Two papers published by Algebraic & Geometric Topology Publications

Geometry and Topology · science forum beginner Joined: 24 May 2006 Posts: 4

Two papers have been published by Algebraic & Geometric Topology
at http://www.msp.warwick.ac.uk/agt/agtcontents6.html

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(1) A family of pseudo-Anosov braids with small dilatation}
by Eriko Hironaka and Eiko Kin

(2) Isovariant mappings of degree 1 and the Gap Hypothesis
by Reinhard Schultz

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Abstracts follow:

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(1) A family of pseudo-Anosov braids with small dilatation}
by Eriko Hironaka and Eiko Kin

URL:
http://www.msp.warwick.ac.uk/agt/ftp/main/2006/agt-06-27.pdf

This paper describes a family of pseudo-Anosov braids with small
dilatation. The smallest dilatations occurring for braids with 3, 4
and 5 strands appear in this family. A pseudo-Anosov braid with
2g+1 strands determines a hyperelliptic mapping class with the same
dilatation on a genus-g surface. Penner showed that logarithms of
least dilatations of pseudo-Anosov maps on a genus-g surface grow
asymptotically with the genus like 1/g, and gave explicit examples
of mapping classes with dilatations bounded above by log 11/g.
Bauer later improved this bound to log 6/g. The braids in this paper
give rise to mapping classes with dilatations bounded above by
log(2+sqrt(3))/g. They show that least dilatations for hyperelliptic
mapping classes have the same asymptotic behavior as for general mapping
classes on genus-g surfaces.

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(2) Isovariant mappings of degree 1 and the Gap Hypothesis
by Reinhard Schultz

URL:
http://www.msp.warwick.ac.uk/agt/ftp/main/2006/agt-06-28.pdf

Unpublished results of S Straus and W Browder state that two notions
of homotopy equivalence for manifolds with smooth group actions -
isovariant and equivariant - often coincide under a condition called
the Gap Hypothesis; the proofs use deep results in geometric topology.
This paper analyzes the difference between the two types of maps from
a homotopy theoretic viewpoint more generally for degree one maps if
the manifolds satisfy the Gap Hypothesis, and it gives a more homotopy
theoretic proof of the Straus-Browder result.

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