Geodesic
Could the Poincar\'e Conjecture Be False?
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 247-251

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Two conjectures are stated which imply that the Poincaré hypothesis (asserting that any simply connected closed compact $3$-manifold is the $3$-sphere) is false. The first one claims that, for certain classes of finitely presented groups, the triviality problem is algorithmically undecidable, and the second one claims that certain embeddings of two-dimensional polyhedra in $3$-manifolds can effectively be constructed. 
@article{TM_2004_247_a17,
     author = {A. B. Sosinskii},
     title = {Could the {Poincar\'e} {Conjecture} {Be} {False?}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {247--251},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_247_a17/}
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A. B. Sosinskii. Could the Poincar\'e Conjecture Be False?. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 247-251. http://geodesic.mathdoc.fr/item/TM_2004_247_a17/