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A loop theorem/Dehn's lemma for some orbifolds

Josh Barnard
2011 Algebraic and Geometric Topology  
The equivariant loop theorem implies the existence of a loop theorem/Dehn's lemma for 3-orbifolds that are good (covered by a 3-manifold). In this note we prove a loop theorem/Dehn's lemma for any locally orientable 3-orbifold (good or bad) whose singular set is labeled with powers of 2. The proof is modeled on the standard tower construction. 57M35 We prove a version of the loop theorem and Dehn's lemma for a certain class of 3orbifolds, namely those which are locally orientable and have
more » ... able and have singular set labeled with powers of 2. As part of a program to extend Waldhausen's theorems on 3-manifolds to 3-orbifolds, Takeuchi and Yokoyama [4, Corollary 6:4] have proven a loop theorem for good 3-orbifolds (ie, those covered by a manifold) using the equivariant loop theorem for 3-manifolds, along with a generalization of normal surface theory. Thus the novelty of the proof given here is two-fold: it extends the loop theorem to certain bad 3-orbifolds, and it uses the more direct techniques of cutting and pasting, in the spirit of Papakyriakopoulos' original tower proof [3]. Our presentation and notation are modeled on those of Hatcher [2] . As a corollary we show that in a covering of an orbifold to which the loop theorem applies, orbifold incompressible 2-suborbifolds lift to orbifold incompressible 2-suborbifolds.
doi:10.2140/agt.2011.11.2815 fatcat:aqui7rdwfbb7nh66cxotuzldqm