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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 12

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 12

  1. Let \(M\) be a closed, orientable 3-manifold such that \[ H_1(M;\mathbb {Z})=\mathbb {Z}\oplus \mathbb {Z}_2. \] Calculate the remaining homology and cohomology groups for \(M\) with \(\mathbb {Z}\) coefficients.
  2. 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, orientable 3-manifold such that
\[
  H_1(M;\Z)=\Z\oplus\Z_2.
\]
Calculate the remaining homology and cohomology groups for \(M\) with
\(\Z\) coefficients.
\item[(b)] 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 Fundamental Groups Homology with Coefficients Orientability and Volume Forms