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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Periodic homotopy and conjugacy idempotents


Author: Jaka Smrekar
Journal: Proc. Amer. Math. Soc. 135 (2007), 4045-4055
MSC (2000): Primary 55P99; Secondary 20F38, 57M07, 57M10.
Published electronically: August 15, 2007
MathSciNet review: 2341957
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Abstract: A self-map $ f$ on the CW complex $ Z$ is a periodic homotopy idempotent if for some $ r\geqslant 0$ and $ p>0$ the iterates $ f^r$ and $ f^{r+p}$ are homotopic. Geoghegan and Nicas defined the rotation index $ RI(f)$ of such a map. They proved that for $ r=p=1$, the homotopy idempotent $ f$ splits if and only if $ RI(f)=1$, while for $ r=0$, the index $ RI(f)$ divides $ p^2$. We extend this to arbitrary $ p$ and $ r$, and generalize various results related to the splitting of homotopy idempotents on CW complexes and conjugacy idempotents on groups.


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Additional Information

Jaka Smrekar
Affiliation: Fakulteta za matematiko in fiziko, Jadranska ulica 19, SI-1111 Ljubljana, Slovenia
Email: jaka.smrekar@fmf.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08900-9
PII: S 0002-9939(07)08900-9
Keywords: Periodic homotopy idempotent, split idempotent, rotation index, eventual coherence, Thompson groups
Received by editor(s): April 26, 2006
Received by editor(s) in revised form: September 6, 2006
Published electronically: August 15, 2007
Additional Notes: The author was supported in part by the MŠZŠ of the Republic of Slovenia research program No. P1-0292-0101-04 and research project No. J1-6128-0101-04, and in part by the DURSI of the Generalitat de Catalunya grant 2004-CRED-00048.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2007 American Mathematical Society