4 Algebraic Foundations
4.1 Octonion Structure
The octonions \(\mathbb {O}\) have 7 imaginary units: \(|\mathrm{Im}(\mathbb {O})| = 7\)
\(\mathrm{dim}(G_2) = 14\)
4.2 Betti Numbers from Octonions
\(b_2= \binom {7}{2}\) (pairs of imaginary octonion units)
\(b_2= 21\)
\(\mathrm{fund}(E_7) = 56\)
\(\mathrm{fund}(E_7) = 2 \cdot b_2+ \mathrm{dim}(G_2) = 42 + 14 = 56\)
\(b_3= 3 \cdot b_2+ \mathrm{dim}(G_2)\)
\(b_3= 77\)
\(b_3= b_2+ \mathrm{fund}(E_7) = 21 + 56 = 77\)
\(H^*= b_2+ b_3+ 1 = 99\)