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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2017 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2017 • Question 1

Let \(B\) be the open ball in \(\mathbb {R}^n\) that is centered at \(0\) and has radius \(1\). If \(y\in B\) and \(\lVert y\rVert <1/2\), define a vector field on \(B\) and use the resulting flow to find a diffeomorphism \(h:B\to B\) with \(h(0)=y\) such that \(h\) is homotopic to the identity map on \(B\). Conclude that if \(N\) is a smooth connected manifold and \(p,q\in N\), then there is a diffeomorphism \(H:N\to N\) such that \(H(p)=q\) and such that \(H\) is homotopic to the identity map on \(N\).

Raw LaTeX

Let \(B\) be the open ball in \(\R^n\) that is centered at \(0\) and has radius
\(1\). If \(y\in B\) and \(\lVert y\rVert<1/2\), define a vector field on \(B\) and use
the resulting flow to find a diffeomorphism \(h:B\to B\) with \(h(0)=y\) such that
\(h\) is homotopic to the identity map on \(B\). Conclude that if \(N\) is a smooth
connected manifold and \(p,q\in N\), then there is a diffeomorphism \(H:N\to N\)
such that \(H(p)=q\) and such that \(H\) is homotopic to the identity map on \(N\).

Topic Tags

Diffeomorphisms and Isotopy Methods Homotopy and Degree on Manifolds Left-Invariant Vector Fields Vector Fields and Flows