Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2010 • Question 19

Question

Math 6510 • Differentiable Manifolds Spring 2010 • Question 19

Let \(X\) be a cell complex obtained from the circle by attaching a 2-cell with a degree 3 attaching map. Prove that \(X\) cannot be embedded into \(\mathbb {R}^3\). You may assume that the embedding is such that there exists a “regular neighborhood”, i.e. a smooth compact manifold \(N\subset \mathbb {R}^3\) such that \(X\hookrightarrow N\) is a homotopy equivalence.

Raw LaTeX

Let \(X\) be a cell complex obtained from the circle by attaching a 2-cell
with a degree 3 attaching map. Prove that \(X\) cannot be embedded into \(\R^3\). You
may assume that the embedding is such that there exists a ``regular
neighborhood'', i.e. a smooth compact manifold \(N\subset\R^3\) such that
\(X\hookrightarrow N\) is a homotopy equivalence.

Topic Tags

CW Complexes and Cell Attachments Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds