Coincidence Theory: The Minimizing Problem

S. A. Bogatyi ; D. L. Gonçalves ; H. Zieschang

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 52-86.

Résumé

We provide a short history of the main contributions about the problem of computing the minimal coincidence number $MC[f_1,f_2]$ where $(f_1,f_2)$ is a pair of maps between two topological spaces and where these maps can be homotopically deformed. The more recent contributions are treated in more detail, including some material not yet published.

Bibliographie

[1] Alsedà Ll., Baldwin S., Llibre J., Swanson R., Szlenk W., “Minimal sets of periods for torus maps”, Europ. Conf. Iteration Theory (ECIT 91) (Lisbon, 1991), eds. J. P. Lampreia et al., World Sci. Publ., River Edge–New York, 1992, 8–11

[2] Anosov D. V., “O chislakh Nilsena otobrazhenii nilmnogoobrazii”, UMN, 40:4 (1985), 133–134 | MR | Zbl

[3] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[4] Arnold V. I., “Dinamika slozhnosti peresechenii”, Izbrannoe–60, Fazis, M., 1996, 489–499 | MR

[5] Arnold V. I., “O nekotorykh zadachakh teorii dinamicheskikh sistem”, Izbrannoe–60, Fazis, M., 1996, 533–551

[6] Arnold V. I., Varchenko A. N., Givental' A. B., Khovanskii A. G., “Singularities of functions, wave fronts, caustics and multidimensional integrals”, Sov. Sci. Rev. C, 4, 1984, 1–92 | MR

[7] Artin M., Mazur B., “On periodic points”, Ann. Math., 81 (1965), 82–99 | DOI | MR | Zbl

[8] Babenko I. K., Bogatyi S. A., “Algebraicheskaya kratnost i periodicheskie tochki otobrazheniya”, Pyatyi Tiraspol. Simpoz. po obschei topologii i ee prilozheniyam, Shtiintsa, Kishinev, 1985, 12–14

[9] Babenko I. K., Bogatyi S. A., “Algebraicheskaya kratnost i periodicheskie tochki otobrazheniya”, Geometriya, differentsialnye uravneniya i mekhanika: Sedmaya Konf. molodykh uchenykh (Moskva, 1985 g.), Izd-vo MGU, M., 1986, 35–48 | MR

[10] Babenko I. K., Bogatyi S. A., “Chisla Lefshetsa, lokalnye indeksy i periodicheskie tochki”, DAN SSSR, 291:3 (1986), 521–524 | MR

[11] Babenko I. K., Bogatyi S. A., “Povedenie indeksa periodicheskikh tochek pri iteratsiyakh otobrazheniya”, Izv. AN SSSR. Ser. mat., 55:1 (1991), 3–31 | MR | Zbl

[12] Baker I. N., “The iteration of entire transcendental functions and the solution of the functional equation $f\{f(z)\}=F(z)$”, Math. Ann., 129 (1955), 174–180 | DOI | MR | Zbl

[13] Baker I. N., “Fixpoints and iterates of entire functions”, Math. Ztschr., 71 (1959), 146–153 | DOI | MR | Zbl

[14] Baker I. N., “The existence of fixpoints of entire functions”, Math. Ztschr., 73 (1960), 280–284 | DOI | MR | Zbl

[15] Beardon A. F., Iteration of rational functions. Complex analytic dynamical systems, Graduate Texts Math., 132, Springer, N. Y. etc., 1991 | MR | Zbl

[16] Becker E., Cardinal J. P., Roy M.-F., Szafraniec Z., Multivariate Bezoutians, Kronecker symbol and Eisenbud–Levine formula, Progr. Math., 143, Birkhäuser, Boston etc., 1996, 79–104 | MR

[17] Begle E., “The Vietoris mapping theorem for bicompact spaces”, Ann. Math., 51 (1950), 534–543 | DOI | MR | Zbl

