The Dror-Whitehead theorem in prohomotopy and shape theories

Author:
S. Singh

Journal:
Trans. Amer. Math. Soc. **268** (1981), 487-496

MSC:
Primary 55P10; Secondary 55P55

DOI:
https://doi.org/10.1090/S0002-9947-1981-0632540-7

MathSciNet review:
632540

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Abstract: Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror's theorems in pro-homotopy and shape theories. More specifically, suppose is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then induces isomorphisms of the homotopy pro-groups, for instance, when and are simple, nilpotent, complete, or -objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.

**[**M. Artin and B. Mazur,**AM**]*Étale homotopy*, Lecture Notes in Math., vol. 100, Springer, Berlin and New York, 1969. MR**0245577 (39:6883)****[**A. K. Bousfield and D. M. Kan,**BK**]*Homotopy limits, completions and localizations*, Lecture Notes in Math., vol. 304, Springer, New York, 1972. MR**0365573 (51:1825)****[**K. Borsuk,**BO**]*Theory of shape*, Monografie Mat., no. 59, PWN, Warsaw, 1975. MR**0418088 (54:6132)****[**E. Dror,**DR**]*A generalization of the Whitehead theorem*, Lecture Notes in Math., vol. 249, Springer, Berlin and New York, 1971, pp. 13-22. MR**0350725 (50:3217)****[**J. Dydak and J. Segal,**DS**]*Shape theory*, Lecture Notes in Math., vol. 688, Springer, Berlin and New York, 1978. MR**520227 (80h:54020)****[**D. A. Edwards and H. M. Hastings,**ED**]*Čech and Steenrod homotopy theories with applications to geometric topology*, Lecture Notes in Math., vol. 542, Springer, New York, 1976. MR**0428322 (55:1347)****[**B. Eckmann and P. Hilton,**EH**]*Group-like structures in general cattegories*. I:*Multiplications and comultiplications*, Math. Ann.**145**(1962), 227-255. MR**0136642 (25:108)****[**D. A. Edwards,**EW**]*Étale homotopy theory and shape*, Algebraic and Geometric Methods in Topology (edited by L. McAuley), Lecture Notes in Math., vol. 428, Springer, Berlin and New York, 1974, pp. 58-107. MR**0375295 (51:11491)****[**A. Grothendieck,**GR**]*Technique de decene . . .*, Sém. Bourbaki, 12 année, 1959-60, Exp. 195, pp. 1-22. MR**2320488 (2008b:01007)****[**P. Hilton, G. Mislin and J. Roitberg,**HI**]*Localization of nilpotent groups and spaces*, North-Holland Math. Studies, North-Holland, Amsterdam; American Elsevier, New York, 1975. MR**0478146 (57:17635)****[**S. T. Hu,**HU**]*Homotopy theory*, Academic Press, New York, 1959. MR**0106454 (21:5186)****[**S. Mardešić,**MA**]*On the Whitehead theorem in shape theory*. I, II, Fund. Math.**91**(1976), 51-64; 93-103. MR**0407798 (53:11568)****[**S. Mac Lane,**ML**]*Homology*, Springer, New York, 1975. MR**1344215 (96d:18001)****[**M. Moszyńska,**MO**]*The Whitehead theorem in the theory of shape*, Fund. Math.**80**(1973), 221-263.**[**K. Morita,**MR**]*The Hurewicz and Whitehead theorems in shape theory*, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A**12**(1974), 246-258. MR**0372848 (51:9052)****[**M. M. Postnikov,**PO**]*Localization of topological spaces*, Uspehi Mat. Nauk**32**(6) (1977), 117-181 = Russian Math. Surveys**32**(6) (1977), 121-184. MR**0515419 (58:24258)****[**M. Raussen,**RA**]*Hurewicz and Whitehead Theorems in pro-categories*, Arch. Math.**30**(1978), 153-164. MR**0494090 (58:13021)****[**S. Singh,**SI**]*Pro-homology and isomorphisms of pro-groups*, J. Pure Appl. Algebra (to appear). MR**639574 (83a:20066)****[**-,**SI**]*Mittag-Leffler for homotopy pro-groups of movable continua*, Glas. Mat. Ser. III (to appear). MR**634301 (83e:55005)****[**E. H. Spanier,**SP**]*Algebraic topology*, McGraw-Hill, New York, 1966. MR**0210112 (35:1007)****[**J. R. Stallings,**ST**]*Homology and central series of groups*, J. Algebra**2**(1965), 170-181. MR**0175956 (31:232)****[**G. W. Whitehead,**WH**]*Elements of homotopy theory*, Graduate Texts in Math., vol. 61, Springer, New York, 1978. MR**516508 (80b:55001)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0632540-7

Keywords:
Pro-homotopy,
shape,
nilpotent spaces,
homology pro-groups,
homotopy pro-groups,
-nilpotent continua,
-structures

Article copyright:
© Copyright 1981
American Mathematical Society