Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 10

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 10

Let \(Y\subset \mathbb {R}^n\) be a compact manifold without boundary. Prove that there is \(\epsilon >0\) with the following property. If \(f,g:X\to Y\) are smooth maps defined on a manifold \(X\) and if \(|f(x)-g(x)|<\epsilon \) for every \(x\in X\), then \(f\) and \(g\) are homotopic.

Raw LaTeX

Let \(Y\subset\mathbb{R}^n\) be a compact manifold without boundary. Prove that
there is \(\epsilon>0\) with the following property. If \(f,g:X\to Y\) are smooth maps
defined on a manifold \(X\) and if \(|f(x)-g(x)|<\epsilon\) for every \(x\in X\), then \(f\) and
\(g\) are homotopic.

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds