\def\pr#1.{\bigskip\noindent{\bf#1.}\enskip}
\def\sm{C^\infty}
\def\tr{\mathop{\rm tr}}
\def\End{\mathop{\rm End}}
\def\k{{\bf k}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\H{{\rm H}}
\def\K{{\rm K}}
\def\rK{\tilde\K}
\def\even{{\rm even}}
\def\odd{{\rm odd}}
\centerline{\bf Spring 2011, Math 276: Index Theory, Homework~4}
\bigskip
\noindent Please submit by whatever date you deem appropriate.
\pr Problem 9: Chern-Weil theory.
Recall that a connection on a vector bundle~$E$ over a manifold~$M$ can be seen as a degree~1
endomorphism~$\nabla$ of the complex $\Omega(M)\otimes_{\sm(M)}\sm(E)$
that satisfies Leibniz identity.
The endomorphism~$\nabla^2$ is $\sm(M)$-linear and therefore can be seen as an element of
$\Omega^2(M)\otimes_{\sm(M)}\End(E)$, which we denote by~$R(\nabla)$
and call the {\it curvature\/} of~$\nabla$.
The $n$th power of~$R(\nabla)$ can be defined as the element of~$\Omega^{2n}(M,\End(E))$
that corresponds to~$\nabla^{2n}$.
We equip $\Omega(M,\End(E))$ with the obvious trace and Lie bracket.
Finally we observe that we can substitute $R(\nabla)$ into any $f\in\k[[x]]$
in the obvious way, where $\k$ is the field of coefficients (i.e., real or complex numbers).
\item{(a)}
Prove that for any $f\in\k[[x]]$ the form $\tr(f(R(\nabla)))$ is closed.
Prove that if $\nabla'$ is a different connection on~$E$,
then the difference $\tr(f(R(\nabla)))-\tr(f(R(\nabla')))$ is exact.
Conclude that any $f\in\k[[x]]$ gives a cohomology class that does not depend on the choice
of~$\nabla$.
Hint: Connections form an affine space and any two connections can be connected by a path.
\item{(b)}
Prove that the formal powers series $\exp(ix/2\pi)\in\C[[x]]$ gives the Chern character.
Prove that the total Chern class is the exponent of the class given by the formal
power series $\log(1+ix/2\pi)\in\C[[x]]$.
Prove that the total Pontryagin class is the exponent of the class given by the formal
power series $\log((1-(x/2\pi)^2)^{1/2})\in\R[[x]]$.
Is there a power series that gives the Euler class?
\pr Problem 10: Chern characters and the topological index of operators on trivial line bundles.
\item{(a)}
Define the {\it odd Chern character\/} as the composition of maps
$\K^\odd(X)\to\rK^\even(SX)\to\H^\even(SX)\to\H^\odd(X)$.
Prove that Chern characters combine into a homomorphism of graded rings
$\K^{\even/\odd}(X)\to\H^{\even/\odd}(X)$.
Extend Chern character to the relative setting and prove that
the diagram consisting of the long exact sequences for K-theory and ordinary cohomology
connected by Chern character is commutative.
\item{(b)}
Use part~(a) to prove that the topological index of every elliptic operator
from any trivial line bundle to itself vanishes.
\bye