= Third homotopy group of a Hopf attachment with a sphere summand
{title2=$\pi_3(X_{d,f})\cong\mathbb Z\oplus\mathbb Z/\gcd(d,f)$}
Map $X_{d,f}$ to $K(\mathbb Z,2)=\mathbb{CP}^\infty$ using its degree-two generator. The <homotopy fibre> is the total space of the corresponding <circle bundle>, is 2-connected, and has $\pi_3$ equal to that of $X_{d,f}$. Over $S^2\vee S^3$ its total space is $P=S^3\cup_{S^1}(S^1\times S^3)$, with $H_3(P)=\mathbb Z A\oplus\mathbb Z B$. The four-cell bundle is trivial; its relative degree-four generator has boundary $dA+fB$, the lifted attaching map. The <long exact sequence in relative homology> and <Hurewicz theorem> therefore give $\pi_3(X_{d,f})=\mathbb Z^2/\langle(d,f)\rangle$. <Smith normal form> gives the displayed expression when $d>0$, including $f=0$.
Back to article page
