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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2006 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2006 • Question 10

For a connected CW complex \(X\), call a connected covering space \(\widetilde {X}\to X\) abelian if it is normal and has abelian deck transformation group. Show that \(X\) has an abelian covering space that is a covering space of every other abelian covering space of \(X\), and that such ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for \(X=S^1\vee S^1\). Carefully state theorems you are using.

Raw LaTeX

For a connected CW complex \(X\), call a connected covering space
\(\widetilde{X}\to X\) abelian if it is normal and has abelian deck transformation group.
Show that \(X\) has an abelian covering space that is a covering space of every other
abelian covering space of \(X\), and that such `universal' abelian covering space is
unique up to isomorphism. Describe this covering space explicitly for
\(X=S^1\vee S^1\). Carefully state theorems you are using.

Topic Tags

CW Complexes and Cell Attachments Classification of Covering Spaces Covering Spaces Free Groups, Graph Coverings, and Presentations