Download Ends of Complexes by Bruce Hughes, Andrew Ranicki PDF

By Bruce Hughes, Andrew Ranicki

The ends of a topological house are the instructions within which it turns into noncompact via tending to infinity. The tame ends of manifolds are relatively fascinating, either for his or her personal sake, and for his or her use within the class of high-dimensional compact manifolds. The ebook is dedicated to the comparable conception 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 stories tame leads to topology. The authors exhibit 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 concerns into a suitable algebraic context, referring to tameness to homological homes and algebraic okay- and L-theory. This ebook will attract researchers in topology and geometry.

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Example text

The locally finite cohomology of W is defined by ∗ Hlf (W ) = H∗ (S lf (W )∗ ) . The cohomology of W at ∞ is defined by ∗ H∞ (W ) = H ∗ (S ∞ (W )) = H∗+1 (i∗ : S lf (W )∗ −→S(W )∗ ) . 19 The various cohomology groups are related by a long exact sequence i r+1 r r . . −→ Hlf (W ) −→ H r (W ) −→ H∞ (W ) −→ . . 14) of a locally finite CW complex W with the dimension of the real vector space 0 (W ; R). H∞ 4 Cellular homology It is well-known that the singular homology groups of a CW complex are isomorphic to the cellular homology groups; it is less well documented (and much harder to prove) that the singular locally finite homology groups of a ‘strongly locally finite’ CW complex are isomorphic to the cellular locally finite homology groups.

0 1 2 ... ... ... ... ... ... ... ... ............................................................................................................................................................................................................................................................................................................................. 4 Let X be a CW complex and let X0 ⊆ X1 ⊆ . . ⊆ Xj ⊆ Xj+1 ⊆ . . ⊆ X be a sequence of subcomplexes.

Wj = ∅ , j=1 . . ⊆ W \Kj ⊆ Wj ⊆ W \Kj−1 ⊆ Wj−1 ⊆ . . (i) The singular locally finite chain complex S lf (W ) is the inverse limit S lf (W ) = lim S(W, W \Kj ) = lim S(W, Wj ) , ←− ←− j j and there are defined short exact sequences 0 −→ lim1 Hr+1 (W, W \Kj ) −→ Hrlf (W ) −→ lim Hr (W, W \Kj ) −→ 0 , ←− ←− j j 1 0 −→ lim Hr+1 (W, Wj ) −→ ←− j Hrlf (W ) −→ lim Hr (W, Wj ) −→ 0 . ←− j 38 Ends of complexes (ii) The singular chain complex at ∞ S ∞ (W ) is homology equivalent to the derived limit coker(i : S(W )−→S lf (W ))∗+1 = lim1 S(W \Kj )∗+1 = lim1 S(Wj )∗+1 , ←− ←− j j so that H∗∞ (W ) = H∗+1 (lim1 S(Wj )) ←− j and there are defined short exact sequences 0 −→ lim1 Hr+1 (W \Kj ) −→ Hr∞ (W ) −→ lim Hr (W \Kj ) −→ 0 , ←− ←− j j 1 0 −→ lim Hr+1 (Wj ) −→ ←− Hr∞ (W ) −→ lim Hr (Wj ) −→ 0 .

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