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The Fundamental Group At Infinity

Part of the Graduate Texts in Mathematics book series (GTM, volume 243)

Let Y be a strongly locally finite path connected CW complex. In this section we discuss various meanings of the vague sentence “Y is connected at infinity”. One possible meaning is that Y has one end. As we saw in Sect. 13.4, this means that for any two proper rays ω and τ in Y, ω ǀ ℕ and τ ǀ ℕ are properly homotopic. Another possible meaning is that Y is strongly connected at infinity by which we mean that any such ω and τ are themselves properly homotopic. A third possible meaning is that the infinite 1-chains over the ring R defined by any such (cellular) ω and τ are properly homologous, in which case we will say that Y is strongly R-homology connected at infinity. Then the distinctions multiply: if Y has more than one end we can ask: is Y strongly connected or strongly R-homology connected at a particular end? To deal with all these matters we need a vocabulary. So we begin again.

Keywords

Fundamental Group Inverse Limit Solid Torus Vertex Group Smash Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media, LLC 2008

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