arXiv subjects

Anthony Narkawicz

Publications and source records attributed to Anthony Narkawicz.

2 recordsLinked to original sources

arXiv2008

Cohomology Jumping Loci and Relative Malcev Completion

Two standard invariants used to study the fundamental group G of the complement X of a hyperplane arrangement are the Malcev completion of G and the cohomology groups of X with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. This is a prosolvable group that is tightly controlled by the cohomology groups of X with coefficients in rank one local systems. The prounipotent radical U_p of the relative completion S_p corresponds to a pronilpotent Lie algebra u_p. We provide an example of a hyperplane complement X for which this algebra is not quadratically presented. In addition, we show that if X is a hyperplane complement and Y is a subtorus of the character torus, then S_p is combinatorially determined for general p in Y. Finally, we show that the relative completion S_p is generally constant over subvarieties of the character torus.

math.AT↗

arXiv2003

The First Cohomology Group H^1(G,M)

This paper characterizes the first cohomology group H^1(G,M) where M is a Banach space (with norm ||.||) that is also a left CG-module such that the elements of G act on M as continuous complex-linear transformations. Of particular interest is the topology on this group induced by the norm topology on M. The first result is that H^1(G,CG) imbeds in H^1(G,M) whenever CG is contained in M which is in turn contained in L^p(G) for some p. This shows immediately that if H^1(G,M)=0, then G has exactly 1 end. Secondly, it is shown that H^1(G,M) is not Hausdorff if and only if there exist f_i in M with norm 1 (||f_i||=1) for all i with the property that ||gf_i-f_i||->0 as i goes to infinity for every g in G. This is then used to show that if ||.|| and M satisfy certain properties and if G satisfies a "strong Folner condition," then H^1(G,M) is not Hausdorff. The second half of the paper gives several applications of these theorems focusing on the free abelian group on n generators. Of particular interest is the case that M is the reduced group C^* algebra of G.

math.OA↗