Still works if only the normal bundle ν(f) is (stably) orientable.

Remark. For any oriented d-manifold X we have H^{k}(X) ≅ H_{d−k}^{lf}(X) (locally finite or Borel-Moore homology). The isomorphism is given by the pairing with the fundamental class [X]_{lf}.

Lemma 1. If p: V → X is a complex vector bundle of dimension n, then

ch(u_{K}(V)) = (−1)^{n}u_{H}(V) ∪ p^{∗}(Td(V̄)^{−1}) ∈ H_{cs}^{even}(V, Q).

Proof. Enough to show for γ^{n} → BU(n) ← BT^{n}. The direct sum L_{1} ⊕ ⋯ ⊕ L_{n} maps to γ^{n} and BT^{n}. The map i_{0}^{∗} ∘ i^{∗} maps ch(u_{K}(γ^{n})) to ∏_{1≤i≤n}(1 − exp(x_{i})), where x_{i} := c_{1}(L_{i}). The same map maps u_{H}(γ^{n}) ∪ p^{∗}(Td(γ̄^{n−1})) to x_{1} ⋯ x_{n} ∪ ∏_{1≤i≤n}(1 − exp(x_{i}))/(−x_{i}). Hence the result follows from the fact that the Todd genus corresponds to the power series x/(1 − exp(−x)).

Lemma 2. K̃(S^{2n}) ≅ K(C^{n}) →^{ch} H_{cs}^{even}(C^{n}, Q) is given by the inclusion Z → Q. The isomorphism with Q is given by the pairing with the fundamental class [C^{n}] = [C^{n}]_{lf}.

Proof. We have ∏_{1≤i≤n}(1 − H_{i}) ∈ K_{(}C^{n}) ≅ Z ∋ 1 and 1 − H_{i} ↦ S^{2}. Hence

ch ( ∏_{1≤i≤n}(1 − H_{i})) = ∏_{1≤i≤n}(1 − exp(x_{i})) = (−1)^{n} ∏_{1≤i≤n}(x_{i} + x_{i}^{2}/2 + x_{i}^{3}/6 + ⋯) = (−1)^{n} ∏_{1≤i≤n} x_{i},

which is a generator of H^{2n}(S^{2n}). Here x_{i} ∈ H^{2}(S^{2}) is a generator.

Lemma 3. If E → X is a real oriented vector bundle over an oriented manifold X, then for any α ∈ H_{cs}^{d}(X) we have ⟨p^{∗}(α) ∪ u_{H}(E), [E]_{lf}⟩ = ⟨α, [X]_{lf}⟩. Here we use the pairing H_{cs}^{d}(X) ⊗ H_{d}^{lf}(X) → Z.

For de Rham cohomology this follows from Fubini’s theorem, for singular cohomology one needs the the homological Thom isomorphism. We will not spell this out here.

Theorem. We have t-ind(σ) = π_{!}(σ), where σ ∈ K(T^{∗}X) →^{π_{!}} K(pt) and π is the constant map T^{∗}X → pt.

Proof. We identify T X and T^{∗}X using the metric. We choose a proper embedding X ⊂ R^{n} with the normal bundle ν_{0} and obtain a proper embedding of almost complex manifolds T X ⊂ C^{n} with the normal bundle ν ≅ q^{∗}(ν_{0}⊗C), where q: T X → X is the projection map of the tangent bundle. We also denote by p: ν → T X the projection map of the normal bundle. We have the following commutative diagram:

K(T X)  ⟶^{π_{!}}  K(pt)  ≅ Z
≅↓^{Th}  ≅↓^{Th}
K(ν)  \underrightarrow{extend by 0}  K(C^{n})
↓^{ch}  ↓^{ch}
H_{cs}^{even}(ν, Q) \underrightarrow{extend by 0} H_{cs}^{even}(C^{n}, Q)

Therefore π_{!}(σ) = ⟨ch(p^{∗}(σ) ∪ u_{K}(ν)), [C^{n}]⟩ = ⟨(p^{∗}(ch(σ)) ∪ ch(u_{K}(ν)), [ν]⟩ = ±⟨p^{∗}(ch(σ)) ∪ u_{H}(ν) ∪ p^{∗}(Td(ν̄)^{−1}), [ν]⟩ and hence π_{!}(σ) = ±⟨ch(σ) ∪ Td(q^{∗}(ν_{0} ⊗ C)^{−1}), [T X]⟩ = ±⟨q^{∗}(Th^{−1}(ch(σ))) ∪ u_{H}(T X) ∪ q^{∗}(Td(T X ⊗ C)), [T X]⟩ = ±⟨Th^{−1}(ch(σ)) ∪ Td(T X ⊗ C), [X]⟩ = t-ind(σ) by the above three Lemmas. In particular, Lemma 3 was applied to both bundles, p and q.

G-index theorem. Suppose a compact Lie group G acts on a closed smooth manifold X and D is a G-invariant elliptic complex. Then a-ind_{G}(D) = t-ind_{G}(D) ∈ R(G).

Here a-ind_{G}(D) := [ker D] − [cokerD] is a virtual representation of G and hence an element in R(G) and t-ind_{G}(D) := π_{!}(σ(D)) for the equivariant push-forward π_{!}: K_{G}(T^{∗}X) → K_{G}(pt) = R(G). To define f_{!}: K_{G}(Y) → K_{G}(Z) for maps with complex stable normal bundle, we first embed Y ⊂ C^{n} G-equivariantly, where C^{n} has a linear G-action and do the rest as before.

We note that every element g ∈ G gives a homomorphism R(G) → C by taking the trace of g in the given (virtual) representation. We denote the resulting complex numbers by a-ind_{G}(D, g) respectively t-ind_{G}(D, g).

Examples. Take D := d (the de Rham operator). We have a-ind_{G}(D, g) = ∑_{0≤i≤n}(−1)^{i} tr(g|_{H^{i}(X)}), where H^{i}(X) is the complex de Rham cohomology. Also t-ind_{G}(D, g) = χ(X^{g}), where X^{g} is the g-fixed set of X. This is Lefschetz fixed point formula.

Application to X = S^{n}. If χ(X^{g}) = 0, then g reverses/preserves orientation if and only if n is even/odd.

Corollary. If G is finite, then χ(X/G) = |G|^{−1} ∑_{g∈G} χ(X^{g}) = χ(X)/|G|+r, where r denotes contributions from non-trivial fixed sets.

Example. For G = Z/2 and X = S^{2} with reflection in the equator we have 1 = χ(S^{2}/(Z/2)) = 2^{−1}(2 + 0). For X = CP^{2} with complex conjugation we have χ(CP^{2}/(Z/2)) = 2^{−1}(3 + 1) = 2. Arnold showed that CP^{2}/(Z/2) is homeomorphic to S^{4}.

Proof of Corollary. By a result of Grothendieck, the projection map induces an isomorphism H^{i}(X/G) = H^{i}(X)^{G} (we are using C-coefficients here). Moreover, dim H^{i}(X)^{G} = |G|^{−1} ∑_{g∈G} tr(g|_{H^{i}(X)}) by the lemma below. Summing over all g, the Lefschetz fixed point formula gives: χ(X/G) = ∑_{0≤i≤n}(−1)^{i} dim H^{i}(X/G) =

∑_{0≤i≤n}(−1)^{i} dim H^{i}(X)^{G} = ∑_{0≤i≤n}(−1)^{i} (|G|^{−1} ∑_{g∈G} tr(g|_{H^{i}(X)})) = |G|^{−1} ∑_{g∈G} χ(X^{g}).

Lemma. If a finite group G acts on a finite dimensional vector space V (over a field of characteristic zero) then dim V^{G} = tr(N), where N(v) := |G|^{−1} ∑_{g∈G} g(v) is the norm element.

Proof. Consider the inclusion i: V^{G} → V, which is a right inverse to N. We have dim(V^{G}) = tr(id_{V^{G}}) = tr(N_{G} ∘ i) = tr(i ∘ N_{G}) = |G|^{−1} ∑_{g∈G} tr(g|_{V}).