[18] Blanchard P., Franks J., “The dynamical complexity of orientation reversing homeomorphisms of surfaces”, Invent. math., 62 (1980), 333–339 | DOI | MR | Zbl

[19] Bliznyakov N. M., “$(0,1)$-matritsy i indeks osoboi tochki vektornogo polya”, Topologich. konf., Baku, 1987, 42

[20] Bogatyi S. A., “Attraktor Khenona”, Topologich. konf., Baku, 1987, 44

[21] Bogatyi S. A., “Indeksy iteratsii mnogoznachnogo otobrazheniya”, C. r. Acad. Bulg. sci., 41:2 (1988), 13–16 | MR

[22] Bogatyi S. A., “Lokalnye indeksy iteratsii golomorfnogo otobrazheniya”, Funktsion. analiz i ego pril., 23:1 (1989), 61–62 | MR

[23] Bogatyi S. A., “Lokalnye indeksy iteratsii golomorfnogo otobrazheniya”, Obschaya topologiya, Izd-vo MGU, M., 1989, 48–61 | MR

[24] Bogatyi S., “On the nonexistence of iterative roots”, Topol. Appl., 76 (1997), 97–123 | DOI | MR

[25] Bogatyi S., “A topological view on a problem on formal series”, Aequat. Math., 56 (1998), 18–26 | DOI | MR | Zbl

[26] Bogatyi S., Gonçalves D. L., Zieschang H., The minimal number of roots of surface mappings and quadratic equations of free groups, Preprint Ruhr-Univ. Bochum, 1998

[27] Bourgin D. G., “Homeomorphisms of the open disk”, Stud. math., 31:4 (1968), 433–438 | MR | Zbl

[28] Bowen R., Franks J., “The periodic points of maps of the disk and the interval”, Topology, 15 (1976), 337–342 | DOI | MR | Zbl

[29] Bowszyc C., “On the Euler–Poincaré characteristic of a map and the existence of periodic points”, Bull. Acad. polon. sci. Ser. sci. Math. Astr. Phys., 17 (1969), 367–372 | MR | Zbl

[30] Brooks R. B. S., “On removing coincidences of two maps when only one, rather than both of them may be deformed by a homotopy”, Pacif. J. Math., 40 (1972), 45–52 | MR | Zbl

[31] Brooks R. B. S., “Certain subgroups of the fundamental group and the number of roots of $f(x)=a$”, Amer. J. Math., 95 (1973), 720–728 | DOI | MR | Zbl

[32] Brooks R. B. S., Brown R. F., Pak J., Taylor D. H., “Nielsen numbers of maps of tori”, Proc. Amer. Math. Soc., 52 (1975), 398–400 | DOI | MR | Zbl

[33] Brooks R., Odenthal C., “Nielsen numbers for roots of maps of aspherical manifolds”, Pacif. J. Math., 170:2 (1995), 405–420 | MR | Zbl

[34] Browder F., “On the fixed point index for continuous mappings of locally connected spaces”, Summa Brasil. Math., 4 (1960), 253–293 | MR | Zbl

[35] Brown M., “On the fixed point index of iterates of planar homeomorphisms”, Proc. Amer. Math. Soc., 108:4 (1995), 1109–1114 | DOI | MR

[36] Brown R., The Lefschetz fixed point theorem, Scott, Foresman Co., Glenview (Ill.)–London, 1971 | MR | Zbl

[37] Brown R., Topological fixed point theory, Preprint. Dep. Math. UCLA, 1997

[38] Brown R., Schirmer H., “Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary”, Topol. Appl., 46 (1992), 65–79 | DOI | MR | Zbl

[39] Brown R., Schirmer H., Nielsen root theory and Hopf degree theory, Preprint. Univ. Calif. Los Angeles and Carleton Univ., Ottawa, 1998 | MR

[40] Chow S.-N., Mallet-Paret J., Yorke J. A., “A periodic orbit index which is a bifurcation invariant”, Geometric dynamics (Rio de Janeiro, 1981), Lect. Notes Math., 1007, Springer-Verl., Berlin etc., 1983, 109–131

