Cohomology ring of a Hopf attachment with a sphere summand ($X_{d,f}=(S^2\vee S^3){\cup }_{d\eta +f{\iota }_3}{D}^4$)

For the Hopf map $\eta$, the indicated four-cell attachment has cellular boundary multiplication by $f$ from dimension four to three. Its integral cohomology is $\mathbb{Z}$ in degrees zero and two, $\ker (f:\mathbb{Z}\to \mathbb{Z})$ in degree three, and $\mathbb{Z}/f\mathbb{Z}$ in degree four. If $x$ generates degree two and $z$ is the four-cell cochain class, then $x^2=dz$, with every other positive-degree product zero. Collapsing the three-sphere reduces the cup-square calculation to the Hopf invariant; the induced map on degree-four cohomology reduces its integer coefficient modulo $f$. When $f=0$ there is an extra free degree-three class, whose products still vanish by dimension.

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 ${\pi }_3(X_{d,f})={\mathbb{Z}}^2/⟨(d,f)⟩$. Smith normal form gives the displayed expression when $d>0$, including $f=0$.