Farber, Michael, Hausmann, Jean-Claude and Schutz, Dirk (2009) On the conjecture of Kevin Walker. Journal of Topology and Analysis, Vol.1 (No.1). pp. 65-86. doi:10.1142/S1793525309000023

Research output not available from this repository, contact author.

Request Changes to record.

Abstract

In 1985 Kevin Walker in his study of topology of polygon spaces [15] raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R3. We also prove that for planar polygon spaces the conjecture holds in several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to Gubeladze.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Journal of Topology and Analysis
Publisher:	World Scientific Publishing Co. Pte. Ltd.
ISSN:	1793-5253
Official Date:	March 2009
Dates:	Date Event March 2009 Published
Volume:	Vol.1
Number:	No.1
Page Range:	pp. 65-86
DOI:	10.1142/S1793525309000023
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

Request changes or add full text files to a record

Repository staff actions (login required)

View Item