By Bruce Hughes, Andrew Ranicki

ISBN-10: 0521055199

ISBN-13: 9780521055192

The ends of a topological area are the instructions during which it turns into noncompact through tending to infinity. The tame ends of manifolds are really fascinating, either for his or her personal sake, and for his or her use within the class of high-dimensional compact manifolds. The booklet is dedicated to the similar concept and perform of ends, facing manifolds and CW complexes in topology and chain complexes in algebra. the 1st half develops a homotopy version of the habit at infinity of a noncompact house. the second one half reviews tame results in topology. The authors express tame ends to have a uniform constitution, with a periodic shift map. They use approximate fibrations to end up that tame manifold ends are the countless cyclic covers of compact manifolds. The 3rd half interprets those topological issues into a suitable algebraic context, concerning tameness to homological homes and algebraic okay- and L-theory. This booklet will entice researchers in topology and geometry.

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**Example text**

Proof For any subspace V ⊆ W there is defined a short exact sequences of chain complexes 0 −→ S ∞ (V ) −→ S ∞ (W ) −→ S ∞ (W, V ) −→ 0 with each Sr∞ (V )−→Sr∞ (W ) a split injection and S ∞ (W, V ) = C(S(W, V )−→S lf (W, V ))∗+1 . 7 (iii). 14 For any space X let gkX : X −→ X × I ; x −→ (x, k) (k = 0, 1) and let DX : g0X g1X : S(X) −→ S(X × I) be a natural chain homotopy, with ∂DX + DX ∂ = g0X − g1X : Sr (X) −→ Sr (X × I) . , Munkres [101, pp. 171–172]) for the acyclic model definition of DX , one sees that DX induces a chain homotopy lf DX : g0X g1X : S lf (X) −→ S lf (X × I) on the locally finite chain level.

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5 below it will be proved that if X and Y are proper homotopy equivalent then X is semistable (resp. has stable π1 ) at ∞ if and only if Y is semistable (resp. has stable π1 ) at ∞. (ii) Let W = Kj be a locally path-connected σ-compact space, which is j expressed as a union of compact subspaces Kj ⊆ Kj+1 . 14 gives an exact sequence 0 −→ lim1 π1 (Wj ) −→ π0 (e(W )) −→ lim π0 (Wj ) −→ 0 , ←− ←− j j with lim π0 (Wj ) = EW the number of ends of W . 25 (iii). 14 0 −→ lim1 π2 (Wj ) −→ π1 (e(W )) −→ lim π1 (Wj ) −→ 0 ←− ←− j j 2.

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