The Fundamental Group At Infinity
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.
KeywordsFundamental Group Inverse Limit Solid Torus Vertex Group Smash Product
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