Homological methods in fixed-point theory of multi-valued maps

Górniewicz, Lech

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Homological methods in fixed-point theory of multi-valued maps

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Lech Górniewicz

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Rozprawy Matematyczne tom/nr w serii: 129 wydano: 1976

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CONTENTS
  Introduction................................................................. 5
I. HOMOLOGY
  1. Preliminaries............................................................. 7
  2. Maps in spaces of finite type............................................. 9
  3. The Čech homology functor with compact carriers........................... 11
  4. Vietoris maps............................................................. 13
  5. Homology of open subsets of Euclidean spaces.............................. 14
II. THE LEFSCHETZ NUMBER
  1. The (ordinary) Lefschetz number........................................... 18
  2. The generalized Lefschetz number.......................................... 20
III. MULTI-VALUED MAPS
  1. Upper semi-continuous and compact multi-valued mapB....................... 24
  2. Admissible maps........................................................... 26
  3. Homotopy and selectors.................................................... 9
  4. Lefschetz maps............................................................ 30
IV. ANB-s, AANR-B and w-AANB-s
  1. ANR-s..................................................................... 32
  2. Approximation Theorem..................................................... 33
  3. AANR-B.................................................................... 34
  4. w-AANR-s.................................................................. 36
V. THE LEFSCHETZ FIXED-POINT THEOREM
  1. The index of coincidence.................................................. 37
  2. The Lefschetz Fixed-Point Theorem for open subsets in $R^n$............... 40
  3. The Lefschetz Fixed-Point Theorem for AANR-s.............................. 41
  4. Neighbourhood fixed-point property........................................ 45
  5. The Lefschetz Fixed-Point Theorem for w-AANR-s............................ 46
  6. Two consequences of the Lefschetz Fixed-Point Theorem..................... 47
VI. FIXED-POINT PROPERTY OP THE TYCHONOFF CUBE
  1. Almost fixed points....................................................... 51
  2. Fixed-point property for infinite products................................ 51

