×

Retracting compact 3-manifolds onto their boundary components. (English) Zbl 0998.57042

Summary: Let \(X\) be a compact 3-manifold and \(A\) a boundary component of \(X\) to which \(X\) retracts. We propose the problem of classifying such \((X,A)\) up to PL homeomorphisms in a suitable class of manifolds. We give a complete solution in a special case. We show in this case there are exactly twenty-four basic pairs \((X,A)\) and all other cases are obtained from these by some obvious modifications. If \(A\) is assumed to be different from a 2-sphere, there are six basic pairs.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
54C15 Retraction
57M99 General low-dimensional topology
57Q99 PL-topology

Keywords:

retraction
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Hempel, J.: Free cyclic actions on \(S^1\times S^1\times S^1\). Proc. Amer. Math. Soc., 48 , 221-227 (1975). · Zbl 0321.57028
[2] Kim, P. K., and Tollefson, J. L.: PL involutions of fibered 3-manifolds. Trans. Amer. Math. Soc., 232 , 221-237 (1977). · Zbl 0376.57021
[3] Kwun, K. W., and Tollefson, J. L.: PL involutions of \(S^1\times S^1\times S^1\). Trans. Amer. Math. Soc., 203 , 97-106 (1975). · Zbl 0321.57029
[4] Stallings, J.: On fibering certain 3-manifolds. Topology of 3-manifold and Related Topics. Prentice-Hall, Englewood Cliffs, N. J., pp. 95-100 (1962). · Zbl 1246.57049
[5] Tao, Y.: On fixed point free involutions of \(S^1\times S^2\). Osaka Math. J., 14 , 145-152 (1962). · Zbl 0105.17302
[6] Tollefson, J. L.: Involutions on \(S^1\times S^1\) and other 3-manifolds. Trans. Amer. Math. Soc., 183 , 139-152 (1973). · Zbl 0276.57001
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.