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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2005 • Question 15

Question

Math 6510 • Differentiable Manifolds Winter 2005 • Question 15

Construct examples of CW-complexes \(X\) with each of the following properties.

  1. For some \(n\), \(H_n(X)\not \cong H^n(X;\mathbb {Z})\).
  2. For some \(n\), \(H_n(X;\mathbb {Z}_2)\ne 0\) and \(H_n(X;\mathbb {Z}_3)\ne 0\) but \(H_n(X;\mathbb {Z}_5)=0\).

Raw LaTeX

Construct examples of CW-complexes \(X\) with each of the following properties.
\begin{enumerate}
\item For some \(n\), \(H_n(X)\not\cong H^n(X;\mathbb{Z})\).
\item For some \(n\), \(H_n(X;\mathbb{Z}_2)\ne0\) and \(H_n(X;\mathbb{Z}_3)\ne0\) but
\(H_n(X;\mathbb{Z}_5)=0\).
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Chain Complexes and Chain Homotopy Cohomology and Cup Products Homology with Coefficients