\def\pr#1.{\bigskip\noindent{\bf#1.}\enskip}
\def\sm{C^\infty}
\def\L{{\rm L}}
\def\K{{\rm K}}
\def\rK{\tilde\K}
\def\T{{\bf T}}
\def\C{{\bf C}}
\def\GL{{\rm GL}}
\def\CP{{\bf CP}}

\centerline{\bf Spring 2011, Math 276: Index Theory, Homework~2}
\bigskip
\leftline{Please submit by February~8 and contact Dmitri Pavlov (pavlov@math) for all questions about homework.}

\pr 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 $a\colon E\to E$ and $b\colon F\to F$ are isomorphic
if there is an isomorphism $c\colon E\to F$ such that $ca=bc$.
Moreover, the sum of $a$~and~$b$ is $a\oplus b\colon E\oplus F\to E\oplus F$.
An automorphism $a\colon E\to E$ is {\it elementary\/} if it is homotopic
to the identity in the space of all automorphisms of~$E$.
\item{(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?
\item{(b)} Suppose $Y\subset X$ is closed.
Define $\K^{-1}(X,Y)$ in exactly the same way as $\K^{-1}(X)$
except that all automorphisms $a\colon E\to E$ 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.
\item{(c)} Define natural maps $\K^{-1}(Y)\to\K^0(X,Y)$
such that the following 6-term sequence is exact:
$$\K^{-1}(X,Y)\to\K^{-1}(X)\to\K^{-1}(Y)\to\K^0(X,Y)\to\K^0(X)\to\K^0(Y).$$
\item{(d)} Give an alternative definition of $\K^0(X,Y)$ in terms of
chain complexes $0\to E_0\to E_1\to\cdots\to E_k\to0$ 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.

\pr Problem~5: Computing~$\K(X)$.
\item{(a)} Suppose that $Y\to X$ is a cofibration of compact Hausdorff spaces.
Prove that $\K^i(X,Y)=\rK^i(X/Y)$ for $i\in\{0,-1\}$.
Here $\rK^{-1}=\K^{-1}$.
\item{(b)}
Suppose $X$ is a compact Hausdorff space.
Prove that $\K^{-1}(X)\cong\rK(SX)\cong[X,\GL_\infty]$,
where $SX$ is the {\it unreduced suspension\/}
of~$X$, which is obtained from the space $X\times[0,1]$
by collapsing $X\times\{0\}$ and $X\times\{1\}$ to points.
Here $\GL_\infty$ is the colimit of groups $\GL(\C^n)$,
where inclusions $\GL(\C^m)\to\GL(\C^n)$ are given by $a\mapsto{a\,0\choose0\,1}$.
\item{(c)} Compute $\K^0$ and $\K^{-1}$ for the complex $n$-dimensional projective spaces~$\CP^n$,
wedges $S^m\vee S^n$ and products $S^m\times S^n$ of spheres.

\pr Problem~6: Sobolev spaces have well defined topologies and the second extreme.
\item{(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~$(\psi,U)$ indexed by a set~$I$
together with embeddings $\phi_i\colon U_i\to\T^d$
and equip $C^k(M)$ with the norm $f\in C^k(M)\mapsto\|f\|_k
=\sum_{i\in I}\|(\psi_if)\circ\phi_i^{-1}\|_k$,
where the Sobolev norm on $C^k(\T^d)$
is $f\in C^k(\T^d)\mapsto\|f\|_k=\sum_{|r|\le k}\sup|\partial_rf|$.
Prove that all norms on $C^k(M)$ induced by different
choices of $(\psi,U,\phi)$ are equivalent to each other and hence
define the same topology on~$C^k(M)$.
\item{(b)} The second extreme: Consider an order~0 differential operator~$D\colon\sm(M)\to\sm(M)$
on the trivial line bundle on a smooth compact manifold~$M$
given by the multiplication by a function~$f\in\sm(M)$.
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~$\L^2(M)$: $\hat D\colon\L^2(M)\to\L^2(M)$.
Discuss which of these operators are Fredholm and compute their index.

\bye