[41] Davey O., Hart E., Trapp K., “Computation of Nielsen numbers for maps of closed surfaces”, Trans. Amer. Math. Soc., 384:8 (1996), 3245–3266 | DOI | MR

[42] Deleanu A., “Théorie des points fixes: sur le rétractes de voisinage des espaces convexoides”, Bull. Soc. math. France., 87 (1959), 235–243 | MR | Zbl

[43] tom Dik T., Gruppy preobrazovanii i teoriya predstavlenii, Mir, M., 1982 | MR | Zbl

[44] Dobreńko R., Jezierski J., “The coincidence Nielsen theory on non-orientable manifolds”, Rocky Mount. J. Math., 23 (1993), 67–85 | DOI | MR | Zbl

[45] Dold A., Lektsii po algebraicheskoi topologii, Mir, M., 1976 | MR

[46] Dold A., “Fixed point indices of iterated maps”, Invent. math., 74 (1983), 419–435 | DOI | MR | Zbl

[47] Dzedzej Z., “Fixed point index for a class of nonacyclic multivalued maps”, Diss. Math., 253 (1985), 1–58 | MR

[48] Eisenbud D., “Note on the topological degree of a smooth mapping”, Queen's Pap. Pure and Appl. Math., 42, 1975, 71–79 | MR

[49] Eisenbud D., “An algebraic approach to the topological degree of a smooth map”, Bull. Amer. Math. Soc., 84:5 (1978), 751–764 | DOI | MR | Zbl

[50] Eisenbud D., Levine H. I., “An algebraic formula for the degree of a $C^{\infty}$ map germ”, Ann. Math., 106 (1977), 19–44 | DOI | MR | Zbl

[51] Epstein D. B. A., “The degree of a map”, Proc. London Math. Soc., ser. 3, 16 (1966), 369–383 | DOI | MR | Zbl

[52] Fadell E., “On a coincidence theorem of F. B. Fuller”, Pacif. J. Math., 15:3 (1965), 825–834 | MR | Zbl

[53] Fadell E., “Recent results in the fixed point theory of continuous maps”, Bull. Amer. Math. Soc., 76 (1970), 10–29 | DOI | MR | Zbl

[54] Fadell E., Husseini S., “The Nielsen number on surfaces”, Topological methods in nonlinear functional analysis (Toronto, 1982), Contemp. Math., 21, Amer. Math. Soc., Providence, R.I., 1983, 59–98 | MR

[55] Fadell E., Husseini S., “On a theorem of Anosov on Nielsen numbers for nilmanifolds”, Nonlinear functional analysis and its applications (Maratea, 1985), NATO Adv. Sci. Inst. C: Math. Phys. Sci., 173, Kluwer Acad. Publ., Dordrecht, 1986, 47–53 | MR

[56] Fel'shtyn A., Hill R., Wong P., “Reidemeister numbers of equivariant maps”, Topol. Appl., 67 (1995), 119–131 | DOI | MR

[57] Franks J., “Some smooth maps with infinitely many hyperbolic periodic points”, Trans. Amer. Math. Soc., 226 (1977), 175–179 | DOI | MR | Zbl

[58] Franks J., “Period doubling and the Lefschetz formula”, Trans. Amer. Math. Soc., 287 (1985), 275–283 | DOI | MR | Zbl

[59] Franks J., Fried D., “The Lefschetz function of a point”, Lect. Notes Math., 1411, 1989, 83–87 | MR | Zbl

[60] Fried D., “Periodic points of holomorphic maps”, Topology, 25:4 (1986), 429–441 | DOI | MR | Zbl

[61] Fuller F. B., “The existence of periodic points”, Ann. Math., 57 (1953), 229–230 | DOI | MR | Zbl

