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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Fall 2023 • Question 1

Question

Math 6520 • Algebraic Topology Fall 2023 • Question 1

Let \(a\) and \(b\) be the two generators of \(\pi _1(S^1 \vee S^1)\) corresponding to the two \(S^1\) summands. Draw a picture of the covering space of \(S^1 \vee S^1\) corresponding to the normal subgroup generated by \(a^2\), \(b^2\), and \((ab)^4\), and prove this covering space is indeed the correct one.

Raw LaTeX

Let \(a\) and \(b\) be the two generators of \(\pi_1(S^1 \vee S^1)\) corresponding to the
two \(S^1\) summands. Draw a picture of the covering space of \(S^1 \vee S^1\)
corresponding to the normal subgroup generated by \(a^2\), \(b^2\), and \((ab)^4\),
and prove this covering space is indeed the correct one.

Topic Tags

Classification of Covering Spaces Covering Spaces Fundamental Groups Homotopy Groups