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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2012 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 2012 • Question 12

Consider the disk with 2 holes \[ P=\{(x,y)\in \mathbb {R}^2\mid \lVert (x,y)\rVert \le 4,\, \lVert (x,y)-(2,0)\rVert \ge 1,\, \lVert (x,y)+(2,0)\rVert \ge 1\}. \] Orient each boundary component counterclockwise. Let \(X\) be the space obtained from \(P\) by identifying all boundary components via orientation preserving homeomorphisms. Find a presentation of \(\pi _1(X)\).

Raw LaTeX

Consider the disk with 2 holes
\[
  P=\{(x,y)\in\R^2\mid \lVert(x,y)\rVert\le4,\,
  \lVert(x,y)-(2,0)\rVert\ge1,\,
  \lVert(x,y)+(2,0)\rVert\ge1\}.
\]
Orient each boundary component counterclockwise. Let \(X\) be the space obtained from
\(P\) by identifying all boundary components via orientation preserving
homeomorphisms. Find a presentation of \(\pi_1(X)\).

Topic Tags

Free Groups, Graph Coverings, and Presentations Fundamental Groups Homotopy Groups Orientability and Volume Forms