[62] Fuller F. B., “The homotopy theory of coincidences”, Ann. Math., 59 (1954), 219–226 | DOI | MR | Zbl

[63] Gonçalves D. L., “Braids groups and Wecken pairs”, Contemp. Math., 72 (1988), 89–97 | MR | Zbl

[64] Gonçalves D. L., “Coincidence of maps between surfaces”, J. Korean Math. Soc., 1999 (to appear) | MR

[65] Gonçalves D. L., “Coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations”, Topol. Appl., 83 (1998), 169–183 | DOI | MR

[66] Gonçalves D. L., “Coincidence theory for maps from a complex into a manifold”, Topol. Appl., 92:1 (1999), 63–77 | DOI | MR | Zbl

[67] Gonçalves D. L., “The coincidence Reidemeister classes of maps on nilmanifolds”, Topol. Meth. Nonlin. Anal. (to appear)

[68] Gonçalves D. L., Jezierski J., “Lefschetz coincidence formula on non-orientable manifolds”, Fund. math., 153 (1997), 1–23 | MR | Zbl

[69] Gonçalves D. L., Zieschang H., “Equations in free groups and coincidence of mappings on surfaces”, Math. Ztschr (to appear)

[70] Górniewicz L., “Homological methods in fixed point theory of multivalued maps”, Diss. Math., 129 (1976), 1–71 | MR

[71] Gottlieb D., “A certain subgroup of the fundamental group”, Amer. J. Math., 87 (1965), 840–856 | DOI | MR | Zbl

[72] Guillamon A., Jarque X., Llibre J., Ortega J., Torregrosa J., “Periods for transversal maps via Lefschetz numbers for periodic points”, Trans. Amer. Math. Soc., 347:12 (1995), 4779–4806 | DOI | MR | Zbl

[73] Halpern B., “Periodic point on tori”, Pacif. J. Math., 83:1 (1979), 117–133 | MR | Zbl

[74] Handel M., 291–296, Topology, 21, no. 3, 1982 | DOI | MR | Zbl

[75] Haeseler F. von, Skordev G., “Borsuk–Ulam theorem, fixed point index and chain approximations for maps with multiplicity”, Pacif. J. Math., 153:2 (1992), 369–396 | MR

[76] Hirsch G., “Quelques théorèmes sur les points fixes des groupes de représentations”, Bull. Soc. Roy. sci. Liège, 12 (1943), 392–407 | MR | Zbl

[77] Hopf H., “Über die algebraische Anzahl von Fixpunkten”, Math. Ztschr., 29 (1929), 493–524 | DOI | MR | Zbl

[78] Hopf H., “Zur Topologie der Abbildungen von Mannigfaltigkeiten, II”, Math. Ann., 102 (1930), 562–623 | DOI | MR

[79] Hurwitz A., “Über Riemannische Fläche mit gegebenen Verzweigungs-punkten”, Math. Ann., 39 (1891), 1–60 | DOI | MR

[80] Isaacs R., “Iterates of fractional order”, Canad. J. Math., 2 (1950), 409–416 | MR | Zbl

[81] Ivanov N. V., “Chisla Nilsena otobrazhenii poverkhnostei”, Zap. nauch. seminarov LOMI, 122, 1982, 56–65 | MR | Zbl

[82] Jezierski J., “The least number of coincidence points on surfaces”, J. Austral. Math. Soc. A, 58 (1995), 27–38 | DOI | MR | Zbl

[83] Jiang B., “Estimation of the Nielsen numbers”, Chin. Math., 5 (1964), 330–339 | MR | Zbl

[84] Jiang B., “On the least number of fixed points”, Amer. J. Math., 102:4 (1980), 749–763 | DOI | MR | Zbl

[85] Jiang B., “The fixed point classes from the differential view point”, Fixed point theory, Lect. Notes Math., 886, Springer-Verl., Berlin etc., 1981, 163–170 | MR

[86] Jiang B., Lectures on Nielsen fixed point theory, Contemp. Math., 14, Amer. Math. Soc., Providence, R.I., 1983 | MR

