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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2004 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2004 • Question 1

Let \(M\subset \mathbb {R}^n\) be a compact manifold without boundary. Prove that there is \(\epsilon >0\) such that for any manifold \(X\) the following holds: any two smooth maps \[ f,g:X\to M \] with \(|f(x)-g(x)|<\epsilon \) for every \(x\in X\) are homotopic. Find an explicit \(\epsilon >0\) for the case when \(M\) is the unit sphere \(S^m\).

Raw LaTeX

Let \(M\subset\mathbb{R}^n\) be a compact manifold without boundary. Prove that
there is \(\epsilon>0\) such that for any manifold \(X\) the following holds: any two smooth
maps
\[
  f,g:X\to M
\]
with \(|f(x)-g(x)|<\epsilon\) for every \(x\in X\) are homotopic. Find an explicit
\(\epsilon>0\) for the case when \(M\) is the unit sphere \(S^m\).

Topic Tags

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