Example. Take D := d + d^{∗} (the signature operator). We have

t-ind_{G}(D, g) = ∑_{F}⟨L(T F) ∏_{0<θ≤π} L_{θ}(N_{k}^{θ}), [F]⟩,

where F runs through all connected components of X^{g}, the normal bundle of F in X is written as ⨁_{0<θ≤π} N_{k}^{θ}, where g acts on N_{k}^{θ} by multiplication with exp(iθ) and L_{θ}(x) := (exp(iθ) exp(2x)+1)/(exp(iθ) exp(2x)−1) ∈ C[[x]].

Lemma. The homomorphisms π_{!} = π_{G}^{X}: K_{G}(T^{∗}X) → R(G) are characterized by the following properties: (A0) Functoriality for homomorphisms ϕ: G′ → G with respect to restriction maps; (A1) π_{G}^{pt} = id_{R(G)}; (A2) Functoriality for G-embeddings of closed manifolds: If f: X′ → X is a G-embedding, then T f: T X′ →

T X is proper and the composition K_{G}(T X′) →^{T f_{!}} K_{G}(T X) →^{π_{G}^{X}} R(G) equals π_{G}^{X′}.

Proof. Pick a G-embedding X → V, where V is a real orthogonal G-representation. We have an embedding j: T X → T V ≅ V ⊗_{R} C. If there are homomorphisms a_{G}^{X}: K_{G}(T X) → R(G) satisfying (A1) and (A2) then the following diagram commutes:

K_{G}(T V)
↗^{j_{!}}  ↓  ↖^{i_{!}}
K_{G}(T X) ⟶^{j_{!}} K_{G}(T V^{∞})  ⟵^{i_{!}^{∞}}  K_{G}(pt) = R(G)
↘_{a_{G}^{X}}  ↓_{a_{G}^{V ^{∞}}}  ↙_{a_{G}^{pt}=id_{R(G)}}
R(G)

The map i_{!} is an isomorphism by Bott periodicity and hence a_{G}^{X} can be computed by going clockwise around the diagram. By definition, this is the map π_{!}.

Theorem. There is an index function a_{G}^{X}: K_{G}(T X) → R(G) for any closed Riemannian manifold X and compact Lie group G such that the properties (A0), (A1) and (A2) in the Lemma above are satisfied and a_{G}^{X}(σ(D)) = a-ind_{G}(D) for any G-invariant elliptic operator D on X.

Corollary. We have a_{G}^{X} = (π_{!})_{G}^{X}, where π: T X → pt and therefore a-ind_{G}(D) = t-ind_{G}(D) for any Gequivariant elliptic complex D.

Reminder: Consider the map a_{1}^{X}: K(T X) → K(pt). K(T X) is generated by triples (E^{0}, E^{1}, α), where E_{i} → X are vector bundles and α: π^{∗}E^{0} → π^{∗}E^{1} is a morphism of bundles that is an isomorphism outside D_{ϵ}(T X). Relations: (1) isomorphism of triples; (2) homotopies of α; (3) additions of triples where α extends to an isomorphism on all of T X.

For G = 1 we have Diff(E^{0}, E^{1}) → Symb(E^{0}, E^{1}) = Hom_{S(T X)}(π^{∗}E^{0}, π^{∗}E^{1}) ∋ α. We embed Diff^{m}(E^{0}, E^{1}) ⊂ Ψ^{m}(E^{0}, E^{1}). The symbol map extends to Ψ^{m}: σ: Ψ^{m}(E^{0}, E^{1}) → Symb(E^{0}, E^{1}). The kernel of σ is Ψ^{m−1}(E^{0}, E^{1})_{S}.

Lemma 1. If D is elliptic, then D_{s}: W_{s}(E^{0}) → W_{s−m}(E^{1}) is Fredholm, i.e., invertible up to compact operators.

