On the proper Steenrod homotopy groups, and proper embeddings of planes into 3-manifolds

Brin, Matthew G.; Thickstun, T. L.

doi:10.1090/S0002-9947-1985-0784012-0

On the proper Steenrod homotopy groups, and proper embeddings of planes into $3$-manifolds
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by Matthew G. Brin and T. L. Thickstun PDF

Trans. Amer. Math. Soc. 289 (1985), 737-755 Request permission

Abstract:

Standard algebraic invariants of proper homotopy type are discussed. These do not naturally fit into long exact sequences. Groups of proper homotopy classes of proper maps of Euclidean spaces and open annuli which do naturally form a long exact sequence are defined, and a diagram relating these groups to the standard algebraic invariants of proper homotopy type is given. The structures defined are used to compare several notions of essentiality for proper maps. Some results and examples are given for proper maps of spaces into manifolds of dimension $2$ and $3$. These results are used to add information to a theorem of Brown and Feustel about properly embedding planes in $3$-manifolds.

References

A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
Edward M. Brown, Proper homotopy theory in simplicial complexes, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 41–46. MR 0356041

On the proper homotopy type of simplicial complexes

E. M. Brown and C. D. Feustel, On properly embedding planes in arbitrary $3$-manifolds, Proc. Amer. Math. Soc. 94 (1985), no. 1, 173–178. MR 781077, DOI 10.1090/S0002-9939-1985-0781077-2
Zvonko Čerin, On various relative proper homotopy groups, Tsukuba J. Math. 4 (1980), no. 2, 177–202. MR 623435, DOI 10.21099/tkbjm/1496159172
T. A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972), no. 3, 181–193. MR 320997, DOI 10.4064/fm-76-3-181-193

On the behavior at infinity of the fundamental group of a homologically trivial manifold

11

David A. Edwards and Harold M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, Vol. 542, Springer-Verlag, Berlin-New York, 1976. MR 0428322
D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI 10.1112/plms/s3-16.1.369
Steve Ferry, A stable converse to the Vietoris-Smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980), no. 2, 369–386. MR 580894, DOI 10.1090/S0002-9947-1980-0580894-1
Ross Geoghegan, A note on the vanishing of $\textrm {lim}^{1}$, J. Pure Appl. Algebra 17 (1980), no. 1, 113–116. MR 560787, DOI 10.1016/0022-4049(80)90025-0
Brayton I. Gray, Spaces of the same $n$-type, for all $n$, Topology 5 (1966), 241–243. MR 196743, DOI 10.1016/0040-9383(66)90008-5
John Hempel, $3$-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025
Y. Kodama and J. Ono, On fine shape theory, Fund. Math. 105 (1979/80), no. 1, 29–39. MR 558127, DOI 10.4064/fm-105-1-29-39
François Laudenbach, Topologie de la dimension trois: homotopie et isotopie, Astérisque, No. 12, Société Mathématique de France, Paris, 1974 (French). With an English summary and table of contents. MR 0356056
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
William S. Massey, Homology and cohomology theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 46, Marcel Dekker, Inc., New York-Basel, 1978. An approach based on Alexander-Spanier cochains. MR 0488016
Michael L. Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983), no. 1, 307–321. MR 690054, DOI 10.1090/S0002-9947-1983-0690054-4

On the Steenrod homology theory

J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327

On the group of a certain linkage

8

J. Brendan Quigley, An exact sequence from the $n$th to the $(n-1)$-st fundamental group, Fund. Math. 77 (1973), no. 3, 195–210. MR 331379, DOI 10.4064/fm-77-3-195-210
J. Brendan Quigley, Equivalence of fundamental and approaching groups of movable pointed compacta, Fund. Math. 91 (1976), no. 2, 73–83. MR 413098, DOI 10.4064/fm-91-2-73-83
N. E. Steenrod, Regular cycles of compact metric spaces, Ann. of Math. (2) 41 (1940), 833–851. MR 2544, DOI 10.2307/1968863

A certain open manifold whose group is unity

6