[87] Jiang B., “Fixed points and braids”, Invent. math., 75 (1984), 69–74 | DOI | MR | Zbl

[88] Jiang B., “Fixed points and braids, II”, Math. Ann., 272 (1985), 249–256 | DOI | MR | Zbl

[89] Jiang B., “Surface maps and braid equations, I”, Lect. Notes Math., 1369, 1989, 125–141 | MR | Zbl

[90] Jiang B., “Estimation of the number of periodic orbits”, Pacif. J. Math., 172:1 (1996), 151–185 | MR | Zbl

[91] Kelly M. R., “Mimimizing the number of fixed points for self-maps of compact surfaces”, Pacif. J. Math., 126 (1987), 81–123 | MR | Zbl

[92] Keppelmann E. C., McCord C. K., “The Anosov theorem for exponential solvmanifolds”, Pacif. J. Math., 170:1 (1995), 143–159 | MR | Zbl

[93] Khimshiashvili G. N., “O lokalnoi stepeni gladkogo otobrazheniya”, Soobsch. Akad. nauk GSSR, 85 (1977), 309–311 | MR

[94] Khovanskii A. G., Malochleny, Fazis, M., 1997 | MR

[95] Kiang T. H., The theory of fixed point classes, Springer-Verl., Berlin etc., 1989 | MR

[96] Kneser H., “Die kleinste Bedeckungszahl innerhalb einer Klasse von Flächen abbildungen”, Math. Ann., 103 (1930), 347–358 | DOI | MR | Zbl

[97] Komiya K., “Congruences for fixed point indices of equivariant maps and iterated maps”, Lect. Notes Math., 1411, 1989, 130–136 | MR | Zbl

[98] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[99] Krasnoselskii M. A., Perov A. I., Povolotskii A. I., Zabreiko P. P., Vektornye polya na ploskosti, Fizmatgiz, M., 1963 | MR

[100] Łȩcki A., Szafraniec Z., “Applications of the Eisenbud–Levine's theorem to real algebraic geometry”, Computational algebraic geometry, Progr. Math., 109, Birkhäuser, Boston etc., 1993, 177–184 | MR

[101] Łȩcki A., Szafraniec Z., “An algebraic method for calculating the topological degree”, Topol. Meth. Nonlin. Anal., 35 (1996), 73–83 | MR

[102] Lefschetz S., “Continuous transformations of manifolds”, Proc. Nat. Acad. Sci. USA, 9 (1923), 90–93 | DOI | Zbl

[103] Lefschetz S., “Intersections of complexes on manifolds”, Proc. Nat. Acad. Sci. USA, 11 (1925), 290–292 | DOI | Zbl

[104] Lefschetz S., “Intersections and transformations of complexes and manifolds”, Trans. Amer. Math. Soc., 29 (1926), 1–49 | DOI | MR

[105] Lefschetz S., “Manifolds with boundary and their transformations”, Trans. Amer. Math. Soc., 29 (1927), 429–462 | DOI | MR | Zbl

[106] Leray J., “Sur les équations et les transformations”, J. math. pures et appl., 9 (1946), 201–248 | MR | Zbl

[107] Llibre J., “Lefschetz numbers for periodic points”, Contemp. Math., 152, 1993, 215–227 | MR | Zbl

[108] Łojasiewicz S., “Solution générale de l`équation fonctionnelle $f(f(\ldots f(x)\ldots))=g(x)$”, Ann. Soc. Polon. Math., 24 (1951), 88–91 | MR

[109] Matsuoka T., “The number of periodic orbits of smooth maps”, Lect. Notes Math., 1411, 1989, 148–154 | MR | Zbl

[110] McCord C. K., “Nielsen numbers and Lefschetz numbers on solvmanifolds”, Pacif. J. Math., 147 (1991), 153–164 | MR | Zbl

