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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 15

Suppose that maps between three spheres forms a fibration. Show that \(k=n-1\).

Diagram description: \(S^k\) maps to \(S^{n+k}\) by \(\iota \), and \(S^{n+k}\) maps down to \(S^n\) by \(p\).

By giving examples, show that such a fibration is indeed possible for some \(n\).

Raw LaTeX

Suppose that maps between three spheres forms a fibration. Show that
\(k=n-1\).

\emph{Diagram description: \(S^k\) maps to \(S^{n+k}\) by \(\iota\), and \(S^{n+k}\) maps
down to \(S^n\) by \(p\).}

By giving examples, show that such a fibration is indeed possible for some \(n\).

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility