By Solomon Lefschetz
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Additional info for Algebraic Topology (Colloquium Pbns. Series, Vol 27)
4. n; F /. Then G has finite decomposition complexity. We refer to  for a proof. 9. 1. Show, using only the definition, that asdim Z2 is exactly 2. 2. N/ (or any Banach space which does not have a finite basis) have infinite asymptotic dimension. 3. U; V / R. 4. 3. 5. Show that the wreath product of finitely generated groups is finitely generated. 6. 3 to the following statement: if G and H both contain an element of infinite order, then asdim G o H D 1. Notes and remarks Topological dimension theory was founded in the twenties of the XXth century and since then has become a fundamental tool in geometric topology.
3 to finite decomposition complexity. 5. Let X and Y be families of metric spaces and let F W X ! Y be a uniformly expansive map. V / has finite decomposition complexity, then X has finite decomposition complexity. Proof. Consider a decomposition strategy for Y and, as in the previous proof, pull it back to obtain a labeled decomposition tree for X. V/, which are not necessarily bounded families. V / has finite decomposition complexity. V/. V /. The resulting tree is a decomposition tree for X.
B/g; 22 Chapter 1. a// < 2D. Define yb similarly. There exist points ya ; yb 2 such that their distances to ya and yb , respectively, are at most D. ya ; yb / Ä 6D: Let p be the path connecting ya , ya , yb and yb , which runs through between ya and yb and is geodesic from ya to ya and from yb to yb . p/ Ä 2N C1 : N Let fPi g2iD0 be equidistant points on p, with P0 and P2N being the endpoints of p. By hyperbolicity of X, x0 is within ı of the geodesic ŒP0 ; P2N 1 or within ı of the geodesic ŒP2N 1 ; PN .
Algebraic Topology (Colloquium Pbns. Series, Vol 27) by Solomon Lefschetz