The stability problem in shape, and a Whitehead theorem in pro-homotopy

Edwards, David A.; Geoghegan, Ross

doi:10.1090/S0002-9947-1975-0413095-6

The stability problem in shape, and a Whitehead theorem in pro-homotopy
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by David A. Edwards and Ross Geoghegan PDF

Trans. Amer. Math. Soc. 214 (1975), 261-277 Request permission

Abstract:

Theorem 3.1 is a Whitehead theorem in pro-homotopy for finite-dimensional pro-complexes. This is used to obtain necessary and sufficient algebraic conditions for a finite-dimensional tower of complexes to be pro-homotopy equivalent to a complex (§4) and for a finite-dimensional compact metric space to be pointed shape equivalent to an absolute neighborhood retract (§5).

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
Karol Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223–254. MR 229237, DOI 10.4064/fm-62-3-223-254
Karol Borsuk, Fundamental retracts and extensions of fundamental sequences, Fund. Math. 64 (1969), 55-85; errata, ibid. 64 (1969), 375. MR 0243520, DOI 10.4064/fm-64-1-55-85
K. Borsuk, On the shape of $\textrm {FANR}$-sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 529–532 (English, with Russian summary). MR 256394
Karol Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970), 221–240. MR 266168, DOI 10.4064/fm-67-2-221-240
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, DOI 10.1007/978-3-540-38117-4
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, Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy, Trans. Amer. Math. Soc. 219 (1976), 351–360. MR 402735, DOI 10.1090/S0002-9947-1976-0402735-4

Stability theorems in shape and pro-homotopy

Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492
Ralph H. Fox, On shape, Fund. Math. 74 (1972), no. 1, 47–71. MR 296914, DOI 10.4064/fm-74-1-47-71
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
Peter Hilton, Homotopy theory and duality, Gordon and Breach Science Publishers, New York-London-Paris, 1965. MR 0198466
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
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

The topology of CW complexes

James Keesling, On the Whitehead theorem in shape theory, Fund. Math. 92 (1976), no. 3, 247–253. MR 431156, DOI 10.4064/fm-92-3-247-253
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
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
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
Dock Sang Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700–712. MR 104721, DOI 10.2307/1970033
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
C. T. C. Wall, Finiteness conditions for $\textrm {CW}$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR 171284, DOI 10.2307/1970382