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On weak shape equivalences

Morón, Manuel A. and Romero Ruiz del Portal, Francisco (1999) On weak shape equivalences. Topology and its Applications, 92 (3). pp. 225-236. ISSN 0166-8641

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Abstract

We prove that weak shape equivalences are monomorphisms in the shape category of uniformly pointed movable continua Sh(M). We use an example of Draper and Keesling to show that weak shape equivalences need not be monomorphisms in the shape category. We deduce that Sh(M) is not balanced. We give a characterization of weak dominations in the shape category of pointed continua, in the sense of Dydak (1979). We introduce the class of pointed movable triples (X,F,Y), for a shape morphism F:X --> Y, and we establish an infinite-dimensional Whitehead theorem in shape theory from which we obtain, as a corollary, that for every pointed movable pair of continua (Y,X) the embedding j: X --> Y is a shape equivalence iff it is a weak shape equivalence.


Item Type:Article
Uncontrolled Keywords:Homotopy; monomorphisms; epimorphisms; weak shape equivalence; shape category of uniformly pointed movable continua; monomorphisms and epimorphisms in categories
Subjects:Sciences > Mathematics > Topology
ID Code:15497
References:

M.F. Atiyah and G.B. Segal, Equivariant K-theory and completion, J. Differential Geom. 3 (1969) 1-18.

K. Borsuk, Theory of Shape, Monografie Matematyczne 59 (PWN, Warsaw, 1975).

K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968) 223-254.

J. Draper and J. Keesling, An example concerning the Whitehead theorem in shape theory, Fund. Math. 92 (1976) 255-259.

J. Dydak, The Whitehead and the Smale theorems in shape theory, Dissertationes Math. 156 (1979) 1-55.

J. Dydak, Epimorphisms and monomorphisms in homotopy, Proc. Amer. Math. Sot. 116 (4) (1992) 1171-1173.

J. Dydak and J. Segal, Shape Theory: An Introduction, Lecture Notes in Math. 688 (Springer, Berlin, 1978).

E. Dyer and J. Roitberg, Homotopy-epimorphisms, homotopy-monomorphisms and homotopyequivalences, Topology Appl. 46 (1992) 119-124.

D.A. Edwards and R. Geoghegan, Compacta weak equivalent to ANR’s, Fund. Math. 90 (1975) 115-124.

D.A. Edwards and R. Geoghegan, Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy, Trans. Amer. Math. Soc. 219 (1976) 351-360.

R. Geoghegan, Elementary proofs of stability theorems in pro-homotopy and shape, General Topology Appl. 8 (1978) 265-281.

D.S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1965) 584.

J.E. Keesling, On the Whitehead theorem in shape theory, Fund. Math. 92 (1976) 247-253.

S. Mardešić and J. Segal, Shape Theory (North-Holland, Amsterdam, 1982).

K. Morita, The Hurewicz and the Whitehead theorems in shape theory, Sci. Rep. Tokyo Kyoiku Daigaku. Sec. A. 12 (1974) 246-258.

M.A. Morón and F.R. Ruiz del Portal, Shape as a Cantor completion process, Math. Z. 225 (1) (1997) 67-86.

M.A. Morón and F.R. Ruiz del Portal, Counting shape and homotopy types among FANR’s: An elementary approach, Manuscripta Math. 79 (1993) 411-414.

M.A. Morón and F.R. Ruiz del Portal, Ultrametrics and infinite dimensional Whitehead theorems in shape theory, Manuscripta Math. 89 (1996) 325-333.

M.A. Morón and F.R. Ruiz del Portal, Spaces of discrete shape and C-refinable maps that induce shape equivalences, J. Math. Soc. Japan 49 (4) (1997) 713-721.

E. Cuchillo-Ibáñez, M.A. Morón, F.R. Ruiz del Portal and J.M.R. Sanjurjo, A topology for the sets of shape morphisms, Topology Appl. 94 (1999).

M. Moszyńska, Uniformly movable compact spaces and their algebraic properties, Fund. Math. 77 (1972) 125-144.

M. Moszyńska, The Whitehead theorem in the theory of shapes, Fund. Math. 80 (1973) 221-263.

R.H. Overton aud J. Segal, A new construction of movable compacta, Glasnik Mat. 6 (1971) 361-363.

W.H. Schikhof, Ultrametric Calculus. An Introduction to p-adic Analysis (Cambridge University Press, 1984).

E. Spanier, Algebraic Topology (McGraw-Hill, New York, 1966).

Deposited On:06 Jun 2012 07:58
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