Fundamental group at infinity: Difference between revisions
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==Definition== | ==Definition== | ||
The '''fundamental group at infinity''' of a [[path-connected space]] is the inverse limit of the fundamental | ===Loose definition=== | ||
The '''fundamental group at infinity''' of a [[path-connected space]] is the inverse limit of the [[fundamental group]]s 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== | ==Facts== | ||
The fundamental group at infinity is ''not'' homotopy-invariant. In fact, there exist [[contractible space]]s 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. | The fundamental group at infinity is ''not'' homotopy-invariant. In fact, there exist [[contractible space]]s 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. | ||
Latest revision as of 20:11, 9 January 2011
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.