The topology of free topological groups

O. V. Sipacheva

Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 2, pp. 99-204.

Résumé

The known explicit descriptions of the topology of free and free Abelian topological groups are collected. Examples of the application of such descriptions by various authors to study the properties of topological groups and some related topological-algebraic objects are given. Various approaches to describing the topology of free topological groups are presented and a general method for topologizing free groups is suggested. The fundamental properties of free topological groups and their structure are considered. The most important results on free topological groups obtained by various authors through the years are mentioned.

Bibliographie

[1] Aleksandrov P. S., Pasynkov B. A., Vvedenie v teoriyu razmernosti. Vvedenie v teoriyu topologicheskikh prostranstv i obschuyu teoriyu razmernosti, Nauka, M., 1973 | MR

[2] Arkhangelskii A. V., “Ob otobrazheniyakh, svyazannykh s topologicheskimi gruppami”, DAN SSSR, 181:6 (1968), 1303–1306

[3] Arkhangelskii A. V., Topologicheskie prostranstva i nepreryvnye otobrazheniya. Zamechaniya o topologicheskikh gruppakh, Izd-vo Mosk. un-ta, M., 1969

[4] Arkhangelskii A. V., “O sootnosheniyakh mezhdu invariantami topologicheskikh grupp i ikh podprostranstv”, Uspekhi mat. nauk, 35:3 (1980), 3–22 | MR

[5] Arkhangelskii A. V., “Klassy topologicheskikh grupp”, Uspekhi mat. nauk, 36:3 (1981), 127–146 | MR

[6] Arkhangelskii A. V., “Lyubaya topologicheskaya gruppa yavlyaetsya faktor-gruppoi nulmernoi topologicheskoi gruppy”, DAN SSSR, 258:5 (1981), 1037–1040 | MR

[7] Arkhangelskii A. V., “Algebraicheskie ob'ekty, porozhdennye topologicheskoi strukturoi”, Itogi nauki i tekhniki. Algebra. Topologiya. Geometriya, 25, VINITI, M., 1987, 141–198 | MR

[8] Belnov V. K., “O razmernosti svobodnykh topologicheskikh grupp”, Tezisy IV Tiraspolskogo simpoziuma po obschei topologii i ee prilozheniyam, Shtiintsa, Kishinev, 1979, 14–15

[9] Borubaev A. A., Chekeev A. A., “O $\tau$-polnote topologicheskikh grupp”, Issled. po topol. 7, Zap. nauch. sem. Sankt-Peterburg. otdel. Mat. In-ta im. V. A. Steklova RAN (POMI), 208, 1993, 103–114

[10] Graev M. I., “Svobodnye topologicheskie gruppy”, Izv. AN SSSR. Ser. matem., 12 (1948), 279–324 | MR | Zbl

[11] Graev M. I., “Teoriya topologicheskikh grupp, I”, Uspekhi mat. nauk, 5:2 (1950), 3–56 | MR | Zbl

[12] Guran I. I., O vlozheniyakh topologicheskikh grupp, Dep. v VINITI No 1483-81, Mosk. un-t, M., 1981 | Zbl

[13] Dumitrashku S. S., Choban M. M., “O svobodnykh topologicheskikh algebrakh s nepreryvnoi signaturoi”, Algebraicheskie i topologicheskie sistemy, Shtiintsa, Kishinev, 1982, 27–53 | MR

[14] Kats G. I., “Izomorfnoe otobrazhenie topologicheskikh grupp v pryamoe proizvedenie grupp, udovletvoryayuschikh pervoi aksiome schetnosti”, Uspekhi mat. nauk, 8:6 (1953), 107–113 | MR | Zbl

[15] Kiriyak L. L., Choban M. M., “Primenenie ravnomernykh struktur v issledovanii svobodnykh topologicheskikh algebr”, Sib. mat. zhurn., 33:5 (1992), 891–904 | MR | Zbl

[16] Maltsev A. I., “K obschei teorii algebraicheskikh sistem”, Mat. sb., 35(77):1 (1954), 3–20

[17] Maltsev A. I., “Svobodnye topologicheskie algebry”, Izv. AN SSSR. Ser. matem., 21:2 (1957), 171–198 | MR | Zbl

[18] Markov A. A., “O svobodnykh topologicheskikh gruppakh”, DAN SSSR, 31:4 (1941), 299–301 | Zbl

