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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 12

Let \(U(n)\) denote the set of unitary matrices (the complex \(n\times n\) matrices \(A\) which satisfy \(A\overline {A}^{T}=I\)). Show that \(U(n-1)\to U(n)\to S^{2n-1}\) is a fiber bundle. (Hint 1: Find explicit local trivializations. Hint 2: State a general theorem whose conclusion is that a smooth map between closed manifolds is a submersion and verify the hypotheses.)

Deduce that \(\pi _r(U(n))\simeq \pi _r(U(n+1))\) for \(r<2n\). What are \(\pi _1(U(3))\), \(\pi _2(U(3))\), \(\pi _3(U(3))\)?

Raw LaTeX

Let \(U(n)\) denote the set of unitary matrices (the complex \(n\times n\) matrices
\(A\) which satisfy \(A\overline{A}^{T}=I\)). Show that
\(U(n-1)\to U(n)\to S^{2n-1}\) is a fiber bundle. (Hint 1: Find explicit local
trivializations. Hint 2: State a general theorem whose conclusion is that a smooth
map between closed manifolds is a submersion and verify the hypotheses.)

Deduce that \(\pi_r(U(n))\simeq\pi_r(U(n+1))\) for \(r<2n\). What are
\(\pi_1(U(3))\), \(\pi_2(U(3))\), \(\pi_3(U(3))\)?

Topic Tags

Fundamental Groups Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles