Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy

Edwards, David A.; Geoghegan, Ross

doi:10.1090/S0002-9947-1976-0402735-4

Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy
HTML articles powered by AMS MathViewer

by David A. Edwards and Ross Geoghegan PDF

Trans. Amer. Math. Soc. 219 (1976), 351-360 Request permission

Abstract:

In Theorem 3.3 and Remark 3.4 conditions are given under which an infinite-dimensional Whitehead theorem holds in pro-homotopy. Applications to shape theory are given in Theorems 1.1, 1.2, 4.1 and 4.2.

References

M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577, DOI 10.1007/BFb0080957
M. F. Atiyah and G. B. Segal, Equivariant $K$-theory and completion, J. Differential Geometry 3 (1969), 1–18. MR 259946, DOI 10.4310/jdg/1214428815
Karol Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223–254. MR 229237, DOI 10.4064/fm-62-3-223-254
James Draper and James Keesling, An example concerning the Whitehead theorem in shape theory, Fund. Math. 92 (1976), no. 3, 255–259. MR 431157, DOI 10.4064/fm-92-3-255-259
David A. Edwards and Ross Geoghegan, Compacta weak shape equivalent to ANR’s, Fund. Math. 90 (1975/76), no. 2, 115–124. MR 394643, DOI 10.4064/fm-90-2-115-124
David A. Edwards and Ross Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975), 521–535. MR 375330, DOI 10.2307/1970939
David A. Edwards and Ross Geoghegan, The stability problem in shape, and a Whitehead theorem in pro-homotopy, Trans. Amer. Math. Soc. 214 (1975), 261–277. MR 413095, DOI 10.1090/S0002-9947-1975-0413095-6
David A. Edwards and Ross Geoghegan, Stability theorems in shape and pro-homotopy, Trans. Amer. Math. Soc. 222 (1976), 389–403. MR 423347, DOI 10.1090/S0002-9947-1976-0423347-2
David A. Edwards and Patricia Tulley McAuley, The shape of a map, Fund. Math. 96 (1977), no. 3, 195–210. MR 488046, DOI 10.4064/fm-96-3-195-210
Ross Geoghegan and R. C. Lacher, Compacta with the shape of finite complexes, Fund. Math. 92 (1976), no. 1, 25–27. MR 418029, DOI 10.4064/fm-92-1-25-27
D. Handel and J. Segal, An acyclic continuum with non-movable suspensions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 171–172 (English, with Russian summary). MR 317266

Mappings between ANR’s that are fine homotopy equivalences

Daniel S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1965), 584. MR 179785, DOI 10.1090/S0002-9939-1965-0179785-7
J. Keesling, A non-movable trivial-shape decomposition of the Hilbert cube, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 9, 997–998 (English, with Russian summary). MR 391100

Images of ANR’s

G. Kozlowski and J. Segal, Locally well-behaved paracompacta in shape theory, Fund. Math. 95 (1977), no. 1, 55–71. MR 442926, DOI 10.4064/fm-95-1-55-71
G. Kozlowski and J. Segal, Movability and shape-connectivity, Fund. Math. 93 (1976), no. 2, 145–154. MR 431158, DOI 10.4064/fm-93-2-145-154
Krystyna Kuperberg, Two Vietoris-type isomorphism theorems in Borsuk’s theory of shape, concerning the Vietoris-Cech homology and Borsuk’s fundamental groups, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 285–313. MR 0383398

Shape theory

The topology of CW complexes

Sibe Mardešić, Shapes for topological spaces, General Topology and Appl. 3 (1973), 265–282. MR 324638, DOI 10.1016/0016-660X(72)90018-9

On the Whitehead theorem in shape theory

Sibe Mardešić and Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), no. 1, 41–59. MR 298634, DOI 10.4064/fm-72-1-41-59
Sibe Mardešić and Jack Segal, Equivalence of the Borsuk and the ANR-system approach to shapes, Fund. Math. 72 (1971), no. 1, 61–68. (errata insert). MR 301694, DOI 10.4064/fm-72-1-61-68
Kiiti Morita, On shapes of topological spaces, Fund. Math. 86 (1975), no. 3, 251–259. MR 388385, DOI 10.4064/fm-86-3-251-259
M. Moszyńska, The Whitehead theorem in the theory of shapes, Fund. Math. 80 (1973), no. 3, 221–263. MR 339159, DOI 10.4064/fm-80-3-221-263

Čech homotopy

6

47

50

Richard B. Sher, Realizing cell-like maps in Euclidean space, General Topology and Appl. 2 (1972), 75–89. MR 303546, DOI 10.1016/0016-660X(72)90039-6
Joseph L. Taylor, A counterexample in shape theory, Bull. Amer. Math. Soc. 81 (1975), 629–632. MR 375328, DOI 10.1090/S0002-9904-1975-13768-2
C. T. C. Wall, Finiteness conditions for $\textrm {CW}$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR 171284, DOI 10.2307/1970382
Karol Borsuk, Some remarks on shape properties of compacta, Fund. Math. 85 (1974), no. 2, 185–195. MR 353246, DOI 10.4064/fm-85-2-185-195
George Kozlowski and Jack Segal, Local behavior and the Vietoris and Whitehead theorems in shape theory, Fund. Math. 99 (1978), no. 3, 213–225. MR 482754, DOI 10.4064/fm-99-3-213-225
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
John Milnor, On spaces having the homotopy type of a $\textrm {CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272–280. MR 100267, DOI 10.1090/S0002-9947-1959-0100267-4
M. Moszyńska, Uniformly movable compact spaces and their algebraic properties, Fund. Math. 77 (1972), no. 2, 125–144. MR 322863, DOI 10.4064/fm-77-2-125-144
M. Moszyńska, Concerning the Whitehead theorem for movable compacta, Fund. Math. 92 (1976), no. 1, 43–55. MR 423348, DOI 10.4064/fm-92-1-43-55
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
S. Spież, Movability and uniform movability, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 43–45 (English, with Russian summary). MR 346741
J. Dydak, Some remarks concerning the Whitehead theorem in shape theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 4, 437–445 (English, with Russian summary). MR 400155