Classifying Space - Examples

Examples

  1. The circle S1 is a classifying space for the infinite cyclic group Z.
  2. The n-torus is a classifying space for Zn, the free abelian group of rank n.
  3. The wedge of n circles is a classifying space for the free group of rank n.
  4. A closed (that is compact and without boundary) connected surface S of genus at least 1 is a classifying space for its fundamental group .
  5. The infinite-dimensional projective space is a classifying space for Z/2Z.
  6. A closed (that is compact and without boundary) connected hyperbolic manifold M is a classifying space for its fundamental group .
  7. A finite connected locally CAT(0) cubical complex is a classifying space of its fundamental group.
  8. is the classifying space BS1 for the circle S1 thought of as a compact topological group.

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