Proof. Pick P ∈ Ψ_{s}^{−m}(E^{1}, E^{0}) such that σ(P D − id) = 0. Thus P D − id is compact. Use Fourier transform to express D as the multiplication by the total symbol of D. For any symbol p we define a pseudo-differential operator P_{p} with symbol p as the Fourier transform of the multiplication by p.

Lemma 2. The G-invariant index is a locally constant function from G-invariant Fredholm operators to representations of G.

Lemma. If p(x, ξ) satisfies |D_{x}^{β}D_{ξ}^{α}| ≤ C_{α,β,K}(1 + |ξ|)^{m−|α|} for all x ∈ K (K is compact), then for each s ∈ Z we have P_{p} ∈ L(W_{s}(E^{0}), W_{s−m}(E^{1})) on closed manifolds X.

Definition. P^{m}(E^{0}, E^{1}) ⊂ Hom(C^{∞}E^{0}, C^{∞}E^{1}) are those operators that can be locally written as P_{p} with p(x, ξ) satsfying the above growth condition and σ_{p}(x, ξ) := lim_{λ→∞} λ^{−m}p(x, λξ) exists fro all ξ ≠ 0.

The map P^{m}(E^{0}, E^{1}) → Symb(E^{0}, E^{1}) is continuous.

Recall that P: C^{∞}(E^{0}) → C^{∞}(E^{1}) is a pseudodifferential operator of order m ∈ R if it can be locally (in X) written as P (u)(x) = (2π)^{−n} ∫_{R^{n}} exp(i(x, ξ))p(x, ξ)û(ξ)dξ, where p lies in the symbol class Symb^{m}(R^{n}, E^{0}, E^{1}) = {p ∈ Hom(π^{∗}E^{0}, π^{∗}E^{1}) | ∀α, β ∀K ⊂ R^{n} ∃C ≥ 0 ∀x ∈ K: |D_{x}^{α}D_{ξ}^{β}p(x, ξ)| ≤ C(1 + |ξ|)^{m−|β|}} (K is compact).

Properties: (a) The symbol map σ: (P^{m}/P^{m−1})(X, E^{0}, E^{1}) → (Symb^{m} / Symb^{m−1})(X, E^{0}, E^{1}) is an isomorphism that preserves compositions. (More generally, σ is an equivalence of the associated graded categories of P and Symb.) (b) For any s ∈ Z (more generally, s ∈ R) and for any P ∈ P^{m} we get an induced bounded operator P_{s}: W_{s} → W_{s−m}. In particular, the intersection of all P^{m} is the space of smoothing operators P^{−∞}. Every element of this space induces an operator P: W_{s}(E^{0}) → C^{∞}(E^{1}).

Lemma. If D ∈ Diff^{m} ⊂ P^{m} is elliptic, then σ(D) is invertible modulo Symb^{−∞}.

Definition. P ∈ P^{m} is elliptic if σ(P) is invertible modulo Symb^{−∞}.

Proof of Lemma. Outside some compact set around X ⊂ T X the inverse σ(D)^{−1} exists and lies in Symb^{−m}. We extend this to q_{0} such that σ(D)q_{0} − 1 ∈ Symb^{−∞} and q_{0}σ(D) − 1 ∈ Symb^{−∞}.

Elliptic regularity. If P is an elliptic pseudodifferential operator, then for any s ∈ R we have ker(P_{s}) ⊂ C^{∞}(E^{0}) and P_{s}(u) ∈ C^{∞}(E^{1}) implies u ∈ C^{∞}(E^{0}). Similarly for the cokernel of P.

Corollary. The index of P_{s} does not depend on s and equals the index of P.

