For a compact symmetric space M, the 2-number #2(M) is the maximal possible cardinality of a subset A2(M) of M such that the point symmetry sx fixes every point of A2(M) for every x ∈ A2(M) (see [B.-Y. Chen and T. Nagano, A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273–297] or [B.-Y. Chen, The 2-ranks of connected compact Lie groups. Taiwanese J. Math. 17 (2013), 815–831] ).
By direct computation we can see that the 2-number #2(M) is in fact equal to the smallest number of cells needed to have a CW-complex structure on M if M is a sphere, a real projective space or a Hermitian symmetric space (cf. [B.-Y. Chen, A new approach to compact symmetric spaces and applications. A report on joint work with Professor T. Nagano, KU Leuven, 1987]).
When M is connected compact symmetric space, the definition of 2-number is equivalent to say that #2M is the maximal possible cardinality #A2 of a subset A2 of M such that, for every pair of points x and y of A2, there exists a closed geodesic of M on which x and y are antipodal to each other. Using the second definition, one can define the 2-number #2M on any Riemannian manifold.
I would like to point out that the 2-number #2(M) is also closely related to the Euler number χ(M). More precisely, we have the following two results proved in [B.-Y. Chen and T. Nagano, A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273–297] :
(1) #2(M) = χ(M) holds for every semi-simple Hermitian compact symmetric space.
(2) #2(M) ≥ χ(M) holds for every compact connected symmetric space.
Our conjecture to this problem is that the 2-number #2(M) is equal to the smallest number of cells needed to have a CW-complex structure on M for every compact connected symmetric space M.
As an extension of a maximal antipodal set A2, C. U. Sanchez defined the p-set Ap(M) in [The index number of an R-space: An extension of a result of M. Takeuchi's, Proc. Amer. Math. Soc. , 125 (1997) 893-900] and used it to define the index number #I(M) for a real flat manifold. Moreover, J. Berndt, S. Console and A. Fino proved in [On index number and topology of flag manifolds, Differential Geom. Appl. 15 (2001), 81-90] that the index number #IM of a real flag manifold M coincides with the smallest number of cells that are needed for a CW-complex structure on a real flag manifold M.
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