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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Fall 2023 • Question 2

Question

Math 6520 • Algebraic Topology Fall 2023 • Question 2

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 \(\mathbb {Z}\) 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

Calculate \(\pi_1(X_n)\).



\item
Calculate the homology and cohomology of \(X_n\) with \(\mathbb{Z}\) coefficients.

\end{enumerate}

Topic Tags

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