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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 12

Let \(X_p=D^4\cup _f S^3\) be the space obtained by attaching a 4-cell to the 3-sphere by an attaching map \(f:\partial D^4\to S^3\) of degree \(p\).

  1. Compute the singular homology of \(H_*(X_p;\mathbb {Z})\) and \(H_*(X_p\times X_q;\mathbb {Z})\) for any \((p,q)\) using the CW-complex cell structure.
  2. Find \(H_*(X_p\times X_q;\mathbb {Z})\) using the Künneth Theorem.

Raw LaTeX

Let \(X_p=D^4\cup_f S^3\) be the space obtained by attaching a 4-cell to the
3-sphere by an attaching map \(f:\partial D^4\to S^3\) of degree \(p\).
\begin{enumerate}
\item Compute the singular homology of \(H_*(X_p;\mathbb{Z})\) and
\(H_*(X_p\times X_q;\mathbb{Z})\) for any \((p,q)\) using the CW-complex cell structure.
\item Find \(H_*(X_p\times X_q;\mathbb{Z})\) using the K\"unneth Theorem.
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Degree Theory Homology with Coefficients Homotopy and Degree on Manifolds