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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2014 • Question 8

Question

Math 6510 • Differentiable Manifolds Spring 2014 • Question 8

Let \(X\) be the space obtained from the disjoint union of a circle \(S^1\) and a cylinder \(S^1\times [0,1]\) by attaching the cylinder along its boundary to the circle with the attaching map on \(S^1\times \{0\}\) having degree 2 and on \(S^1\times \{1\}\) degree 3. Find a presentation of the fundamental group of \(X\).

Raw LaTeX

Let \(X\) be the space obtained from the disjoint union of a circle \(S^1\) and
a cylinder \(S^1\times[0,1]\) by attaching the cylinder along its boundary to the
circle with the attaching map on \(S^1\times\{0\}\) having degree 2 and on
\(S^1\times\{1\}\) degree 3. Find a presentation of the fundamental group of \(X\).

Topic Tags

CW Complexes and Cell Attachments Free Groups, Graph Coverings, and Presentations Fundamental Groups Homotopy and Degree on Manifolds