Topological Methods in Group Theory pp 369-409 | Cite as

# 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.

## Keywords

Fundamental Group Inverse Limit Solid Torus Vertex Group Smash Product## Preview

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