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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added the following statement; which appears as Lemma 2.1 in
Let
S4⟶BSpin(4)↓πBSpin(5)be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space S4≃Spin(5)/Spin(4) (this Prop.).
Then the fiber integration of the triple cup power of the Euler class χ∈H4(BSpin(4),ℤ) (see this Prop) is twice the second Pontryagin class:
π*(χ3)=2p2∈H4(BSpin(5),ℤ).added the following statement, but for the moment without good referencing:
The integral cohomology ring of the classifying space BSpin(5) is spanned by two generators
the first fractional Pontryagin class 12p1
the linear combination 12p2−12(p1)2 of the half the second Pontryagin class with half the cup product-square of the first Pontryagin class:
every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces Λ2+V and Λ2−V must have a 5-dimensional subspace invariant under the action of Sp(V)
I get that the invariant subspace at the start of this quote is generated by the element J preserved by SU(4), but it’s not clear what the subspace is. Is it the real span? Why the claim that one or both of Λ2+V and Λ2−V have an invariant subspace?
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