𝐾-groups generated by 𝐾-spaces

Nummela, Eric C.

doi:10.1090/S0002-9947-1975-0352319-0

$K$-groups generated by $K$-spaces
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by Eric C. Nummela PDF

Trans. Amer. Math. Soc. 201 (1975), 279-289 Request permission

Abstract:

A $K$-group $G$ with identity $e$ is said to be generated by the $K$-space $X$ if $X$ is a subspace of $G$ containing $e,X$ algebraically generates $G$, and the canonical morphism from the Graev free $K$-group over $(X,e)$ on-to $G$ is a quotient morphism. An internal characterization of the topology of such a group $G$ is obtained, as well as a sufficient condition that a subgroup $H$ of $G$ be generated by a subspace $Y$ of $H$. Several illuminating examples are provided.

References

R. M. Dudley, Continuity of homomorphisms, Duke Math. J. 28 (1961), 587–594. MR 136676, DOI 10.1215/S0012-7094-61-02859-9
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
M. I. Graev, Free topological groups, Izvestiya Akad. Nauk SSSR. Ser. Mat. 12 (1948), 279–324 (Russian). MR 0025474

$k$-groups

John Mack, Sidney A. Morris, and Edward T. Ordman, Free topological groups and the projective dimension of a locally compact abelian group, Proc. Amer. Math. Soc. 40 (1973), 303–308. MR 320216, DOI 10.1090/S0002-9939-1973-0320216-6
A. Markoff, On free topological groups, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 9 (1945), 3–64 (Russian, with English summary). MR 0012301
M. C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273–298. MR 251719, DOI 10.1090/S0002-9947-1969-0251719-4
John Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272–284. MR 77122, DOI 10.2307/1969609

Homological algebra of $K$-modules

N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075, DOI 10.1307/mmj/1028999711
Rainer M. Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. (Basel) 22 (1971), 545–555. MR 300277, DOI 10.1007/BF01222616
David Wigner, Algebraic cohomology of topological groups, Trans. Amer. Math. Soc. 178 (1973), 83–93. MR 338132, DOI 10.1090/S0002-9947-1973-0338132-7