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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2020 • Question 10

Question

Math 6510 • Differentiable Manifolds Spring 2020 • Question 10

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1\times \{p\}\) in \(T\).

  1. Calculate \(\pi _1(X_n)\).
  2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) and \(\mathbb {Z}_m\)-coefficients.

Raw LaTeX

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the
torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from
\(\partial D\) to \(S^1\times\{p\}\) in \(T\).

\begin{enumerate}
\item[(a)] Calculate \(\pi_1(X_n)\).
\item[(b)] Calculate the homology and cohomology of \(X_n\) with \(\Z\) and
\(\Z_m\)-coefficients.
\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Fundamental Groups Homology with Coefficients