Kähler-like manifolds, Weyl spinor particles and E-infinity high energy physics

The Kähler-like fuzzy manifold of E-infinity theory is introduced. The Betti numbers and other topological invariants of this manifold are determined. In particular it is found that the Euler characteristic is equal to 26 + k = 26.18033989 while the absolute value of the inverse signature is equal to the electromagnetic fine structure times D(10) = 10. The relation to Weyl spinor is discussed.