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• Tags
  • #homotopy
• Refs:
  • #todo/add-references
• Links:
  • Moore space
  • Eckman-Hilton
  • cohomotopy

Eilenberg-MacLane spaces

Questions

• Why do Eilenberg-MacLane spaces have complicated higher cohomology?
  • Dually, why do spheres have higher complicated homotopy?
• \({\mathbf{B}}G \simeq K(G, 1)\) and \({\mathbf{B}}^n G \simeq K(G, n)\) when \(n\geq 2\) for \(G\) an abelian discrete group.
  • What is \({\mathbf{B}}^n\)?
  • What is \(\pi_* {\mathbf{B}}G\) for \(G\) nonabelian and nondiscrete? #todo/questions

Results

• \({\Omega}^n {\mathbf{B}}^n G \simeq G\).

Uniqueness of E-M Spaces

If \(X\) is a space with one nontrivial homology group \(G\) in degree \(k\), so that \(X\) satisfies \begin{align*}\pi_i(X) = \cases{G,~i=k\\0,~\text{otherwise}}\end{align*} Then \(X \simeq K(G, k)\).

*Note: two spaces with isomorphic homotopy groups may *not* be homotopy-equivalent in general - this is one exception.*

Construction