Proof of elliptic regularity. Denote by m the order of P. Let Q_{0} ∈ P^{−m} satisfy σ(P Q_{0}) − 1 ∈ Symb^{−∞} and σ(Q_{0}P)−1 ∈ Symb^{−∞}. Consider R := 1−P Q_{0}. We know that σ_{0}(R) = 0, hence R ∈ P^{−1}. Denote by T_{n} the sum 1 + R + ⋯ + R^{n−1}. We have (1 − R)T_{n} = 1 − R^{n} ≡ 1 (mod P^{−n}). The operator Q_{n} := Q_{0}T_{n} ∈ P^{−m}. We have P Q_{n} − 1 = P Q_{0}T_{n} − 1 = (1 − R)T_{n} − 1 ∈ P^{−n}. Let u ∈ W_{s} satisfy P_{s}(u) = 0. Suppose u ∈ W_{s} satisfies P_{s}(u) = 0. Then u = ((Q_{n})_{s−m}P_{s} − 1)(u) ∈ W_{s−n}. This is true for any n, hence u ∈ C^{∞}(E^{0}). To prove the other statement we need to construct the actual parametrix, i.e., consruct T_{∞} and Q_{∞}.

Definition. For a class (E^{0}, E^{1}, α) ∈ K_{G}(T X) define a_{G}^{X}(E^{0}, E^{1}, α) := ind_{G}(P_{s}), where P ∈ P^{m} is Ginvariant satisfies σ_{m}(P)(x, ξ) = |ξ|^{m}α(x, ξ/|ξ|) for a fixed m ∈ R.

Proof of Definition. Step 1: The G-index of P_{s} is independent of P ∈ P^{m}. If σ_{m}(P) = σ_{m}(P′), then (P − P′)_{s}: W_{s} → W_{s−m} is compact. Hence ind(P_{s}) = ind(P′_{s}).

Fix a closed G-manifold X, G-vector bundles E^{0} and E^{1} on X with G-invariant metrics. We assume that G and X are compact. For any m ∈ R we have a map from the set of isomorphisms from π^{∗}E^{0} to π^{∗}E^{1} (where π: ST X → X) to Symb^{m}(E^{0}, E^{1}) that sends an isomorphism α to α_{m}, where α_{m}(x, ξ) = |ξ|^{m}α(x, ξ/|ξ|).

The index function a_{G}^{X} is the composition of the above map, the map that lifts a symbol to a pseudodifferential operator, and the composition P_{e}^{m}(E^{0}, E^{1}) → Fred(W_{s}(E^{0}), W_{s−m}(E^{0})) → R(G).

Lemma. The above construction is independent of the choice of m, P, and s.

Proof. Elliptic regularity implies that the index of P_{s} is independent of s, hence the above construction is independent from P. The symbol map σ_{m} is surjective because pseudodifferential operators modulo smoothing pseudodifferential operators form a sheaf (the Schwarz kernel of a pseudodifferential operator has singularities only on the diagonal). Suppose P ∈ P^{m} and Q ∈ P^{n}, then there is R ∈ P^{m−n} such that R = R^{∗} and σ_{m}(P) = σ_{n}(Q)σ_{m−n}(R). (We construct R by restricting to the unit sphere bundle.) The index vanishes for self-adjoint operators. Now ind(P) = ind(QR) = ind(Q) + ind(R) = ind(Q). The index function also respects equivalence relations on triples [E^{0}, E^{1}, α].

We now look at some generalizations of the index theorem. For simplicity we assume G = 1.

Example. There is an operator P: C^{∞}(S^{1}) → C^{∞}(S^{1}) such that P (z ↦ z^{n}) is z ↦ nz^{n} for n ≤ 0 and nz^{n−1} for n > 0 (here we identify S^{1} = U(1)). This is an elliptic pseudodifferential operator of index 1. Recall that K_{cs}(T S^{1}) ≅ Z.

Proof. Observe that P_{0} = −∂ is a self-adjoint differential operator of index 0. Its spectrum consists of all integers with multiplicity 1. Its symbol satisfies σ_{P_{0}}(z, ξ) = ξ. The operator P_{0}^{2} has positive spectrum (zero has multiplicity 1, squares of other positive integers have multiplicity 2). Now we take P_{1} = (P_{0}^{2})^{1/2} has eigenvalue 0 with multiplicity 1 and other positive integers with multplicity 2). The symbol of P_{1} satisfies σ_{P_{1}}(z, ξ) = |ξ|. Now define P := (z ↦ z^{−1})(P_{0} + P_{1})/2 + (P_{0} − P_{1})/2. The symbol of P satisfies σ_{P} (z, ξ) = z^{−1}ξ for ξ ≥ 0 and σ_{P} (z, ξ) = ξ for ξ ≤ 0. Hence P is elliptic and its index is 1. The kernel of P is one-dimensional and P is surjective.

