Skip to main content

Section 10.6 Exercises

Exercise 10.28.

Show that the composition of two (Serre or Hurewicz) fibrations is a fibration.

Exercise 10.29.

Let \(p\colon E\to B\) be a fibration and \(g\colon A\to B\) any map. Show that the projection \(A\times_B E=\{(a,x)\in A\times E: g(a)=p(x)\}\to A\) is a fibration, and that its fibre over \(a\) is homeomorphic to the fibre of \(p\) over \(g(a)\text{.}\) Deduce that the restriction \(p^{-1}(U)\to U\) to any subspace \(U\subseteq B\) is a fibration.

Exercise 10.30.

Show that if \(p\colon E\to B\) and \(p'\colon E'\to B'\) are fibrations, then so is \(p\times p'\colon E\times E'\to B\times B'\text{.}\) In particular, the projection \(B\times F\to B\) is a fibration for every space \(F\text{.}\)

Exercise 10.31.

Let \(p\colon E\to B\) be a Serre fibration with \(B\) path-connected and \(E\neq\varnothing\text{.}\)
  1. Show that \(p\) is surjective.
  2. Show that if one fibre \(F=p^{-1}(b)\) is weakly contractible, then \(p\) is a weak homotopy equivalence.
  3. Show that \(p\colon[0,1]\to[0,1]\text{,}\) \(p(t)=\max(0,2t-1)\text{,}\) is not a Serre fibration, although all of its fibres are contractible.

Exercise 10.32.

The antipodal action of \(\mathbb{Z}/2\) on \(S^\infty\) gives a covering, and hence a fibre bundle, \(\mathbb{Z}/2\to S^\infty\to\mathbb{R}P^\infty\text{.}\) Use it to compute \(\pi_n(\mathbb{R}P^\infty)\) for all \(n\geq1\text{.}\)

Exercise 10.33.

Complex projective spaces.
  1. Use the bundle \(S^1\to S^{2n+1}\to\mathbb{C}P^n\) of Example 10.15 to show that \(\pi_1(\mathbb{C}P^n)=0\text{,}\) \(\pi_2(\mathbb{C}P^n)\cong\mathbb{Z}\text{,}\) and \(\pi_i(\mathbb{C}P^n)\cong\pi_i(S^{2n+1})\) for \(i\geq3\text{.}\)
  2. Show that \(S^2\) and \(S^3\times\mathbb{C}P^\infty\) have isomorphic homotopy groups in every degree, but are not homotopy equivalent.

Exercise 10.34.

Show that the inclusion of the fibre \(S^3\hookrightarrow S^7\) of the quaternionic Hopf fibration (Example 10.17) is null-homotopic. Deduce that \(\pi_n(S^7)\to\pi_n(S^4)\) is injective for every \(n\text{,}\) and hence (using \(\pi_7(S^7)\cong\mathbb{Z}\text{,}\) Corollary 11.12) that \(\pi_7(S^4)\) is infinite.
PrevTopNext