The rational cohomology of a homogeneous space G/K admits a homomorphism from H∗(BK) induced from the classifying map G/K→BK of the principal K-bundle G→G/K. Assume the Lie group is K connected, so that π1(BK)=0; then the space G/K is formal in the sense of rational homotopy theory if and only if H∗(G/K) is a free module over the image A of the map H∗(BK)→H∗(G/K). In this event, one actually has a graded A-algebra decomposition

H∗(G/K)≅A⊗ΛˆP

for ΛˆP an exterior subalgebra on rkG−rkK generators,

isomorphic to the image of H∗(G/K)→H∗(G), and consequently

the rational vector space dimension of H∗(G/K) is given by

dimQH∗(G/K)=(dimQA)⋅2rkG−rkK.The ring factorization above is too good to hold in general, but one can measure its failure with the

deficiency, a constant defined as follows. In general the quotient map G→G/K still induces a map H∗(G/K)→H∗(G) whose image is an exterior algebra ΛˆP, but the dimension of ˆP is only bounded above by rkG−rkK, whereas in the formal case it is equal. The deficiency is the difference from this upper bound:

df(G/K):=rkG−rkK−dimQˆP,

so that formality is deficiency 0.Now, it happens that if df(G/K)∈{1,2}, then even though the ring structure on H∗(G/K) is more recalcitrant than in the formal case, the dimension condition on H∗(G/K) adduced above

continues to hold.

Does this dimension condition hold in greater generality? If not, what’s a counterexample?

**Answer**

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