Fundamental group at infinity

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Definition

Loose definition

The fundamental group at infinity of a path-connected space is the inverse limit of the fundamental groups of complements of compact subsets. For a compact space, the fundamental group at infinity is trivial.

Note that this loose definition is not quite precise, because the choice of basepoint does affect the choice of map, and if complements of compact sets are not path-connected, then we need to be careful about choosing the fundamental group.

Facts

The fundamental group at infinity is not homotopy-invariant. In fact, there exist contractible spaces whose fundamental group at infinity does not vanish. Thus, the fundamental group at infinity is a tool to distinguish between non-homeomorphic spaces which are homotopy-equivalent.