= Solution
Let $R=MU^*(\mathrm{pt})$ be the <coefficient ring of complex cobordism>. We use its standard coefficient theorem: $R\cong\mathbb Z[b_1,b_2,\ldots]$ with $|b_j|=-2j$. In particular $R$ is concentrated in nonpositive even degrees. Let $H$ be the <quaternionic tautological line bundle> on $\mathbb{HP}^2$, regarded as a complex rank-two bundle by right multiplication by a fixed imaginary quaternion. Set
$$
q=c_2^{MU}(H)\in MU^4(\mathbb{HP}^2).
$$
We will prove
$$
\boxed{MU^*(\mathbb{HP}^2)\cong R[q]/(q^3),\qquad |q|=4.}
$$
The argument must establish the multiplication as well as the additive groups, since collapse of a <spectral sequence> alone does not settle ring extensions.
The <quaternionic projective space> has one cell in each dimension $0,4,8$, giving integral <cohomology> $\mathbb Z$ in exactly these three degrees. The <sphere bundle> of $H$ is the quaternionic Hopf fibration
$$
S^3\longrightarrow S^{11}\longrightarrow\mathbb{HP}^2.
$$
Write $u=e(H_{\mathbb R})=c_2(H)$ for its ordinary degree-four class, fixing its sign by this equality. The <Gysin sequence of a sphere bundle> gives surjections, in fact isomorphisms,
$$
H^0(\mathbb{HP}^2;\mathbb Z)\xrightarrow{\ \smile u\ }H^4(\mathbb{HP}^2;\mathbb Z),
\qquad
H^4(\mathbb{HP}^2;\mathbb Z)\xrightarrow{\ \smile u\ }H^8(\mathbb{HP}^2;\mathbb Z).
$$
Indeed the corresponding neighboring <cohomology> groups of $S^{11}$ vanish. Both source and target groups are $\mathbb Z$, so these maps make $u$ and $u^2$ generators. There are no higher <cohomology> groups. This proves the <cohomology ring of quaternionic projective space> in the form $H^*(\mathbb{HP}^2;\mathbb Z)=\mathbb Z[u]/(u^3)$.
Apply the multiplicative <Atiyah-Hirzebruch spectral sequence> for <complex cobordism>:
$$
E_2^{s,t}=H^s(\mathbb{HP}^2;R^t)\Longrightarrow MU^{s+t}(\mathbb{HP}^2),
\qquad d_r:E_r^{s,t}\to E_r^{s+r,t-r+1}.
$$
The skeletal construction obtains this sequence from the <long exact sequences> of successive skeleta; the quotients are wedges of spheres, and the first <differential> is the ordinary cellular <differential> with coefficient groups $R^t$. Its multiplication on the associated graded group is the ordinary <cup product> with these coefficients. The finite eight-dimensional skeleton makes the filtration finite, so there is no convergence ambiguity.
Only columns $s=0,4,8$ and even rows $t$ are nonzero. If $r$ is odd, the target column $s+r$ is odd and zero. If $r$ is even, the target coefficient degree $t-r+1$ is odd and zero. The same observations exclude incoming <differentials>. Consequently $E_\infty=E_2$, with associated graded $R$-module freely generated by $1,u,u^2$ in filtrations $0,4,8$.
The canonical map from <complex cobordism> to integral <cohomology> preserves the chosen complex orientation and hence sends $q$ to $u$. The class $q$ vanishes on the zero-dimensional skeleton since $R^4=0$, and its leading skeletal term is $u$ in filtration four. Multiplicativity makes the leading term of $q^2$ the generator $u^2$ in filtration eight. Thus $1,q,q^2$ lift the associated graded basis.
This resolves all module extensions explicitly. For a class of total degree $d$, subtract its coefficient $a_0\in R^d$ in filtration zero. From the remainder subtract $a_1q$, where $a_1\in R^{d-4}$ is its leading coefficient in filtration four, and then subtract $a_2q^2$ with $a_2\in R^{d-8}$. The remaining filtration is beyond dimension eight and is zero. Conversely a relation $a_0+a_1q+a_2q^2=0$ has zero leading coefficient in each of the three successive filtrations, forcing all three $a_j$ to vanish. Therefore
$$
MU^*(\mathbb{HP}^2)=R\,1\oplus R\,q\oplus R\,q^2.
$$
Finally $q^3$ has degree twelve. Every associated graded contribution in that total degree would have coefficient degree $12-s>0$ for $s=0,4,8$, and all such coefficients vanish. Thus $MU^{12}(\mathbb{HP}^2)=0$ and $q^3=0$. The polynomial map $R[t]/(t^3)\to MU^*(\mathbb{HP}^2)$ taking $t$ to $q$ is surjective by the basis argument and injective by its uniqueness. This proves the <complex cobordism ring of the quaternionic projective plane> as a graded ring over the stated coefficient ring, including the absence of any additional coefficient-dependent relation.
Back to article page