[19] Markov A. A., “O svobodnykh topologicheskikh gruppakh”, Izv. AN SSSR. Ser. matem., 9:1 (1945), 3–64

[20] Pestov V. G., O stroenii i vlozhenii topologicheskikh grupp, Dep. v VINITI. 03.04.81. No 1495-81, Tomskii un-t, Tomsk, 1981

[21] Pestov V. G., “Nekotorye svoistva svobodnykh topologicheskikh grupp”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1982, no. 1, 35–37 | MR | Zbl

[22] Pestov V. G., “Okrestnosti edinitsy v svobodnykh topologicheskikh gruppakh”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1985, no. 3, 8–10 | MR | Zbl

[23] Pestov V. G., “Svobodnye topologicheskie abelevy gruppy i dvoistvennost Pontryagina”, Vestnik Mosk. Un-ta. Ser. 1, Matematika, mekhanika, 1986, no. 1, 3–5 | MR | Zbl

[24] Reznichenko E. A., Sipacheva O. V., “Faktornye otobrazheniya na slova ogranichennoi dliny v svobodnykh topologicheskikh gruppakh”, Obschaya topologiya. Otobrazheniya, proizvedeniya i razmernost prostranstv, Izd-vo Mosk. un-ta, M., 1995, 98–119

[25] Sipacheva O. V., “Topologii na svobodnykh gruppakh”, V Tirasp. simp. po obschei topologii i ee pril., Shtiintsa, Kishinev, 1985, 220–222

[26] Sipacheva O. V., “Opisanie topologii svobodnykh topologicheskikh grupp bez ispolzovaniya universalnykh ravnomernykh struktur”, Obschaya topologiya. Otobrazheniya topologicheskikh prostranstv, Izd-vo Mosk. un-ta, M., 1986, 122–130 | MR

[27] Sipacheva O. V., “Kompakty s nepreryvnoi operatsiei Maltseva i retrakty topologicheskikh grupp”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1991, no. 1, 33–36 | MR | Zbl

[28] Sipacheva O. V., “Ob odnom klasse svobodnykh lokalno vypuklykh prostranstv”, Mat. sb., 194:3 (2003), 25–52 | MR | Zbl

[29] Sipacheva O. V., Uspenskii V. V., “Svobodnye topologicheskie gruppy bez malykh podgrupp”, Vestnik Mosk. un-ta. Ser. 1, Matematika, mekhanika, 1987, no. 4, 21–24 | MR

[30] Tkachenko M. G., O topologii svobodnykh po Markovu grupp, Proc. V Symp. on General Topology, Prague, 1982

[31] Tkachenko M. G., “O nulmernykh topologicheskikh gruppakh”, Trudy Leningradskoi mezhdunarodnoi konf. po topologii i ee prilozheniyam, Nauka, L., 1982, 113–118

[32] Tkachenko M. G., “O polnote svobodnykh abelevykh topologicheskikh grupp”, DAN SSSR, 269 (1983), 299–303 | MR | Zbl

[33] Tkachenko M. G., “O svoistve Suslina v svobodnykh topologicheskikh gruppakh nad bikompaktami”, Mat. zametki, 34:4 (1983), 601–607 | MR | Zbl

[34] Tkachenko M. G., “O polnote topologicheskikh grupp”, Sib. mat. zhurn., 25:1 (1984), 146–158 | MR | Zbl

[35] Tkachenko M. G., “O topologii svobodnykh grupp nad bikompaktami”, Otobrazheniya i funktory, Izd-vo Mosk. un-ta, M., 1984, 122–137 | MR

[36] Tkachenko M. G., “O spektralnom razlozhenii svobodnykh topologicheskikh grupp”, Uspekhi mat. nauk, 39:2(236) (1984), 191–192 | MR | Zbl

[37] Tkachenko M. G., “O nulmernosti svobodnykh topologicheskikh grupp”, Dokl. Bolgarskoi akad. nauk, 38:2 (1985), 173–174 | MR | Zbl

[38] Tkachenko M. G., “O nekotorykh svoistvakh svobodnykh topologicheskikh grupp”, Mat. zametki, 37:1 (1985), 110–118 | MR | Zbl