[111] McCord C. K., “Estimating Nielsen numbers on infrasolvmanifolds”, Pacif. J. Math., 154 (1992), 345–368 | MR | Zbl

[112] McCord C. K., “Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, I”, Topol. Appl., 43 (1992), 249–261 ; “II”, 75 (1997), 81–92 | DOI | MR | Zbl | DOI | MR | Zbl

[113] McCord C. K., “A Nielsen theory for intersection numbers”, Fund. math., 152 (1997), 117–150 | MR | Zbl

[114] Marzantowicz W., Przygodzki P. M., Finding periodic points of a map by use of a $K$-adic expansion, Preprint, 1998 ; Discrete Contin. Dynam. Systems, 5:3 (1999), 495–514 | MR | Zbl | DOI | Zbl

[115] Mukherjea K., “Coincidence theory for manifolds with boundary”, Topol. Appl., 46 (1992), 23–39 | DOI | MR | Zbl

[116] Narasimhan C. C., “The periodic behaviour of Morse–Smale diffeomorphisms on compact surfaces”, Trans. Amer. Math. Soc., 248 (1979), 145–169 | DOI | MR | Zbl

[117] Nielsen J., Collected mathematical papers, Birkhäuser, Basel, 1986, 223–341 | DOI | MR | Zbl

[118] Nikishin N. A., “O nepodvizhnykh tochkakh diffeomorfizmov dvumernoi sfery, sokhranyayuschikh orientirovannuyu ploschad”, Funktsion. analiz i ego pril., 8:1 (1974), 84–85 | MR | Zbl

[119] O'Neil B., “Essential sets and fixed points”, Amer. J. Math., 75 (1953), 497–509 | DOI | MR

[120] Oprea J., Gottlieb groups, groups actions and rational homotopy, Lect. Notes Ser., 29, Seoul Nat. Univ., 1995 | MR | Zbl

[121] Palamodov V. P., “O kratnosti golomorfnogo otobrazheniya”, Funktsion. analiz i ego pril., 1:3 (1967), 54–65 | MR | Zbl

[122] Peitgen H. O., “On the Lefschetz number for iterates of continuous mappings”, Proc. Amer. Math. Soc., 54 (1976), 441–444 | DOI | MR | Zbl

[123] Pelikan S., Slaminka E. E., “A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds”, Ergod. Theory and Dyn. Syst., 7:3 (1987), 463–479 | MR | Zbl

[124] Pontryagin L. S., Nepreryvnye gruppy, Nauka, M., 1973 | MR | Zbl

[125] Powers M. J., “Fixed point theorems for non-compact approximative ANR's”, Fund. math., 75:1 (1972), 61–68 | MR | Zbl

[126] Powers M. J., “Lefschetz fixed point theorems for a new class of multi-valued maps”, Pacif. J. Math., 42:1 (1972), 211–220 | MR | Zbl

[127] Reich L., “On power series transformations in one indeterminate having iterative roots of given order and with given multiplier”, Europ. Conf. Iteration Theory (ECIT 91) (Lisbon, 1991), eds. J. P. Lampreia et al., World Sci. Publ., River Edge–New York, 1992, 210–216

[128] Reidemeister K., “Automorphismen von Homotopiekettenringen”, Math. Ann., 112 (1936), 586–593 | DOI | MR

[129] Schirmer H., “Mindestzahlen von Koinzidenzpunkten”, J. reine und angew. Math., 194:1–4 (1955), 21–39 | MR | Zbl

[130] Schirmer H., “Effluent and noneffluent fixed points on dendrites”, Pacif. J. Math., 75 (1978), 359–552 | MR

[131] Schirmer H., “An index and a Nielsen number for $n$-valued multifunctions”, Fund. math., 124 (1984), 207–219 | MR | Zbl

[132] Schwaiger J., “Roots of formal power series in one variable”, Aequat. Math., 29 (1985), 40–43 | DOI | MR | Zbl

