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## Fundamenta Mathematicae

1999 | 162 | 1 | 65-89
Tytuł artykułu

### A Lefschetz-type coincidence theorem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: $I_{fg} = λ _{fg}$, that is, the coincidence index is equal to the Lefschetz number. It follows that if $λ_{fg} ≠ 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with "point-like" (acyclic) and "sphere-like" values.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
65-89
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-07-15
Twórcy
autor
• Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, U.S.A.
Bibliografia
• [1] E. G. Begle, The Vietoris Mapping Theorem for bicompact spaces, Ann. of Math. 81 (1965), 82-99.
• [2] G. E. Bredon, Topology and Geometry, Springer, 1993.
• [3] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Chicago, 1971.
• [4] R. F. Brown and H. Schirmer, Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary, Topology Appl. 46 (1992), 65-79.
• [5] R. F. Brown and H. Schirmer, Correction to "Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary", ibid. 67 (1995), 233-234.
• [6] V. R. Davidyan, Coincidence points of two maps, Russian Acad. Sci. Sb. Math. 40 (1980), 205-210.
• [7] V. R. Davidyan, On coincidence of two maps for manifolds with boundary, Russian Math. Surveys 38 (1983), no. 2, 176.
• [8] A. Dawidowicz, Spherical maps, Fund. Math. 127 (1987), 187-196.
• [9] A. Dold, Lectures on Algebraic Topology, Springer, 1980.
• [10] A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965), 1-8.
• [11] A. Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976), 215-244.
• [12] A. Dold, A coincidence-fixed-point index, Enseign. Math. (2) 24 (1978), 41-53.
• [13] A. N. Dranishnikov, Absolute extensors in dimension n and dimension-raising n-soft maps, Russian Math. Surveys 39 (1984), no. 5, 63-111.
• [14] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214-222.
• [15] L. Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976).
• [16] L. Górniewicz, Fixed point theorems for mutivalued maps of subsets of Euclidean spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 111-115.
• [17] L. Górniewicz and A. Granas, Some general theorems in coincidence theory I, J. Math. Pures Appl. 60 (1981), 361-373.
• [18] L. Górniewicz and A. Granas, Topology of morphisms and fixed point problems for set-valued maps, in: Fixed Point Theory and Applications, M. A. Thera and J.-B. Baillon (eds.), Pitman Res. Notes Math. Ser. 252, Longman Sci. Tech., Harlow, 1991, 173-191.
• [19] V. G. Gutev, A fixed-point theorem for $UV^n$ usco maps, Proc. Amer. Math. Soc. 124 (1996), 945-952.
• [20] B. Halpern, A general coincidence theory, Pacific J. Math. 77 (1978), 451-471.
• [21] D. S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1969), 584.
• [22] W. Kryszewski, Remarks on the Vietoris Theorem, Topol. Methods Nonlinear Anal. 8 (1996), 383-405.
• [23] S. Lefschetz, Algebraic Topology, Amer. Math. Soc. Colloq. Publ. 27, Amer. Math. Soc., Providence, RI, 1942.
• [24] K. Mukherjea, Coincidence theory for manifolds with boundary, Topology Appl. 46 (1992), 23-39.
• [25] M. Nakaoka, Coincidence Lefschetz numbers for a pair of fibre preserving maps, J. Math. Soc. Japan 32 (1980), 751-779.
• [26] B. O'Neill, A fixed point theorem for multi-valued functions, Duke Math. J. 24 (1957), 61-62.
• [27] S. N. Patnaik, Fixed points of multiple-valued transformations, Fund. Math. 65 (1969), 345-349.
• [28] H. Schirmer, Fixed points, antipodal points and coincidences of n-acyclic valued multifunctions, in: Topological Methods in Nonlinear Functional Analysis, Contemp. Math. 21, Amer. Math. Soc., Providence, RI, 1983, 207-212.
• [29] J. W. Vick, Homology Theory. An Introduction to Algebraic Topology, Academic Press, New York, 1973.
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