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)

 
 

 

Free products of topological groups which are $ k\sb{\omega }$-spaces


Author: Edward T. Ordman
Journal: Trans. Amer. Math. Soc. 191 (1974), 61-73
MSC: Primary 22A05
DOI: https://doi.org/10.1090/S0002-9947-1974-0352320-6
MathSciNet review: 0352320
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let G and H be topological groups and $ G \ast H$ their free product topologized in the manner due to Graev. The topological space $ G \ast H$ is studied, largely by means of its compact subsets. It is established that if G and H are $ {k_\omega }$-spaces (respectively: countable CW-complexes) then so is $ G \ast H$. These results extend to countably infinite free products. If G and H are $ {k_\omega }$-spaces, $ G \ast H$ is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group $ \pi (G \ast H)$. If $ {G_1}$ and $ {H_1}$ are quotients (continuous open homomorphic images) of G and H, then $ {G_1} \ast {H_1}$ is a quotient of $ G \ast H$.


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

  • [1] C. H. Dowker, Topology of metric complexes, Amer. J. Math. 74 (1952), 555-577. MR 13, 965. MR 0048020 (13:965h)
  • [2] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [3] M. I. Graev, Free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279-324; English transl., Amer. Math. Soc. Transl. (1) 8 (1962), 305-364. MR 10, 11. MR 0025474 (10:11d)
  • [4] -, On free products of topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), 343-354. (Russian) MR 12, 158. MR 0036768 (12:158c)
  • [5] A. Hulanicki, Isomorphic embeddings of free products of compact groups, Colloq. Math. 16 (1967), 235-241. MR 35 #300. MR 0209402 (35:300)
  • [6] B. L. Madison, Congruences in topological semigroups, Second Florida Symposium on Automata and Semigroups, University of Florida, Gainesville, Fl., 1971, Part II.
  • [7] M. C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273-298. MR 40 #4946. MR 0251719 (40:4946)
  • [8] E. Michael, Bi-quotient maps and Cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 2, 287-302. MR 39 #6277. MR 0244964 (39:6277)
  • [9] S. A. Morris, Free products of topological groups, Bull. Austral. Math. Soc. 4 (1971), 17-29. MR 43 #410. MR 0274647 (43:410)
  • [10] -, Local compactness and free products of topological groups. I (to appear).
  • [11] -, Local compactness and free products of topological groups. II (to appear).
  • [12] E. T. Ordman, Free products of topological groups with equal uniformities. I, Colloq. Math. (to appear). MR 0374319 (51:10519)
  • [13] -, Free products of topological groups with equal uniformities.II, Colloq. Math. (to appear).
  • [14] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133-152. MR 35 #970. MR 0210075 (35:970)
  • [15] J. H. C. Whitehead, Combinatorial homotopy. I. Bull. Amer. Math. Soc. 55 (1949), 213-245. MR 11, 48. MR 0030759 (11:48b)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22A05

Retrieve articles in all journals with MSC: 22A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0352320-6
Keywords: Free product of topological groups, k-space, $ {k_\omega }$-space, CW-complex, quotient of topological groups, nonmetrizable group
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society