If X is almost complex then K(X) ≅ K_{cs}(T X). This isomorphism pulls back a bundle to T X and multiplies it by the Thom class. Composing this isomorphism with the index map we get a map K(X) → Z, which sends W ∈ K(X) to the index of P such that σ_{m}(P) = q^{∗}W ⊗ u_{K}(T X). For complex X we can use P = ¯∂ ⊗ (W, ∇).

Corollary. If X is complex then the general Atiyah-Singer index theorem follows from twisted Hirzebruch-Riemann-Roch.

Remark. The same is true if X has a spin^{C}-structure: The map W ∈ K(X) ↦ σ_{D⊗(W,∇)} = q^{∗}W ⊗ σ_{D} ∈ K_{cs}(T X) is an isomorphism, where D: C^{∞}(S^{+}) → C^{∞}(S^{−}) is the Dirac operator.

Corollary. As above, but for twisted Dirac operators.

Let X → F → B be a bindle of smooth closed manifolds over a compact Hausdorff space B. The analytic index is now a map K(T_{|}F) → [B, Fred(H)] ≅ K(B). The topological index is a map K(T_{|}F) → K(B). The isomorphism [B, Fred(H)] → K(B) is given by the index bundle construction. If T_{|}F has a spin^{C}-structure, then we have an isomorphism K(F) → K(T_{|}F).

For spin^{C}-families the family index theorem says that two pushforward maps are equal.

The analog of the index theorem for ordinary cohomology is the following diagram, whose commutativity boils down to Fubini’s theorem:

H_{dR}^{∗}(F) ⟶^{∫_{X}} H_{dR}^{∗−d}(B)
↓≅  #  ↓≅
H^{∗}(F, R) ⟶^{p_{!}} H^{∗−d}(B, R)

Thus Fubini’s theorem is the index theorem for ordinary cohomology.

This point of view can be generalized to K-theory. Feynman-Kac formula: The value of the heat kernel exp(−tD^{2}(b))(x, y) is the integral with respect to the Wiener measrure of the parallel transport homomorphism ∥^{S}(γ), where γ: [0, t]: X is any path in X. In terms of Euclidean field theories this amounts to going from dimension 0|1 to dimension 1|1.

Non-existent index theorem: If X → F → B is family of string manifolds (p_{1}/2 = 0), then there is a non-existent push-forward for 2|1-dimensional EFTs over F to 2|1-dimensional EFTs over B using the two-dimensional Feynman integral. The index theorem should say that this push-forward is the same as the push-forward for TMF.

Theorem. The function a_{G}^{X}: K_{G}(T X) → R(G) satisfies conditions (A0), (A1), and (A2).

Corollary. We have a_{G}^{X} = π_{!}, where π: T X → pt and hence a-ind_{G} = t-ind_{G}.

Proof. Need to show (A2), i.e., for every embedding i: X → Y, where X and Y are closed G-manifolds, the

