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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 16

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 16

Let \(M\) be a closed 4-manifold, \(S\) a surface with \(\chi (S)\le 0\) and \(p:M\to S\) a fiber bundle. Show that there exists a closed surface \(F\) with \(\pi _n(M)=\pi _n(F)\) for \(n\ge 2\).

Raw LaTeX

Let \(M\) be a closed 4-manifold, \(S\) a surface with \(\chi(S)\le0\) and
\(p:M\to S\) a fiber bundle. Show that there exists a closed surface \(F\) with
\(\pi_n(M)=\pi_n(F)\) for \(n\ge2\).

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups Tangent Bundles and Vector Bundles