Some frequently used constructions (functors) and tools in AT

7k words • 15minutes

• unique up to (weak) homotopy equivalence;
• product: K(G, n) × K(H, n) ≃ K(G × H, n)
• representation results: Let G be an Ab-grp, X a CW complex, then ⟨X, K(G, n)⟩ ↔︎ Hⁿ(X; G), [f] ↦ f^*α where α ∈ Hⁿ(K; G)) = Hom(H_n(K), G) is given by the inverse of Hurewitz isomorphism G → H_n(K).
• cohomology: $H^*(K(\mathbb{Z},n);\mathbb{Q})=\left.\begin{cases} \mathbb{Q}[x_n]&,n\text{ is even}\\ \mathbb{Q}[x_n]/(x_n^2)&,n\text{ is odd} \end{cases}\right.$, using SSS with loopspace fibration, see page 11 of AT-all.

• motivations of (complex) universal bundles: K(ℤ, 1) = S¹, K(ℤ/2, 1) = ℝℙ^∞, K(ℤ, 2) = ℂℙ^∞.
• (co)homology of groups: Hⁿ(G) = Hⁿ(K(G, n))

• BG is the CW complex such that there is a principle G-bundle EG → BG, with EG weakly contractible
• construction (Milnor): for a topo-grp G, there is a accenting chain via topological join: G ⊂ G^⋆2 ⊂ ⋯ ⊂ G^⋆n⋯ Let EG = ∪G^⋆n, BG = EG/G.

• uniqueness up to homotopy equivalence
• bijection: $[B,BG]\leftrightarrow Bun_{G}(B)=\left.\begin{cases}\text{principal } G-\\\text{bundles over B}\end{cases}\right\} /\cong$

definition M ⋆ N = M × [0, 1] × N/∼, where (x, 0, y₁) ∼ (x, 0, y₂), (x₁, 1, y) ∼ (x₂, 1, y)

basic results

• if M is m-connented, N is n-connected, then M ⋆ N is (m + n + 2)-connected.
• S⁰ ⋆ S⁰ = S¹, S^m ⋆ Sⁿ = S^{m + n + 1}.
• M^⋆n is always (n − 2)-connected.

applications

• Milnor’s construction of classifying space: EG = ∪G^⋆n, BG = EG/G