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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 11

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 11

  1. Let \(M\) be a closed, connected, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_2. \] Calculate the remaining homology and cohomology groups with \(\mathbb {Z}\) coefficients.
  2. Show that the Euler characteristic of any closed 3-manifold is zero.
  3. Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that \(H_1(M;\mathbb {Z})\) is infinite.

Raw LaTeX

\begin{enumerate}
\item[(a)] Let \(M\) be a closed, connected, orientable 3-manifold such that
\[
  H_1(M;\Z)=\Z\oplus\Z_2.
\]
Calculate the remaining homology and cohomology groups with \(\Z\) coefficients.
\item[(b)] Show that the Euler characteristic of any closed 3-manifold is zero.
\item[(c)] Now assume that \(M\) is a closed, non-orientable 3-manifold. Show that
\(H_1(M;\Z)\) is infinite.
\end{enumerate}

Topic Tags

Cohomology and Cup Products Euler Characteristic and Lefschetz Fixed Point Theorem Homology with Coefficients Orientability and Volume Forms