composition K_{G}(T X) ⟶^{(T i)_{!}} K_{G}(T Y) ⟶^{a_{G}^{Y}} R(G) equals the map a_{G}^{X}: K_{G}(T X) → R(G). Step 1: Excision: If U ⊂ Y is open and U ⊂ Y′ is open, then the two obvious triagles commute. Step 2: If p: V → X is a real vector bundle with the zero section i_{0}: X → V and V = P ×_{O(n)} R^{n} for some principal O(n)-bundle P → X. For α ∈ K_{G}(T X) we have a_{G}^{V}((T i_{0})_{!}α) = a_{G}^{V}(α⊗u_{K}(V ⊗C)) = a_{G}^{X}(α)a_{G×O(n)}^{R^{n}}(β) = a_{G}^{X}(α) = a_{G}^{X}(α)a_{O(n)}^{R^{n}}(β). Here β is the image of 1 under the map (T j)_{!}: R(O(n)) → K_{O(n)}(T R^{n}). Fact: If H acts freely on Z, then K_{H}(Z) ≅ K(Z/H). By normalization (Step 3) a_{O(n)}^{R^{n}}(β) = 1. The normalization axiom says that 1 is mapped to 1 under the composition R(O(n)) ⟶^{(T j)_{!}} K_{O(n)}(T R^{n}) ⟶^{a_{O(n)}^{R^{n}}} R(O(n)). Lemma: Normalization follows from multiplicativity and the following computation of a_{O(n)}^{S^{n}}(S^{n}). We have a_{G}^{X}(ρ_{X} = σ_{1}(d_{X})) = ⊕_{0≤i≤n}(−1)^{i}H_{dR}^{i}(X) ∈ R(G). Multiplicativity: Given an action of G on an H-principal bundle P → X with connection and a left H-manifold F we have a_{G}^{W} (αβ) = a_{G}^{X}(α) ⊗ a_{G×H}^{F}(β) ∈ R(G) if a_{G×H}^{F}(β) ∈ R(G) ≤ R(G × H).

Knot concordance and L^{2}-index theorem for 4-manifolds with boundary.

Consider the knot K_{n} (the n-twist knot) consisting of n full twists. K_{0} is the unknot, for n > 0 we get a non-trivial knot.

Definition. A knot in S^{3} is smoothly slice if it bounds a smooth embedded disc in S^{4}. It is topologically slice if it bounds a disc with a normal bundle.

There are topologically sliced knots that are not smoothly slice (Freedman, Donaldson).

Theorem. (a) K_{n} is algebraically slice if and onlu if 4n + 1 is a square (Casson-Gordon). (b) K_{n} is slice if and only if n = 0 or n = 2. (Fintushel-Stern, Cochran-Orr).

Remark. In higher dimensions a knot is slice if and only if it is algebraically slice.

Definition. Given a Seifert surface F for a knot K consider the Seifert pairing S_{F}: H_{1}(F) ⊗ H_{1}(F) → Z given by computing the linking number of a and b^{↑}. Here the arrow denotes the “pushing” operation.

Definition. A knot K is algebraically slice if there is a Seifert surface F such that S_{F} has a Lagrangian subspace.

Theorem (Levine, 1960s). The factormonoid of the monoid of oriented knots by the submonoid of (topologically or smoothly) slice knots is an abelian group.

Proof. Consider a knot K and its inverse −K (reversed mirror). Then K# − K is slice (actually a ribbon knot).

The groups defined above are still unknown. However, for algebraically sliced knots the group was compute by Levine: Z^{∞} × Z/2^{∞} × Z/4^{∞}. Lemma. If K is slice, then it is algebraically slice. Proof. Just draw some pictures.

Levine-Tristran signatures. For a Seifert surface F of K and ω ∈ S^{1} define the hermitian form h_{ω} = (1 − ω̄)S_{F} + (1 − ω)S_{F}^{t} and denote by σ_{ω}(K) the signature of h_{ω}. This is a piecewise-constant element of L^{∞}(S^{1}, C). (Note: The determinant of h_{ω} corresponds to the Alexander polynomial of K.)

Lemma. The function σ_{ω} does not depend on F, in fact it only depends on S^{0}(K), the 0-surgery of K. We have σ_{ω} = σ(N, C_{ω}), where N is a 4-manifold with boundary S^{0}(K) and π_{1}(N) ≅ Z and C_{ω} denotes the flat line bundle with holonomy ω.

Lemma. The Lagrangian in H_{1}(F), where ∂F = K_{m(m+1)}, is generated by γ_{m} = mh_{1} + h_{2}, where S_{F} =

( \begin{matrix}−1 & 0 \\ −1 & m(m + 1) \end{matrix} ). Remark: 4n + 1 = (2m + 1)^{2} if and only if n = m(m + 1).