[39] Tkachenko M. G., “Strogaya kollektivnaya normalnost i schetnaya kompaktnost v svobodnykh topologicheskikh gruppakh”, Sib. mat. zhurn., 28:5 (1987), 167–177 | MR | Zbl

[40] Uspenskii V. V., “Topologicheskaya gruppa, porozhdennaya lindelëfovym $\Sigma$-prostranstvom, obladaet svoistvom Suslina”, DAN SSSR, 265:4 (1982), 823–826 | MR | Zbl

[41] Uspenskii V. V., “O podgruppakh svobodnykh topologicheskikh grupp”, DAN SSSR, 285:5 (1985), 1070–1072 | MR | Zbl

[42] Uspenskii V. V., “O nepreryvnykh obrazakh lindelëfovykh topologicheskikh grupp”, DAN SSSR, 285:4 (1985), 824–827 | MR | Zbl

[43] Uspenskii V. V., “Topologicheskie gruppy i kompakty Dugundzhi”, Mat. sb., 1989, no. 8, 1092–1118 | MR | Zbl

[44] Uspenskii V. V., “Svobodnye topologicheskie gruppy metrizuemykh prostranstv”, Izv. AN SSSR. Ser. matem., 54:6 (1990), 1295–1319 | MR

[45] Khyuitt E., Ross K., Abstraktnyi garmonicheskii analiz, T. 1, Nauka, M., 1975

[46] Choban M. M., “O nekotorykh voprosakh teorii topologicheskikh grupp”, Obschaya algebra i diskretnaya geometriya. Matematicheskie nauki, Shtiintsa, Kishinev, 1980, 120–135 | MR

[47] Choban M. M., “K teorii topologicheskikh algebraicheskikh sistem”, Trudy MMO, 48, 1985, 106–149 | MR

[48] Choban M. M., “Obschie usloviya suschestvovaniya svobodnykh ob'ektov”, Acta Comment. Univ. Tartuensis, 836 (1989), 157–171 | MR

[49] Shakhmatov D. B., “Prekalibry $\sigma$-kompaktnykh topologicheskikh grupp”, Mat. zametki, 39:6 (1986), 859–868 | MR | Zbl

[50] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR

[51] Arhangel'skii A. V\@., “Topological invariants in algebraic environment”, Recent Progress in General Topology, 2, North-Holland, Amsterdam, 2002, 1–57 | MR

[52] Arhangel'skii A. V., Okunev O. G., and Pestov V. G., “Free topological groups over metrizable spaces”, Topol. and Its Appl., 33:1 (1989), 63–76 | DOI | MR

[53] Borges C. R., “Free topological groups”, J. Austral. Math. Soc. Ser. A, 23 (1977), 360–365 | DOI | MR | Zbl

[54] Choban M. M., “Some topics in topological algebra”, Topol. Appl., 54:1–3 (1993), 183–202 | DOI | MR | Zbl

[55] Choban M. M., “Algebraical equivalences of topological spaces”, Matematica, 2001, no. 1(35), 12–36 | MR | Zbl

[56] Comfort W. W., “Problems on topological groups and other homogeneous spaces”, Open Problems in Topology, eds. J. van Mill and G. M. Reed, North-Holland, Amsterdam, 1990, 313–347 | MR

[57] Comfort W. W., Hoffman K.-H., and Remus D., “Topological groups and semigroups”, Recent Progress in General Topology, North-Holland, Amsterdam, 1992, 57–144 | MR | Zbl

[58] Eda K., Ohta H., and Yamada K., “Prime subspaces in free topological groups”, Topol. and Its Appl., 62 (1995), 163–171 | DOI | MR | Zbl

[59] Enochs E. E., “Homotopy groups of compact Abelian groups”, Proc. Amer. Math. Soc., 15:6 (1964), 878–881 | DOI | MR | Zbl

[60] Fay T., Ordman E., and Thomas B. V. S., “The free topological group over rationals”, Gen. Topol. and Appl., 10:1 (1979), 33–47 | DOI | MR | Zbl

[61] Galindo J. and Hernández S., “Pontryagin–van Kampen reflexifity for free abelian topological groups”, Forum Math., 11 (1999), 399–415 | DOI | MR | Zbl

[62] Gartside P. M., Reznichenko E. A., and Sipacheva O. V., “Maltsev and Retral Spaces”, Topol. and Its Appl., 80 (1997), 115–129 | DOI | MR | Zbl

