A stable converse to the VietorisSmale theorem with applications to shape theory
Author:
Steve Ferry
Journal:
Trans. Amer. Math. Soc. 261 (1980), 369386
MSC:
Primary 55R65; Secondary 54C56, 55P55, 57N20, 57Q05, 57Q10
MathSciNet review:
580894
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Our main result says that if is a map between finite polyhedra which has kconnected homotopy fiber, then there is an n such that is homotopic to a map with kconnected pointinverses. This result is applied to give an algebraic characterization of compacta shape equivalent to locally nconnected compacta. We also show that a compactum can be ``improved'' within its shape class until its homotopy theory and strong shape theory are the same with respect to finite dimensional polyhedra.
 [Bo]
Karol
Borsuk, Theory of shape, PWN—Polish Scientific
Publishers, Warsaw, 1975. Monografie Matematyczne, Tom 59. MR 0418088
(54 #6132)
 [B]
Morton
Brown, Some applications of an approximation
theorem for inverse limits, Proc. Amer. Math.
Soc. 11 (1960),
478–483. MR 0115157
(22 #5959), http://dx.doi.org/10.1090/S00029939196001151574
 [Ch]
T.
A. Chapman, Lectures on Hilbert cube manifolds, American
Mathematical Society, Providence, R. I., 1976. Expository lectures from the
CBMS Regional Conference held at Guilford College, October 1115, 1975;
Regional Conference Series in Mathematics, No. 28. MR 0423357
(54 #11336)
 [ChS]
T.
A. Chapman and L.
C. Siebenmann, Finding a boundary for a Hilbert cube manifold,
Acta Math. 137 (1976), no. 34, 171–208. MR 0425973
(54 #13922)
 [Co]
Marshall
M. Cohen, A course in simplehomotopy theory, SpringerVerlag,
New YorkBerlin, 1973. Graduate Texts in Mathematics, Vol. 10. MR 0362320
(50 #14762)
 [CD]
Donald
Coram and Paul
Duvall, Approximate fibrations and a movability condition for
maps, Pacific J. Math. 72 (1977), no. 1,
41–56. MR
0467745 (57 #7597)
 [DT]
Albrecht
Dold and René
Thom, Quasifaserungen und unendliche symmetrische Produkte,
Ann. of Math. (2) 67 (1958), 239–281 (German). MR 0097062
(20 #3542)
 [D]
James
Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass.,
1966. MR
0193606 (33 #1824)
 [DS]
J.
Dydak and J.
Segal, Strong shape theory, Dissertationes Math. (Rozprawy
Mat.) 192 (1981), 39. MR 627528
(82j:54077)
 [EG]
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 0375330
(51 #11525)
 [EH]
David
A. Edwards and Harold
M. Hastings, Čech and Steenrod homotopy theories with
applications to geometric topology, Lecture Notes in Mathematics, Vol.
542, SpringerVerlag, BerlinNew York, 1976. MR 0428322
(55 #1347)
 [F]
Steve
Ferry, Homotopy, simple homotopy and compacta, Topology
19 (1980), no. 2, 101–110. MR 572578
(81j:57010), http://dx.doi.org/10.1016/00409383(80)900014
 [F]
Steve
Ferry, Shape equivalence does not imply CE
equivalence, Proc. Amer. Math. Soc.
80 (1980), no. 1,
154–156. MR
574526 (81d:55013), http://dx.doi.org/10.1090/S00029939198005745261
 [KO]
Y.
Kodama and J.
Ono, On fine shape theory, Fund. Math. 105
(1979/80), no. 1, 29–39. MR 558127
(81d:57014)
 [KS]
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 0482754
(58 #2808)
 [K]
Józef
Krasinkiewicz, Local connectedness and pointed 1movability,
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.
25 (1977), no. 12, 1265–1269 (English, with
Russian summary). MR 0500986
(58 #18468)
 [L]
R.
C. Lacher, Celllike mappings and their
generalizations, Bull. Amer. Math. Soc.
83 (1977), no. 4,
495–552. MR 0645403
(58 #31095), http://dx.doi.org/10.1090/S000299041977143218
 [MR]
Sibe
Mardešić and T.
B. Rushing, 𝑛shape fibrations, Proceedings of the
1978 Topology Conference (Univ. Oklahoma, Norman, Okla., 1978), II, 1978,
pp. 429–459 (1979). MR 540505
(80m:54028)
 [Q]
J.
Brendan Quigley, An exact sequence from the 𝑛th to the
(𝑛1)st fundamental group, Fund. Math. 77
(1973), no. 3, 195–210. MR 0331379
(48 #9712)
 [Si]
L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension , Thesis, Princeton University, 1965.
 [Sm]
Stephen
Smale, A Vietoris mapping theorem for
homotopy, Proc. Amer. Math. Soc. 8 (1957), 604–610. MR 0087106
(19,302f), http://dx.doi.org/10.1090/S00029939195700871069
 [S]
Edwin
H. Spanier, Algebraic topology, McGrawHill Book Co., New
YorkToronto, Ont.London, 1966. MR 0210112
(35 #1007)
 [W]
C.
T. C. Wall, Finiteness conditions for
𝐶𝑊complexes, Ann. of Math. (2) 81
(1965), 56–69. MR 0171284
(30 #1515)
 [Wa]
John
J. Walsh, Monotone and open mappings on
manifolds. I, Trans. Amer. Math. Soc. 209 (1975), 419–432.
