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Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on manifolds"),

López de Medrano, S., Involutions on manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 59. Berlin-Heidelberg-New York: Springer-Verlag, ix, 102 p. (1971). ZBL0214.22501.

but I couldn't find anything on general Grassmannians.

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    $\begingroup$ "Involutions on manifolds" deals with fake projective spaces, i.e. closed manifolds that are homotopy equivalent to projective spaces. Are you asking about classification of closed manifolds that are homotopy equivalent to Grassmannians? $\endgroup$
    – Igor Belegradek
    Jul 10, 2019 at 20:42
  • $\begingroup$ @IgorBelegradek Yes, precisely. $\endgroup$
    – Kafka91
    Jul 11, 2019 at 7:17
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    $\begingroup$ Probably there is no hope for a complete classification, and in fact, this is the situation for almost all simply-connected manifolds, e.g the product of three spheres is already too hard except possibly for some low dimensional ones. You can get an idea of how complex things are from the classification for the product of two spheres in arxiv.org/abs/0904.1370. $\endgroup$
    – Igor Belegradek
    Jul 11, 2019 at 14:15

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