[63] Guran I. I., “Topology on an infinite symmetric group and condensations”, Comment. Math. Univ. Carolin., 22:2 (1981), 311–316 | MR | Zbl

[64] Hofmann K. H., “An essay on free compact groups”, Lect. Notes Math., 915, 1982, 171–197 | MR | Zbl

[65] Hofmann K. H. and Morris S. A., “Free compact groups. I: Free compact Abelian groups”, Topol. and Its Appl., 23:1 (1986), 41–64 | DOI | MR | Zbl

[66] Hunt D. C. and Morris S. A., “Free subgroups of free topological groups”, Proc. Second Internat. Conf. on the Theory of Groups, Lecture Notes in Mathematics, 372, ed. M. F. Newman, Springer, Berlin, 1974, 377–387 | MR

[67] Joiner C., “Free topological groups and dimension”, Trans. Amer. Math. Soc., 220 (1976), 401–418 | DOI | MR | Zbl

[68] Kakutani S., “Free topological groups and infinite direct product of topological groups”, Proc. Imp. Acad. Tokyo, 20 (1944), 595–598 | DOI | MR | Zbl

[69] Leiderman A., Morris S. A., and Pestov V., “The free Abelian topological group and the free locally convex space on the unit interval”, J. London Math. Soc., 56 (1997), 529–538 | DOI | MR | Zbl

[70] Mack J., Morris S. A., and Ordman E. T., “Free topological groups and the projective dimension of locally compact Abelian groups”, Proc. Amer. Math. Soc., 40 (1973), 303–308 | DOI | MR | Zbl

[71] Michael E., “Local compactness and Cartesian product of quotient maps and $k$-spaces”, Ann. Inst. Fourier, 18 (1968), 281–286 | MR | Zbl

[72] Montgomery D. and Zippin L., Topological Transformation Groups, New York, 1955 | MR

[73] Morris S. A., “Varieties of topological groups”, Bull. Austral. Math. Soc., 1:2 (1969), 145–160 | DOI | MR | Zbl

[74] Morris S. A., “Varieties of topological groups, II”, Bull. Austral. Math. Soc., 2:1 (1970), 1–13 | DOI | MR | Zbl

[75] Morris S. A., “Varieties of topological groups, III”, Bull. Austral. Math. Soc., 2:2 (1970), 165–178 | DOI | MR | Zbl

[76] Morris S. A., “Free compact Abelian groups”, Mat. Čas., 22:2 (1972), 141–147 | MR | Zbl

[77] Morris S. A., “Varieties of topological groups: A survey”, Colloq. Math., 46:2 (1982), 147–165 | MR | Zbl

[78] Morris S. A. and Thompson H. B., “Invariant metrics on free topological groups”, Bull. Austral. Math. Soc., 9:1 (1973), 83–88 | DOI | MR | Zbl

[79] Morris S. A. and Thompson H. B., “Free topological groups with no small subgroups”, Proc. Amer. Math. Soc., 46:3 (1974), 431–437 | DOI | MR | Zbl

[80] Nakayama T., “Note on free topological groups”, Proc. Imp. Acad. Tokyo, 19 (1943), 471–475 | DOI | MR | Zbl

[81] Nummela E. C., “Uniform free topological groups and Samuel compactifications”, Topol. and Its Appl., 13:1 (1982), 77–83 | DOI | MR | Zbl

[82] Nyikos P. J., “Subsets of ${}^\omega\omega$ and the Fréchet–Urysohn and $\alpha_i$-properties”, Topol. and Its Appl., 48 (1992), 91–116 | DOI | MR | Zbl

[83] Okunev O. G., “A method for constructing examples of $M$-equivalent spaces”, Topol. and Its Appl., 36:2 (1990), 157–171 ; “A method for constructing examples of $M$-equivalent spaces”, Topology Appl., 49:2 (1993), 191–192 | DOI | MR | Zbl | DOI | MR | Zbl

[84] Okunev O. and Tamano K., “Lindelöf powers and products of function spaces”, Proc. Amer. Math. Soc., 124:9 (1996), 2905–2916 | DOI | MR | Zbl

[85] Pestov V. G., “Universal arrows to forgetful functors from categories of topological algebra”, Bull. Austral. Math. Soc., 48 (1993), 209–249 | DOI | MR | Zbl

