A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space G/K admits a homomorphism from H(BK) induced from the classifying map G/KBK of the principal K-bundle GG/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)AΛˆP
for ΛˆP an exterior subalgebra on rkGrkK 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)2rkGrkK.

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 GG/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 rkGrkK, whereas in the formal case it is equal. The deficiency is the difference from this upper bound:
df(G/K):=rkGrkKdimQˆP,
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

Attribution
Source : Link , Question Author : jdc , Answer Author : Community

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