[133] Scott G. P., “Braid groups and the group of homeomorphisms of a surface”, Proc. Cambridge Phil. Soc., 68 (1969), 605–617 | DOI | MR

[134] Shub M., Sullivan D., “A remark on the Lefschetz fixed point formula for differentiable maps”, Topology, 13:2 (1974), 189–191 | DOI | MR | Zbl

[135] Siegberg H.-W., Skordev G., “Fixed point index and chain approximation”, Pacif. J. Math., 102 (1982), 455–486 | MR | Zbl

[136] Simon C., “A bound for the fixed-point index of an area-preserving map with applications to mechanics”, Invent. math., 26 (1974), 187–200 | DOI | MR | Zbl

[137] Sklyarenko E. G., “O teoreme Vetorisa–Begla”, DAN SSSR, 149:2 (1963), 264–267 | Zbl

[138] Sklyarenko E. G., “O nekotorykh prilozheniyakh teorii puchkov v obschei topologii”, UMN, 19:6 (1964), 47–70 | MR | Zbl

[139] Skordev G. S., “Kointsidentnost otobrazhenii kompaktnykh grupp”, C. r. Acad. Bulg. sci., 28:11 (1975), 1451–1454 | MR | Zbl

[140] Skordev G. S., “On a coincidence of mappings of compact spaces in topological groups”, Fund. math., 104 (1979), 111–125 | MR | Zbl

[141] Skordev G. S., “Fixed point index for open sets in Euclidean spaces”, Fund. math., 121 (1984), 41–58 | MR | Zbl

[142] Skordev G. S., “The multiplicativity property of the fixed point index for multivalued maps”, Serdica Bulg. Math. J., 15 (1989), 160–170 | MR | Zbl

[143] Slaminka E. E., “Area preserving homeomorphisms of two manifolds”, Contemp. Math., 81 (1988), 153–167 | MR | Zbl

[144] Steinlein H., “Ein Satz über den Leray–Schauderschen Abbildungsgrad”, Math. Ztschr., 126 (1972), 176–208 | DOI | MR | Zbl

[145] Szafraniec Z., “Topological degree and quadratic forms”, J. Pure and Appl. Algebra, 139:1 (1999) | MR

[146] Targonski Gy., Topics in iteration theory, Stud. Math., 6, Vandenhoeck and Rupprecht, Göttingen, 1981 | MR | Zbl

[147] Turaev V. G., “Chisla Nilsena i nepodvizhnye tochki otobrazhenii buketov okruzhnostei”, Zap. nauch. seminarov LOMI, 122, 1982, 135–136 | MR | Zbl

[148] Vick J., Homology theory, Graduate Text Math., 145, Springer-Verl., Berlin etc., 1994 | MR | Zbl

[149] Vinogradov I. M., Osnovy teorii chisel, Nauka, M., 1972 | MR

[150] Wecken F., “Fixpunktklassen, I”, Math. Ann., 117 (1941), 659–671 ; “II”, 118 (1942), 216–234 ; “III”, 544–577 | DOI | MR | DOI | MR

[151] Wong P., Reidemeister number, Hirsch rank and coincidences on solvmanifolds, Preprint, Bates College (USA), 1997

[152] Zabreiko P. P., Krasnoselskii M. A., “Iteratsii operatorov i nepodvizhnye tochki”, DAN SSSR, 196 (1971), 1006–1009 | MR

[153] Tsishang Kh., Fogt E., Koldevai Kh.-D., Poverkhnosti i razryvnye gruppy, Nauka, M., 1988, dobavlenie | MR

[154] Zieschang H., “Alternierende Produkte in freien Gruppen”, Abh. math. Sem. Univ. Hamburg, 27 (1964), 13–31 | DOI | MR | Zbl

[155] Zieschang H., Vogt E., Coldewey H. D., Surfaces and planar discontinuous groups, Lect. Notes Math., 835, Springer-Verl., Berlin etc., 1980 | MR | Zbl