[86] Pestov V., “Topological groups: Where to from here?”, Topol. Proc., 24 (1999), 421–502 | MR

[87] Pestov V. and Yamada K., “Free topological groups on metrizable spaces and inductive limits”, Topol. and Its Appl., 98 (1999), 291–301 | DOI | MR | Zbl

[88] Porst H. E., “Zur Struktur freier topologischer Gruppen”, Rostock. Math. Kolloq., 1991, no. 44, 5–20 | MR | Zbl

[89] Przymusiński T., “Collectionwise normality and extensions of continuous functions”, Fundam. Math., 98:1 (1978), 75–81 | MR | Zbl

[90] Rachev S. T. and Rüschendorf L., Mass Transportation Problems. V. I: Theory. V. II: Applications, Springer, New York, Berlin, Heidelberg, 1998 | Zbl

[91] Reznichenko E. A. and Uspenskii V. V., “Pseudocompact Mal'tsev spaces”, Topol. and Its Appl., 86 (1998), 83–104 | DOI | MR | Zbl

[92] Rudin M. E., Lectures on Set Theoretic Topology, Amer. Math. Soc., Providence, Rhode Island, 1975 | MR | Zbl

[93] Samuel P., “On universal mappings and free topological groups”, Bull. Amer. Math. Soc., 54:6 (1948), 591–598 | DOI | MR | Zbl

[94] Shakhmatov D. B., “Zerodimensionality of free topological groups and topological groups with noncoinciding dimensions”, Bull. Pol. Acad. Sci. Math., 37:7–12 (1989), 497–506 | MR | Zbl

[95] Shakhmatov D. B., “A survey of current research and open problems in the dimension theory of topological groups”, Q A in General Topology, 8 (1990), 101–128 | MR | Zbl

[96] Shakhmatov D. B., “Imbeddings into topological groups preserving dimensions”, Topol. and Its Appl., 36 (1990), 181–204 | DOI | MR | Zbl

[97] Sipacheva O. V., “On free topological groups with the inductive limit topologies”, Annals of the New York Acad. Sci., 788, The New York Acad. Sci., New York, 1996, 188–196 | MR | Zbl

[98] Sipacheva O. V., “Free topological groups of spaces and their subspaces”, Topol. and Its Appl., 101 (2000), 181–212 | DOI | MR | Zbl

[99] Sipacheva O. V., “$\tau$-Locally invariant groups”, Topol. and Its Appl., 107 (2000), 169–182 | DOI | MR | Zbl

[100] Sipacheva O. V. and Tkachenko M. G., “Thin and bounded subsets of free topological groups”, Topol. and Its Appl., 36 (1990), 143–156 | DOI | MR | Zbl

[101] Swierczkowski S., “Topologies in free algebras”, Proc. London Math. Soc., 14:55 (1964), 566–576 | DOI | MR | Zbl

[102] Thompson H. B., “A remark on free topological groups with no small subgroups”, J. Austral. Math. Soc., 18:4 (1974), 482–484 | DOI | MR | Zbl

[103] Tkačenko M. G., “On topologies of free groups”, Czechoslovak Math. J., 34:4 (1984), 541–551 | MR | Zbl

[104] Tkačenko M. G., “Free topological groups and inductive limits”, Topol. and Its Appl., 60 (1994), 1–12 | DOI | MR | Zbl

[105] Tkačenko M. G., “Topological groups for topologists: Parts I, II”, Bol. Soc. Mat. Mexicana, 5 (1999), 735–753; 6 (2000), 1–45 | MR

[106] Uspenskii V. V., “The Mal'tsev operation on countably compact spaces”, Comment. Math. Univ. Carol., 30:2 (1989), 395–402 | MR

[107] Williams S. W., “Box products”, Handbook of Set-Theoretic Topology, eds. K. Kunen, J. E. Vaughan, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1984, 169–200 | MR

[108] Yamada K., “Characterizations of a metrizable space $X$ such that every $A_n(X)$ is a $k$-space”, Topol. and Its Appl., 49 (1993), 75–94 | DOI | MR | Zbl

[109] Yamada K., “Tightness of free Abelian topological groups and of finite products of sequential fans”, Topol. Proc., 22 (1997), 363–381 | MR | Zbl