Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The surjectivity of the canonical homomorphism from singular homology to Cech homology

Authors: Katsuya Eda and Kazuhiro Kawamura
Journal: Proc. Amer. Math. Soc. 128 (2000), 1487-1495
MSC (1991): Primary 55N10, 55N05
Published electronically: December 8, 1999
MathSciNet review: 1712917
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $X$ be a locally $n$-connected compact metric space. Then, the canonical homomorphism from the singular homology group $H_{n+1}(X)$ to the Cech homology group $\check{H}_{n+1}(X)$ is surjective. Consequently, if a compact metric space $X$ is locally connected, then the canonical homomorphism from $H_1(X)$ to ${\check H}_1(X)$ is surjective.

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

  • 1. C. H. Dowker, Homology groups of relations, Ann. Math. 56 (1952), 84-95. MR 13:967d
  • 2. J. Dydak, On algebraic properties of continua, Bull. de l'acad.Pol. Sci. 27 (1979), 717-721. MR 82e:55020
  • 3. K. Eda and K. Kawamura, The fundamental groups of 1-dimensional spaces, Topology Appl. 87 (1998), 163-172. MR 99b:55022
  • 4. K. Eda and K. Sakai, A factor of singular homology, Tsukuba J. Math. 15 (1991), 351-387. MR 93d:55008
  • 5. S. Ferry, A stable converse to the vietoris-smale theorem with applications to shape theory, Trans. Amer. Math. Soc. 261 (1980), 369-386. MR 82c:55018
  • 6. H. Sagan, Space-filling curves, Springer, 1994. MR 95h:00001
  • 7. S. Marde\v{s}i\'{c}, Equivalence of singular and \v{C}ech homology for ANR-s application to unicoherence, Fund. Math. 46 (1958), 29-45. MR 20:5472
  • 8. -, Comparison of singular and \v{C}ech homology in locally connected spaces, Michigan J. Math. 6 (1959), 151-166. MR 21:4415
  • 9. S. Marde\v{s}i\'{c} and J. Segal, Shape theory $($the inverse system approach$)$, North Holland, 1982. MR 84b:55020
  • 10. E. H. Spanier, Algebraic topology, McGraw-Hill, 1966. MR 35:1007

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 55N10, 55N05

Retrieve articles in all journals with MSC (1991): 55N10, 55N05

Additional Information

Katsuya Eda
Affiliation: School of Science and Engineering, Waseda University, Tokyo 169-0072, Japan

Kazuhiro Kawamura
Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba 305, Japan

Keywords: Singular homology, \v Cech homology, canonical homomorphism, surjective
Received by editor(s): July 29, 1997
Published electronically: December 8, 1999
Communicated by: Ralph Cohen
Article copyright: © Copyright 2000 American Mathematical Society