Capped grope

In mathematics, a grope is a construction used in 4-dimensional topology, introduced by Štan'ko (1971) and named by Cannon (1978) "because of its multitudinous fingers". Capped gropes were used by Freedman (1984) as a substitute for Casson handles, that[vague] work better for non-simply-connected 4-manifolds.

A capped surface in a 4-manifold is roughly a surface together with some 2-disks, called caps, whose boundaries generate the fundamental group of the surface. A capped grope is obtained by repeatedly replacing the caps of a capped surface by another capped surface. Capped surfaces and capped gropes are studied in Freedman & Quinn (1990).


  • Cannon, J. W. (1978), "The recognition problem: what is a topological manifold?", Bulletin of the American Mathematical Society, 84 (5): 832–866, doi:10.1090/S0002-9904-1978-14527-3, ISSN 0002-9904, MR 0494113
  • Freedman, Michael Hartley (1984), "The disk theorem for four-dimensional manifolds", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), Warszawa: PWN, pp. 647–663, MR 0804721
  • Freedman, Michael Hartley; Quinn, Frank (1990), Topology of 4-manifolds, Princeton Mathematical Series, 39, Princeton University Press, ISBN 978-0-691-08577-7, MR 1201584
  • Štan'ko, M. A. (1971), "Approximation of the imbedding of compacta in a codimension larger than two", Doklady Akademii Nauk SSSR, 198: 783–786, ISSN 0002-3264, MR 0284994