User:Jcmckeown/sandbox

There are homotopy fiber sequences

$$ \mathbb{S}^{2n-1} \to B U(n-1) \to B U(n) $$

Concretely, a point of $$BU(n-1)$$ is given by a point of the base space $$BU(n)$$ classifying a complex vector space $$V$$, together with a unit vector $$u$$ in $$V$$; together they classify $$ u^\perp < V $$ while the splitting $$V = (\mathbb{C} u) \oplus u^\perp $$, trivialized by $$u$$, realizes the map $$ B U(n-1) \to B U(n) $$ representing direct sum with $$\mathbb{C}$$.

Applying the Gysin Sequence, one has a long exact homology sequence

$$ H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\to} H^{p+2n} ( BU(n) ) \overset{j^*}{\to} H^{p+2n} (BU(n-1)) \overset{\partial}{\to} H^{p+1}(BU(n)) \dots $$

where $$\eta$$ is the fundamental class of the fiber $$\mathbb{S}^{2n-1}$$. By properties of the Gysin Sequence, $$j^*$$ is a multiplicative homomorphism; by induction, $$H^*BU(n-1)$$ is generated by elements with $$ p < -1 $$, where $$\partial$$ must be zero, and hence where $$j^*$$ must be surjective. It follows that $$j^*$$ must always be surjective, and hence that $$\smile d_{2n}\eta $$ must always be injective. On the other hand, $H^{p+2n}(BU(n-1))$ is a free module, and so the short exact sequence

$$ 0 \to H^p ( BU(n) ) \overset{\smile d_{2n} \eta}{\to} H^{p+2n} ( BU(n) ) \overset{j^*}{\to} H^{p+2n} (BU(n-1)) \to 0 $$

splits. We therefore have $$H^*(BU(n)) = H^*(BU(n-1))[c_{2n}]$$ where $$c_{2n} = d_{2n} \eta$$.