5
$\begingroup$

Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H_*(\Lambda X, \mathbb{Q})$ is degree-wise finite-dimensional? Is it known at least for type A (i.e. classical) flag varieties?

This is known to be true for some concrete examples, such as complex projective spaces (Ziller et al.), and complete flag varieties of rank 2 (Burfitt-Grbić), but I was unable to find any statements for the general case.

$\endgroup$

1 Answer 1

9
$\begingroup$

Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (Proposition 9 of chapter IV of Homologie singulière des espaces fibrés. Applications. Ann. of Math., 54, 1951, p. 425-505.)

So the only fact you need about generalized flag varieties is that they are simply-connected compact manifolds.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.