Proof. Just draw some pictures.

Theorem. If K is topologically slice then any genus 1 Seifert surface contains an embedded circle γ such that (1) ⟨γ⟩ ⊂ H_{1}F is a Lagrangian; (2) ∫_{S^{1}} σ_{ω}(γ) = 0.

Lemma. The Lagrangian γ_{m} in H_{1}(F), where ∂F = K_{m(m+1)}, is an (m, m + 1)-torus knot. In particular, (Julia Bergner) ∫_{S^{1}} σ_{ω}(γ_{m}) ≠ 0 and hence K_{m(m+1)} is not topologically slice.

Signature theorem. If (W, g) is a compact Riemannian 4-manifold with product metric near ∂W = M, then (Atiyah-Patodi-Singer) σ(W) = (1/3) ∫_{W} p_{1}(W, g) − η(M, g). Here σ(W) equals the signature of (H_{2}(W, C), λ).

Proof. If D := ∗d − d∗: Ω^{even}(M) → Ω^{even}(M) is the self-adjoint signature operator, then define η(s) := ∑_{λ∈Spec(D)\{0}⊂R} sign(λ)|λ|^{−s}. This is a holomorphic function for ℜs > −1/2. Set η(M, g) := η(0).

Twisted signature theorem. If W is as above and ρ: π_{1}(W) → U(n), then σ(W, ρ) = (1/3) ∫_{W} p_{1}(W, g) − η(M, g, ρ). Here σ(W, ρ) is the signature of (H_{2}(W̃) ⊗_{π_{1}W,ρ} C^{n}, λ̃).

Corollary. The value of σ(W, ρ) − σ(W) = η(M, g, ρ|_{M}) − η(M, g) only depends on (M, ρ|_{M}).

Example. For M = S^{0}K and ρ_{ω}: π_{1} → H_{1}M = Z → U(1) we have η̃(M, ρ_{ω}) = σ_{W} (F) = σ((1 − ω̄)S_{F} + (1 − ω)S_{F}^{t}).

L^{2}-signature theorem. (Atiyah and Ramachandran.) If ρ: π_{1}W → Γ is a representation, then σ_{Γ}^{(2)}(W, ρ) = (1/3) ∫_{W} p_{1}(W, g) − η_{Γ}(M, g_{M}, ρ_{M}). Here σ_{Γ}^{(2)} is the NΓ-signature of (H_{2}(W̃ ⊗_{π_{1}W,ρ} l^{2}Γ), λ̃).

We look at the representation of the metabelian quotient π_{1}(S^{0}K)/π_{1}(S^{0}K)′′, which maps to the group Γ, which is the crossed product of Q(t)/Q[t^{±1}] and Z. Now we look at the right-translation action of Γ on l^{2}(Γ). The closure of the algebra generated by this action in the σ-weak topology is the group von Neumann algebra NΓ. Now we do everything as before except that we twist by an infinite-dimensional bundle and the dimension now refers to the real-valued dimension over type II_{1} factor.

Example. For Γ = Z we obtain σ_{Z}^{(2)}(W, ρ) = ∫_{S^{1}} σ(W, ρ_{ω}) ∈ R because the trace on NZ = L^{∞}(S^{1}) is the integration. Hence ∫_{S^{1}} σ_{ω} = η̃_{Z}^{(2)}(S^{0}K, id_{Z}).

Final steps. (1) In the setting of the theorem ∫_{S^{1}} σ_{ω}(γ) = η̃_{Γ}^{(2)}(S^{0}K, ρ_{γ}), where ρ_{γ}: π_{1}(S^{0}K) → Γ := Q(t)/Q[t^{±1}] × Z. (2) (Cochran, Orr, Teichner): If K is topologically slice, then there is L ⊂ H_{1}F such that η̃_{Γ}^{(2)}(S^{0}K, ρ_{γ}) = 0 for all γ ∈ L.