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# A simple homotopy-theoretical proof of the Sullivan conjecture

• ##### Jeffrey Strom
05/2011;
Source: arXiv

ABSTRACT We give a new proof, using comparatively simple techniques, of the Sullivan
conjecture: the space of pointed maps from the classifying space of the cyclic
group of order \$p\$ to any finite-dimensional CW complex \$K\$ is contractible.

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ABSTRACT: Let G be a tower of finitely generated nilpotent groups and let G(p) denote its localization at a prime p. It is shown that if lim1 (G) is nontrivial, then it contains at least two elements whose images are equal in lim1 (G(p)) for every prime p. Are there, in fact, infinitely many elements with this property? For abelian towers, the answer is yes. It is shown that the answer is also yes for certain nonabelian towers. Other questions considered in this paper include the problem of characterizing the lim1 term of towers of countable abelian groups, towers whose lim1 terms are countably infinite, and the extent to which the lim1 term depends upon the actual group structures in the tower.
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