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 129

Liczba stron

66

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CXXIX

Daty

wydano

1976

Twórcy

autor

Lech Górniewicz

Bibliografia

[1] R. F. Arens and J. Eells, Jr., On embedding uniform and topological spaces. Pacific J. Math. 6 (1956), pp. 397-403.
[2] M. Artin and B. Mazur, On periodic points, Ann. of Math. 81 (1965), pp. 82-99.
[3] E. G. Begle, The Vietoris Mapping Theorem for bicompact spaces, ibid., 51 (1950), pp. 534—543; 544-550.
[4] K. Borsuk, Sur un continu aeyclique qui se laisse transformer topologiquement en lui même sans points invariants, Fund. Math. 24 (1934), pp. 51-58.
[5] K. Borsuk, Theory of Retracts, Warszawa 1967.
[6] N. Bourbaki, Topologie generale, Moscow 1968.
[7] C. Bowszyc, Some theorems in the Theory of Fixed Points, Doctoral Dissertation, University of Warsaw, 1969.
[8] D. G. Bourgin, Cones and Vietoris-Begle type theorems, Trans. Amer. Math. Soc. 174 (1972), pp. 155-183.
[9] D. G. Bourgin, A degree for nonacyclic multi-valued transformations, Bull. Amer. Math. Soc. 80 (1974), pp. 59-61.
[10] F. E. Browder, Fixed points theorems on infinite dimensional manifolds, Trans. Amer. Math. Soc. 2 (1965), pp. 179-194.
[11] F. E. Browder, On the fixed points index for continuous mappings of locally connected spaces, Summa Brasil. Math. 4 (1960), pp. 253-293.
[12] F. E. Browder, Asymptotic fixed point theorems, Math. Ann. 1 (1971), pp. 38-60.
[13] R. F. Brown, Notes on Leray's index theory, Advances in Math. 7 (1971), pp. 1-28.
[14] R. F. Brown, The Lefschete fixed point theorem, Glenviev, Illinois, London, 1971.
[15] R. F. Brown, On a homotopy converse to the Lefschetz fixed point theorem, Pacific. J. Math. 17 (1966), pp. 407-411.
[16] J. Bryszewski and L. Górniewicz, Multi-valued maps of subsets of Euclidean spaces, Fund. Math., to appear.
[17] B. D. Calvert, The local fixed point index for multi-valued transformations in Banach spaces. Math. Ann. 190 (1970), pp. 119-128.
[18] M. H. Clapp, On a generalization of absolute neighbourhood retracts, Fond. Math. 70 (1971), pp. 117-130.
[19] M. Davies, A note on Borsuk's Antipodal-Point Theorem, Quart. J. Math. Oxford Ser. 7 (1956), pp. 293-300.
[20] A. Dold, Fixed point index and fixed point theorem for Euclidean neighbourhood retracts, Topology 4 (1965), pp. 1-8.
[21] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 58 (1946), pp. 214-222.
[22] S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton 1952.
[23] E. Fadell, Recent results in the fixed point theory of continuous maps, Bull Amer. Math. Soc. 1 (1970), pp. 10-29.
[24] E. Fadell, On a coincidence theorem of F. B. Fuller, Pacific J. Math. 15 (1965), pp. 825-834.
[25] E. Fadell, Some examples in the fixed point theory of polyhedra, ibid., 33 (1970), pp. 89-100.
[28] P. B. Fuller, The existence of periodic points, Ann. of Math. 59 (1954), pp. 229-230.
[27] P. B. Fuller, The homotopy theory of coincidences, ibid., 57 (1953), pp. 219-226.
[28] A. Gmurczyk, On approximative retracts. Bull. Acad. Polon. Sci. 16 (1968), pp. 9-14.
[29] L. Górniewicz, Fixed point theorem for multi-valued mappings of approximative ANR-8, ibid., 18 (1970), pp. 431-436.
[30] L. Górniewicz, On non-acyclic multi-valued maps of subsets of Euclidean spaces, ibid., 6 (1972), pp. 379-385.
[31] L. Górniewicz, Some consequences of the Lefschetz fixed point theorem for multi-valued maps, ibid., 2 (1973), pp. 165-170.
[32] L. Górniewicz, Fixed point theorems for multi-valued maps of special ANR-s, University of Gdańsk, 1 (1971), pp. 39-49.
[33] L. Górniewicz, Asymptotic fixed point theorems for multi-valued maps, Symp. of Topology, Tbilisi 1972.
[34] L. Górniewicz, Some fixed point theorems for multi-valued maps, Doctoral Dissertation, Polish Acad, of Sci., Inat. of Math., 1970.
[35] L. Górniewicz, Remarks on the Lefschetz fixed point theorem, Bull. Acad. Polon. Sci. 11 (1973), pp. 993-999.
[36] L. Górniewicz, On a Coincidence Theorem, ibid., 2 (1974), pp. 277-283.
[37] L. Górniewicz, A Lefschetz-type fixed point theorem, Fund. Math. 88 (1975), pp. 103-115.
[38] L. Górniewicz and A. Granas, Fixed point theorems for multi-valued maps of ANR-s, J. Math. Pures Appl. 49 (1970), pp. 381-395.
[39] A. Granas, Sur la notion du degré topologique pour une certaine classe de transformations multivalentas dans espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), pp. 181-194.
[40] A. Granas, Theorem on antipodes and theorems on fixed points for a certain class of multi-valued maps in Banach spaces, ibid., pp. 271-275.
[41] A. Granas, Generalizing the Hopf-Lefschetz fixed point theorem for non-compact ANR-s, Symp. of Infinite Dim. Topology, Baton-Rouge 1967.
[42] A. Granas, Fixed point theorems for approximative ANR-s, Bull. Acad. Polon. Sci. 16 (1968), pp. 15-19.
[43] A. Granas, Some theorems in fixed point theory. The Leray-Schauder Index and Lefschetz number, ibid., pp. 131-137.
[44] A. Granas, Topics in the fixed point theory, Sem. J. Leray, Paris, 1969/70.
