zbMATH — the first resource for mathematics

Properly $$3$$-realizable groups. (English) Zbl 1062.57001
A finitely presented group $$G$$ is properly 3-realizable if there exists a compact 2-polyhedron $$K$$ with fundamental group $$G$$ whose universal cover has the proper homotopy type of a 3-manifold with boundary.
The authors show that if $$H$$ and $$G$$ are finitely presented groups where $$H$$ is a subgroup of finite index of $$G$$, then $$G$$ is properly 3-realizable if and only if $$H$$ is.
They also show that if $$G$$ is properly 3-realizable and $$K$$ is a 2-polyhedron with fundamental group $$G$$ then the universal cover of $$K\vee S^2$$ has the proper homotopy type of a 3-manifold.
Their last result is that the fundamental group of a finite graph of groups with properly 3-realizable vertex groups and finite cyclic edge groups is properly 3-realizable.

MSC:
 57M07 Topological methods in group theory 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M10 Covering spaces and low-dimensional topology 57M20 Two-dimensional complexes (manifolds) (MSC2010)
Full Text:
References:
 [1] Hans-Joachim Baues and Antonio Quintero, Infinite homotopy theory, \?-Monographs in Mathematics, vol. 6, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 0983.55001 [2] M. Cárdenas, T. Fernández, F. F. Lasheras, and A. Quintero, Embedding proper homotopy types, Colloq. Math. 95 (2003), no. 1, 1 – 20. · Zbl 1038.55008 [3] M. Cárdenas, F.F. Lasheras. On properly $$3$$-realizable groups. Preprint. · Zbl 1086.57001 [4] M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449 – 457. · Zbl 0572.20025 [5] D. B. A. Epstein, Ends, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 110 – 117. · Zbl 1246.57006 [6] R. Geoghegan. Topological Methods in Group Theory. Book in preparation. · Zbl 1141.57001 [7] Bruce Hughes and Andrew Ranicki, Ends of complexes, Cambridge Tracts in Mathematics, vol. 123, Cambridge University Press, Cambridge, 1996. · Zbl 0876.57001 [8] Francisco F. Lasheras, Universal covers and 3-manifolds, J. Pure Appl. Algebra 151 (2000), no. 2, 163 – 172. · Zbl 0976.57003 [9] Francisco F. Lasheras, A note on fake surfaces and universal covers, Topology Appl. 125 (2002), no. 3, 497 – 504. · Zbl 1012.57004 [10] J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999. · Zbl 0923.55001 [11] C. T. C. Wall , Homological group theory, London Mathematical Society Lecture Note Series, vol. 36, Cambridge University Press, Cambridge-New York, 1979. · Zbl 0409.00004 [12] C. T. C. Wall, Finiteness conditions for \?\?-complexes, Ann. of Math. (2) 81 (1965), 56 – 69. · Zbl 0152.21902 [13] J. H. C. Whitehead, Simple homotopy types, Amer. J. Math. 72 (1950), 1 – 57. · Zbl 0040.38901 [14] Perrin Wright, Formal 3-deformations of 2-polyhedra, Proc. Amer. Math. Soc. 37 (1973), 305 – 308. · Zbl 0253.57001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.