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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Fall 2024 • Question 4

Question

Math 6520 • Algebraic Topology Fall 2024 • Question 4

We can regard \(\pi _1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0) \to \) \((X,x_0)\). Let \([S^1,X]\) be the set of homotopy classes of maps \(S^1 \to X\) with no condition on basepoints. Thus there is a natural map \(\psi : \pi _1(X,x_0) \to [S^1,X]\) obtained by ignoring basepoints. Show that \(\psi \) is onto if \(X\) is path connected.

Raw LaTeX

We can regard \(\pi_1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0) \to\)
\((X,x_0)\). Let \([S^1,X]\) be the set of homotopy classes of maps \(S^1 \to X\) with no condition on
basepoints. Thus there is a natural map \(\psi : \pi_1(X,x_0) \to [S^1,X]\) obtained by ignoring
basepoints. Show that \(\psi\) is onto if \(X\) is path connected.

Topic Tags

Chain Complexes and Chain Homotopy Fundamental Groups Homotopy Equivalence, Retracts, and Contractibility Homotopy Groups