[45] A. Granas, Points fixes pour les applications compactes en topologie et analyse fonctionnelle, Sem. de Math. Superieures, Montreal 1973.
[46] A. Granas, The Leray-Schauder Index and the fixed point theory for arbitrary ANR-s, Bull. Soc. Math. France 100 (1972), pp. 209-288.
[47] A. Granas and J. W. Jaworowski, Some theorems on multi-valued maps of subsets of the Euclidean, space, Bull. Acad. Polon. Sci. 6 (1965), pp. 277-283.
[48] O. Hajek, Homological fixed point theorems, Comment. Math. Univ. Carolinae 5 (1964), pp. 13-31, 6 (1965), pp. 157-164.
[49] B. Halpern, Algebraic topology and set valued maps, Lee. Not. Math. 171 (1970), pp. 23-33.
[50] B. Halpern, Fixed points for iterates. Pacific J. Math. 25 (1968), pp. 255-275.
[51] W. Hurewicz and H. Wallman, Domension Theory, Princeton 1941.
[52] C. Himmelberg, Fixed points of compact multi-valued maps, J. Math. Anal. Appl. 38 (1972), pp. 205-209.
[53] J. W. Jaworowski, Continuous homology properties of approximative retracts, Bull. Acad. Polon. Sci. 18 (1970), pp. 359-362.
[54] J. W. Jaworowski, Some consequences of the Vietoris Mapping Theorem, Fund. Math. 45 (1958), pp. 261-272.
[55] J. W. Jaworowski, Set-valued fixed point theorems for approximative retracts, Lec. Not. Math. 171 (1970), pp. 34-39.
[56] J. W. Jaworowski and M. Powers, Λ-spaces and fixed point theorems, Fund. Math. 64 (1969), pp. 157-162.
[57] S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J. 8(1941), pp. 457-459.
[58] S. Kinoshita, On some contractible continua without fixed point property, Fund. Math. 49 (1953), pp. 96-98.
[59] R. Knill, Cones, products and fixed points, ibid., 60 (1967), pp. 35-46.
[60] R. Knill, The Lefschetz fixed point theorem for noncompact locally connected spaces, Trans. Amer. Math. Soc. 174 (1972), pp. 185-198.
[61] R. Knill, Q-simplical spaces, Illinois J. Math. 14 (1970), pp. 40-51.
[62] K. Kuratowski, Topology, Warszawa 1966.
[63] S. Lefschetz, Topics in Topology, New Jersey 1942.
[64] J. Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), pp. 221-233.
[65] J. Leray, Fixed point index and Lefschetz number, Sym. on Inf. Dim. Topology, Baton-Rouge 1967.
[66] W. Lopez, An example in the fixed point theory of polyhedra, Bull. Amer. Math. Soc. 13 (1967), pp. 922-924.
[67] S. Mazur, Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält, Studia Math. 2 (1930), pp. 7-9.
[68] M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), pp. 497-511.
[69] M. Nakaoka, Note on the Lefschetz fixed point theorem, Osaka J. Math. 6 (1969), pp. 135-142.
[70] H. Noguchi, A generalization of absolute neighbourhood retracts, Ködai Math. Sem. Rep. 1 (1953), pp. 20-22.
[71] B. O'Neill, Essential sets and fixed points. Amer. J. Math. 75 (1953), pp. 497-509.
[72] B. O'Neill, Induced homology homomorphisme for set-valued maps, Pacific J. Math. 7 (1957), pp. 1179-1184.
[73] M. Powers, Multi-valued mappings and Lefschetz fixed point theorems, Proc. Cambridge Philos. Soc. 68 (1970), pp. 619-630.
[74] M. Powers, Fixed point theorems for non-compact approximative ANR-s, Fund. Math. 76 (1972), pp. 61-68.
[75] M. Powers, Lefschetz fixed point theorem for a new class of multi-valued maps, Pacific J. Math. 42 (1972), pp. 211-220.
[76] C. J. Rhee, Homotopy functors determined by set-valued maps. Math. Zeitschr. 113 (1970), pp. 154-158.
[77] J. Schauder, Der fixpunktsatzen in funktionalraumens, Studia Math. 2 (1930), pp. 171-180.
[78] H. Schirmer, A Kakutani type coincidence theorem, Fund. Math. 69 (1970) pp. 219-226.
[79] E. G. Skljarenko. On a theorem of Vietoris and Begle, Dokl. Akad. Nauk SSSR 149 (1963), pp. 264-267.
[80] G. Skordev, Fixed points of maps of AANR-s, Bull. Acad. Polon. Sci. 21 (1973), pp. 173-180.
[81] E. Spanier, Algebraic topology, New York 1966.
[82] W. L. Strother, Multi homotopy, Duke Math. J. 22 (1955), pp. 281-285.
[83] W. L. Strother, Fixed points, fixed sets and M-retracts, ibid., 22 (1955), pp. 551-556.
[84] C. Teleman, Elemente de topologie si varictati diferentiable, Bucaresti 1964.
[85] R. B. Thompson, A unified approach to local and global fixed point indices, Advances in Math. 3 (1969), pp. 1-71.
{86] R. B. Thompson, On the semicomplexes of F. Browder, Bull. Amer. Math. Soc. 73 (1967), pp. 531-536.
[87] R. B. Thompson, A metatheorem for fixed point theories, Comment. Math. Univ. Carolinae, 11 (1970), pp. 813-815.
[88] R. B. Thompson, Retracts of semicomplexes, Illinois J. Math. 15 (1971), pp. 258-272.
[89] L. Vietoris, Über den höheren Zusammenhang kompakter Baume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), pp. 454-472.
[90] T. van der Walt, Fixed and almost fixed points, Amsterdam 1963.
[91] S. A. Williams, An index for set-valued maps in infinite dimensional spaces, Proc. Amer. Math. Soc. 31 (1972), pp. 557-563.

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Homological methods in fixed-point theory of multi-valued maps

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