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\centerline{\bf Spring 2011, Math 276: Index Theory, Homework~3}
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\noindent Please submit by whatever date you deem appropriate.

\pr Problem 7: The Euler class.
\item{(a)}
Suppose $E$ is an oriented real vector bundle
over a closed oriented manifold~$M$
and $s$ is a section of this bundle that is transversal to the zero section.
Prove that the Poincar\'e dual of the manifold of zeroes of~$s$ is equal to~$e(E)$.
Discuss the statement in the absence of orientations.
\item{(b)}
Find an oriented vector bundle with vanishing Euler class
but without a non-vanishing section.

\pr Problem 8: Topological index on odd-dimensional manifolds.
Prove that the topological index of any elliptic differential
operator on an odd-dimensional manifold~$M$ is zero.
Hint: Consider the involution on the total space of~$T^*M$ given by negation.

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