Homology of a product: over a field, H_*(X×Y) = H_*(X) ⊗ H_*(Y). Over ℤ an extra Tor-correction appears. Gives H*(Tⁿ) = Λⁿ (exterior algebra on n degree-1 generators) and underlies the ring structure of H*(BG).
Homology of a product: over a field, H_*(X×Y) = H_*(X) ⊗ H_*(Y). Over ℤ an extra Tor-correction appears. Gives H*(Tⁿ) = Λⁿ (exterior algebra on n degree-1 generators) and underlies the ring structure of H*(BG).