Spring 2011, Math 276: Index Theory, Homework 2

Please submit by February 8 and contact Dmitri Pavlov (pavlov@math) for all questions about homework.

Problem 4: The 6-term exact sequence. Suppose X is a compact Hausdorff space. In this problem we consider automorphisms of vector bundles over X. Two automorphism a: E → E and b: F → F are isomorphic if there is an isomorphism c: E → F such that ca = bc. Moreover, the sum of a and b is a ⊕ b: E ⊕ F → E ⊕ F. An automorphism a: E → E is elementary if it is homotopic to the identity in the space of all automorphisms of E.

(a) Prove that the quotient K^{−1}(X) of the commutative monoid of isomorphism classes of automorphisms of vector bundles over X by the submonoid of isomorphism classes of elementary automorphisms is a commutative group. Describe the negation map of this group. Prove that homotopic automorphisms represent the same element in this group. Is the converse of this statement true?

(b) Suppose Y ⊂ X is closed. Define K^{−1}(X, Y) in exactly the same way as K^{−1}(X) except that all automorphisms a: E → E of a vector bundle E on X must restrict to the identity automorphism on Y and all homotopies must stay in the space of such automorphisms. Prove the analogues of the statements in part (a) for these relative groups.

(c) Define natural maps K^{−1}(Y) → K^{0}(X, Y) such that the following 6-term sequence is exact:

K^{−1}(X, Y) → K^{−1}(X) → K^{−1}(Y) → K^{0}(X, Y) → K^{0}(X) → K^{0}(Y).

(d) Give an alternative definition of K^{0}(X, Y) in terms of chain complexes 0 → E_{0} → E_{1} → ⋯ → E_{k} → 0 of vector bundles over X that are exact over Y. Hint: Combine E_{i} with the same parity of i together and choose hermitian inner products.

Problem 5: Computing K(X).

(a) Suppose that Y → X is a cofibration of compact Hausdorff spaces. Prove that K^{i}(X, Y) = K̃^{i}(X/Y) for i ∈ {0, −1}. Here K̃^{−1} = K^{−1}.

(b) Suppose X is a compact Hausdorff space. Prove that K^{−1}(X) ≅ K̃(SX) ≅ [X, GL_{∞}], where SX is the unreduced suspension of X, which is obtained from the space X × [0, 1] by collapsing X × {0} and X × {1} to points. Here GL_{∞} is the colimit of groups GL(C^{n}), where inclusions GL(C^{m}) → GL(C^{n}) are given by a ↦ \binom{a 0}{0 1}.

(c) Compute K^{0} and K^{−1} for the complex n-dimensional projective spaces CP^{n}, wedges S^{m} ∨ S^{n} and products S^{m} × S^{n} of spheres.

Problem 6: Sobolev spaces have well defined topologies and the second extreme.

(a) Suppose M is a compact d-dimensional smooth manifold. Recall that one way to define Sobolev spaces of M is to choose a partition of unity (ψ, U) indexed by a set I together with embeddings ϕ_{i}: U_{i} → T^{d} and equip C^{k}(M) with the norm f ∈ C^{k}(M) ↦ ∥f∥_{k} = ∑_{i∈I} ∥(ψ_{i}f) ∘ ϕ_{i}^{−1}∥_{k}, where the Sobolev norm on C^{k}(T^{d}) is f ∈ C^{k}(T^{d}) ↦ ∥f∥_{k} = ∑_{|r|≤k} sup |∂_{r}f|. Prove that all norms on C^{k}(M) induced by different choices of (ψ, U, ϕ) are equivalent to each other and hence define the same topology on C^{k}(M).

(b) The second extreme: Consider an order 0 differential operator D: C^{∞}(M) → C^{∞}(M) on the trivial line bundle on a smooth compact manifold M given by the multiplication by a function f ∈ C^{∞}(M). Suppose that all zeros of f are isolated. Compute the kernel and the cokernel of D. Do the same for the extension of D to L^{2}(M): D̂: L^{2}(M) → L^{2}(M). Discuss which of these operators are Fredholm and compute their index.