Skip Navigation


Journal of the London Mathematical Society Advance Access originally published online on November 20, 2007
Journal of the London Mathematical Society 2007 76(3):757-776; doi:10.1112/jlms/jdm069
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
76/3/757    most recent
jdm069v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Basu, S.
Right arrow Articles by Vorobjov, N.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2007 London Mathematical Society

On the number of homotopy types of fibres of a definable map

Saugata Basu

School of Mathematics
Georgia Institute of Technology
Atlanta
GA 30332
USA
saugata.basu{at}math.gatech.edu

Nicolai Vorobjov

Department of Computer Science
University of Bath
Bath BA2 7AY
United Kingdom

In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map in terms of the format of its graph. In particular, we show that if a semi-algebraic set SsubRm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and {pi}: Rm+n->Rn is the projection on a subspace, then the number of different homotopy types of fibres of {pi} does not exceed s2(m+1)n(2m nd)O(nm). As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials with bounded additive complexity. We also prove single exponential upper bounds on the radii of balls guaranteeing local contractibility for semi-algebraic sets defined by polynomials with integer coefficients.

2000 Mathematics Subject Classification 14P10, 14P25, 32B20.

The first author was supported in part by an NSF Career Award 0133597 and a Alfred P. Sloan Foundation Fellowship. The second author was supported in part by the European RTN Network RAAG (contract HPRN-CT-2001-00271).

Received May 18, 2006; revised February 26, 2007; published online November 20, 2007.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.