Aleš Vavpetič

Large Scale Versus Small Scale

Recent research in coarse geometry revealed similarities between certain concepts of large scale geometry and topology. It is less known that a small scale analog of topology has been developed much earlier in the form of the uniform category. This paper is devoted to an exposition of analogies between basic concepts of topology (paracompactness, covering dimension), important ideas of coarse geometry (Property A of G. Yu, asymptotic dimension of M. Gromov), and notions from the uniform category (l 1-property, the uniform dimension).

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. That embedding can be realized by either Rips complexes or analogs of Roe’s antiČech approximations of spaces. In that model coarse n-connectedness of K = {K1 → K2 → . . .} means that for each k there is m > k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n. The asymptotic dimension being at most n means that for each k there is m > k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex. Property A of G.Yu is equivalent to the condition that for each k and for each ǫ > 0 there is m > k such that the bonding map from |Kk| to |Km| has a contiguous approximation g : |Kk| → |Km| which sends simplices of |Kk| to sets of diameter at most ǫ. Date: June 8, 2009. 2000 Mathematics Subject Classification. Primary 54F45; Secondary 55M10.

Property A and asymptotic dimension

The purpose of this note is to characterize the asymptotic dimension

Coarse amenability versus paracompactness

asdim(X)

Compact maps and quasi-finite complexes

of metric spaces

Co)homology of Lie algebras via algebraic Morse theory

X

Symplectic groups are

in terms similar to Property A of Yu: If

N

(X,d)

-determined 2-compact groups

is a metric space and

On the mod

n\ge 0