MR
0375326 (51 #11521), http://dx.doi.org/10.1090/S00029947197503753260
 [Wa]
, Light open and open mappings on manifolds. II (preprint).
 [Wh]
J. H. C. Whitehead, Simplicial spaces, nucleii, and mgroups, Proc. London Math. Soc. 45 (1939), 243327.
 [Wi]
David
Wilson, Open mappings on manifolds and a counterexample to the
Whyburn conjecture, Duke Math. J. 40 (1973),
705–716. MR 0320989
(47 #9522)
 [Wi]
David
C. Wilson, On constructing monotone and 𝑈𝑉¹
mappings of arbitrary degree, Duke Math. J. 41
(1974), 103–109. MR 0339250
(49 #4011)
 [HI]
L.
S. Husch and I.
Ivanšić, Shape domination and embedding up to
shape, Compositio Math. 40 (1980), no. 2,
153–166. MR
563539 (81h:55007)
 [Bo]
 K. Borsuk, Theory of shape, Monografie Mat., vol.59, Polish Science Publ., Warszawa, 1975. MR 0418088 (54:6132)
 [B]
 M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478483. MR 0115157 (22:5959)
 [Ch]
 T. A. Chapman, Notes on Hilbert cube manifolds, CBMS Regional Conf. Ser. in Math., no. 28, Amer. Math. Soc., Providence, R. I., 1976. MR 0423357 (54:11336)
 [ChS]
 T. A. Chapman and L. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976), 171208. MR 0425973 (54:13922)
 [Co]
 M. M. Cohen, A course in simple homotopy theory, SpringerVerlag, New York, 1970. MR 0362320 (50:14762)
 [CD]
 D. Coram and P. Duvall, Approximate fibrations and a movability condition for maps, Pacific J. Math. 72 (1977), 4156. MR 0467745 (57:7597)
 [DT]
 A. Dold and R. Thom, Quasifaserungen und unendliche symmetrische Producte, Ann. of Math. (2) 67 (1958), 239281. MR 0097062 (20:3542)
 [D]
 J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
 [DS]
 J. Dydak and J. Segal, Strong shape theory (preprint). MR 627528 (82j:54077)
 [EG]
 D. Edwards and R. Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975), 521535. MR 0375330 (51:11525)
 [EH]
 D. Edwards and H. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math., vol. 542, SpringerVerlag, Berlin and New York, 1976. MR 0428322 (55:1347)
 [F]
 S. Ferry, Homotopy, simple homotopy, and compacta. Topology (to appear). MR 572578 (81j:57010)
 [F]
 , Shape equivalence does not imply CE equivalence, Proc. Amer. Math. Soc. (to appear). MR 574526 (81d:55013)
 [KO]
 Y. Kodama and Y. Ono, On fine shape theory. I, II, Fund. Math. (to appear). MR 558127 (81d:57014)
 [KS]
 G. Kozlowski and J. Segal, Local behavior and the Vietoris and Whitehead theorems in shape theory, Fund. Math. 99 (1978), 213225. MR 0482754 (58:2808)
 [K]
 J. Krasinkiewicz, Local connectedness and pointed 1movability, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977). MR 0500986 (58:18468)
 [L]
 R. Lacher, Celllike mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 495552. MR 0645403 (58:31095)
 [MR]
 S. Mardesič and T. B. Rushing, nshape fibrations (preprint). MR 540505 (80m:54028)
 [Q]
 J. Quigley, An exact sequence from the nth to the fundamental group, Fund. Math. 77 (1973), 195210. MR 0331379 (48:9712)
 [Si]
 L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension , Thesis, Princeton University, 1965.
 [Sm]
 S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604610. MR 0087106 (19:302f)
 [S]
 E. Spanier, Algebraic topology, McGrawHill, New York, 1966. MR 0210112 (35:1007)
 [W]
 C. T. C. Wall, Finiteness conditions for CW complexes, Ann. of Math. (2) 81 (1965), 5569. MR 0171284 (30:1515)
 [Wa]
 J. Walsh, Monotone and open mappings on manifolds. I, Trans. Amer. Math. Soc. (to appear). MR 0375326 (51:11521)
 [Wa]
 , Light open and open mappings on manifolds. II (preprint).
 [Wh]
 J. H. C. Whitehead, Simplicial spaces, nucleii, and mgroups, Proc. London Math. Soc. 45 (1939), 243327.
 [Wi]
 D. Wilson, Open mappings on manifolds and a counterexample to the Whyburn conjecture, Duke Math. J. 40 (1973), 705716. MR 0320989 (47:9522)
 [Wi]
 , On constructing monotone and mappings of arbitrary degree, Duke Math. J. 41 (1974), 103109. MR 0339250 (49:4011)
 [HI]
 L. Husch and I. Ivansič, Shape domination and imbedding up to shape (preprint). MR 563539 (81h:55007)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
55R65,
54C56,
55P55,
57N20,
57Q05,
57Q10
Retrieve articles in all journals
with MSC:
55R65,
54C56,
55P55,
57N20,
57Q05,
57Q10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005808941
PII:
S 00029947(1980)05808941
Keywords:
VietorisSmale theorem,
map,
strong shape theory,
Hilbert cube manifold,
finiteness obstruction
Article copyright:
© Copyright 1980
American Mathematical Society
