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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 7

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 7

Let \(X\) be the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(1/n\). Show \(X\) doesn’t have a universal cover (show that every connected covering space of \(X\), including \(X\) itself, has non-trivial \(\pi _1\)).

Raw LaTeX

Let \(X\) be the union of circles in \(\R^2\) with centers at
\((0,\frac{1}{2n})\) and radius \(1/n\). Show \(X\) doesn't have a universal cover
(show that every connected covering space of \(X\), including \(X\) itself, has
non-trivial \(\pi_1\)).

Topic Tags

Classification of Covering Spaces Covering Spaces Fundamental Groups Homotopy Groups