4,104 research outputs found
Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups
We extend each higher Johnson homomorphism to a crossed homomorphism from the
automorphism group of a finite-rank free group to a finite-rank abelian group.
We also extend each Morita homomorphism to a crossed homomorphism from the
mapping class group of once-bounded surface to a finite-rank abelian group.
This improves on the author's previous results [Algebr. Geom. Topol. 7
(2007):1297-1326]. To prove the first result, we express the higher Johnson
homomorphisms as coboundary maps in group cohomology. Our methods involve the
topology of nilpotent homogeneous spaces and lattices in nilpotent Lie
algebras. In particular, we develop a notion of the "polynomial straightening"
of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page
The graded Grothendieck group and the classification of Leavitt path algebras
This paper is an attempt to show that, parallel to Elliott's classification
of AF -algebras by means of -theory, the graded -group classifies
Leavitt path algebras completely. In this direction, we prove this claim at two
extremes, namely, for the class of acyclic graphs (graphs with no cycles) and
comet and polycephaly graphs (graphs which each head is connected to a cycle or
a collection of loops).Comment: 30 pages more polishe
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