Section 11.7 Exercises

Show that for \(q \geq 1\) the suspension \(\Sigma \colon \pi_q(X) \to \pi_{q+1}(\Sigma X)\) of Definition 11.8 is a group homomorphism, and that it is natural, that is, \((\Sigma g)_* \circ \Sigma = \Sigma \circ g_*\) for every based map \(g \colon X \to Y\text{.}\)

Show that the degree defines an isomorphism \(\pi_n(S^n) \to \mathbb{Z}\) for \(n \geq 1\text{.}\) Deduce that two maps \(S^n \to S^n\) are homotopic if and only if they have the same degree. Hint: Use Corollary 11.12. For unbased homotopies, note that \(\pi_1(S^n)\) acts trivially on \(\pi_n(S^n)\text{.}\)

Let \(X\) be a connected CW complex. Show that \(\Sigma \colon \pi_1(X) \to \pi_2(\Sigma X)\) is surjective and factors through the abelianization \(\pi_1(X)^{\mathrm{ab}}\text{.}\) Deduce that \(\pi_2(\Sigma \mathbb{R}P^2)\) is cyclic of order at most \(2\text{.}\) (By the Hurewicz theorem, Theorem 14.2, it is \(\mathbb{Z}/2\text{.}\))

Define a spectrum \(S^{-1}\) by \((S^{-1})_0 = *\) and \((S^{-1})_n = S^{n-1}\) for \(n \geq 1\text{,}\) with structure maps the homeomorphisms \(\Sigma S^{n-1} \cong S^n\) (and the inclusion of the basepoint for \(n = 0\)). Show that \(\pi_q(S^{-1}) \cong \pi_{q+1}^s\) for all \(q \in \mathbb{Z}\text{.}\) In particular \(\pi_{-1}(S^{-1}) \cong \mathbb{Z}\text{,}\) so, unlike suspension spectra of CW complexes, spectra can have nonzero homotopy groups in negative degrees.