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 | 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. | ||
==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. | ||
Revision as of 16:30, 3 December 2007
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.
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.