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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2020 • Question 9

Question

Math 6510 • Differentiable Manifolds Summer 2020 • Question 9

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex. Suppose that inclusion \(A\subset X\) is \(\pi _1\)-injective (for some, hence all, basepoints in \(A\)). Let \(p:\widetilde {X}\to X\) be the universal cover of \(X\). Prove that each component of \(p^{-1}(A)\) is simply connected.

Raw LaTeX

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex.
Suppose that inclusion \(A\subset X\) is \(\pi_1\)-injective (for some, hence all,
basepoints in \(A\)). Let \(p:\widetilde{X}\to X\) be the universal cover of \(X\).
Prove that each component of \(p^{-1}(A)\) is simply connected.

Topic Tags

CW Complexes and Cell Attachments Covering Spaces Fundamental Groups Homotopy Groups