Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An algebraic property of the Čech cohomology groups which prevents local connectivity and movability

Author: James Keesling
Journal: Trans. Amer. Math. Soc. 190 (1974), 151-162
MSC: Primary 55B05
MathSciNet review: 0367973
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let C denote the category of compact Hausdorff spaces and $ H:C \to HC$ be the homotopy functor. Let $ S:C \to SC$ be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and $ {H^n}(X)$ denote n-dimensional Čech cohomology with integer coefficients. Let $ {A_x} = {\text{char}}\;{H^1}(X)$ be the character group of $ {H^1}(X)$ considering $ {H^1}(X)$ as a discrete group. In this paper it is shown that there is a shape morphism $ F \in {\text{Mor}_{SC}}(X,{A_X})$ such that $ {F^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping $ f:X \to {A_X}$ such that $ S(f) = F$ and thus that $ {f^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. This result is applied to show that if X is locally connected, then $ {H^1}(X)$ has property L. Examples are given to show that X may be locally connected and $ {H^n}(X)$ not have property L for $ n > 1$. The result is also applied to compact connected topological groups.

In the last section of the paper it is shown that if X is compact and movable, then for every integer n, $ {H^n}(X)/{\operatorname{Tor}}\;{H^n}(X)$ has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that $ {H^n}(X)$ itself may not have property L even if X is a finite-dimensional movable continuum.

References [Enhancements On Off] (What's this?)

  • [1] K. Borsuk, On a locally connected non-movable continuum, Bull. Acad. Polon. Sci. Sét. Sci. Math. Astronom. Phys. 17 (1969), 425-430. MR 41 #1012. MR 0256356 (41:1012)
  • [2] A. M. Gleason and R. S. Palais, On a class of transformation groups, Amer. J. Math. 79 (1957), 631-648. MR 19, 663. MR 0089367 (19:663d)
  • [3] W. Holsztyński, An extension and axiomatic characterization of Borsuk's theory of shape, Fund. Math. 70 (1971), no. 2, 157-168. MR 43 #8080. MR 0282368 (43:8080)
  • [4] P. J. Huber, Homotopical cohomology and Čech cohomology, Math. Ann. 144 (1961), 73-76. MR 24 #A3646. MR 0133821 (24:A3646)
  • [5] J. Keesling, Shape theory and compact connected abelian topological groups, Trans. Amer. Math. Soc. (to appear). MR 0345064 (49:9803)
  • [6] -, Continuous functions induced by shape morphisms, Proc. Amer. Math. Soc. 41 (1973), 315-320. MR 0334141 (48:12460)
  • [7] S. Mardesič and J. Segal, Movable compacta and ANR-systems, Bull. Polon. Acad. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 649-654. MR 44 #1026. MR 0283796 (44:1026)
  • [8] -, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), 41-59. MR 45 #7686. MR 0298634 (45:7686)
  • [9] R. Overton and J. Segal, A new construction of movable compacta, Glasnik Mat. 6 (26) (1971), 361-363. MR 0322796 (48:1157)
  • [10] L. S. Pontrjagin, Continuous groups, 2nd ed., GITTL, Moscow, 1954; English transl., Topological groups, Gordon and Breach, New York, 1966. MR 17, 171; 34 #1439.
  • [11] W. Scheffer, Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562-567. MR 46 #288. MR 0301130 (46:288)
  • [12] E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [13] N. Steenrod, Universal homology groups, Amer. J. Math. 58 (1936), 661-701. MR 1507191

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55B05

Retrieve articles in all journals with MSC: 55B05

Additional Information

Keywords: Shape functor, Čech cohomology, compact space, compact connected topological group, local connectivity, movability, property L, continuous homomorphism
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society