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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Summer 2022 • Question 3

Question

Math 6520 • Algebraic Topology Summer 2022 • Question 3

Let \(X\) be a CW complex consisting of one vertex p, two edges a and b attached to \(p\) along their boundaries, and two faces \(f_1\) and \(f_2\), attached along \(ab^2\) and \(ba^2\), respectively.

  1. Find \(\pi _1(X)\). Is it a finite group?
  2. Compute the homology groups \(H_i(X;\mathbb {Z})\) of \(X\) for all \(i\).

Raw LaTeX

Let \(X\) be a CW complex consisting of one vertex p, two edges a and b attached to \(p\) along their boundaries, and
two faces \(f_1\) and \(f_2\), attached along \(ab^2\) and \(ba^2\), respectively.
\begin{enumerate}

\item
Find \(\pi_1(X)\). Is it a finite group?


\item
Compute the homology groups \(H_i(X;\mathbb{Z})\) of \(X\) for all \(i\).

\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Fundamental Groups Homology with Coefficients Homotopy Groups