Extensions of the Euler–Satake characteristic for nonorientable 3-orbifolds and indistinguishable examples

Carroll, Ryan; Seaton, Christopher

doi:10.2140/involve.2013.6.345

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Extensions of the Euler–Satake characteristic for nonorientable $3$-orbifolds and indistinguishable examples

Ryan Carroll and Christopher Seaton

Vol. 6 (2013), No. 3, 345–368

DOI: 10.2140/involve.2013.6.345

Abstract

We compute the $F_{ℓ}$ -Euler–Satake characteristics of an arbitrary closed, effective $3$ -dimensional orbifold $Q$ where $F_{ℓ}$ is a free group with $ℓ$ generators. We focus on the case of nonorientable orbifolds, extending previous results for the case of orientable orbifolds. Using these computations, we determine examples of distinct $3$ -orbifolds $Q$ and $Q^{'}$ such that $χ_{Γ}^{ES} (Q) = χ_{Γ}^{ES} (Q^{'})$ for every finitely generated discrete group $Γ$ .

Keywords

orbifold, $3$-orbifold, Euler–Satake characteristic, orbifold Euler characteristic

Mathematical Subject Classification 2010

Primary: 57R18, 57R20

Secondary: 22A22, 57S17

Milestones

Received: 10 August 2012

Accepted: 10 October 2012

Published: 8 September 2013

Communicated by Colin Adams

Authors

Ryan Carroll
Department of Mathematics University of California Santa Cruz, CA 95064 United States

Christopher Seaton
Department of Mathematics and Computer Science Rhodes College 2000 North Parkway Memphis, TN 38112 United States

Vol. 6, No. 3, 2013