Rational Points, R-equivalence and Étale Homotopy of Algebraic Varieties

We study a generalisation of the anabelian section conjecture of Grothendieck by substituting the arithmetic fundamental group with a relative version of the étale homotopy type. We show that the map associating homotopy fixed points to rational points factors though R-equivalence for projective varieties defied over fields of characteristic zero. We show that over p-adic fields rational points are homotopy equivalent in this sense if and only if they are étale-Brauer equivalent. We also show that over the real field rational points on projective varieties are homotopy equivalent if and only if they are in the same connected component. We prove that a natural homotopy version of the section and Shafarevich-Tate conjectures over number fields is equivalent to its well-established analogues in the special case of curves and abelian varieties. We also prove this conjecture for generalised Châtelet surfaces and some other basic class of varieties.