Central theorems for cohomologies of certain solvable groups
Hisashi Kasuya

TL;DR
This paper establishes that the cohomology of certain solvable groups can be computed via algebraic group rational cohomology, generalizing previous results and providing new proofs for known theorems.
Contribution
It introduces a unifying framework linking group cohomology of solvable groups with algebraic group rational cohomology, extending prior work on solvmanifolds.
Findings
Cohomology of torsion-free virtually polycyclic groups is computable via algebraic groups.
Continuous cohomology of simply connected solvable Lie groups aligns with algebraic group rational cohomology.
Provides a simplified proof of the Dekimpe-Igodt cohomology vanishing theorem.
Abstract
We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certian results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a simple proof of the surprising cohomology vanishing theorem given by Dekimpe-Igodt.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometry and complex manifolds · Advanced Algebra and Geometry
