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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2010 • Question 14

Question

Math 6510 • Differentiable Manifolds Fall 2010 • Question 14

  1. State the universal coefficient theorem for cohomology.
  2. Suppose \(X\) is a space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=\mathbb {Z}/2\mathbb {Z}\) and \(H_i(X)=0\) for \(i>1\). Compute \(H^i(X;\mathbb {Z}/2\mathbb {Z})\) for all \(i\).

Raw LaTeX

\begin{enumerate}
\item[(a)] State the universal coefficient theorem for cohomology.
\item[(b)] Suppose \(X\) is a space such that \(H_0(X)=\Z\),
\(H_1(X)=\Z/2\Z\) and \(H_i(X)=0\) for \(i>1\). Compute \(H^i(X;\Z/2\Z)\) for all \(i\).
\end{enumerate}

Topic Tags

Cohomology and Cup Products Fundamental Groups Homology with Coefficients Universal Coefficient Theorem