# A Lefschetz-type coincidence theorem

Fundamenta Mathematicae (1999)

- Volume: 162, Issue: 1, page 65-89
- ISSN: 0016-2736

## Access Full Article

top## Abstract

top## How to cite

topSaveliev, Peter. "A Lefschetz-type coincidence theorem." Fundamenta Mathematicae 162.1 (1999): 65-89. <http://eudml.org/doc/212413>.

@article{Saveliev1999,

abstract = {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.},

author = {Saveliev, Peter},

journal = {Fundamenta Mathematicae},

keywords = {Lefschetz coincidence theory; Lefschetz number; coincidence index; fixed point; multivalued map; Vietoris mapping},

language = {eng},

number = {1},

pages = {65-89},

title = {A Lefschetz-type coincidence theorem},

url = {http://eudml.org/doc/212413},

volume = {162},

year = {1999},

}

TY - JOUR

AU - Saveliev, Peter

TI - A Lefschetz-type coincidence theorem

JO - Fundamenta Mathematicae

PY - 1999

VL - 162

IS - 1

SP - 65

EP - 89

AB - 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.

LA - eng

KW - Lefschetz coincidence theory; Lefschetz number; coincidence index; fixed point; multivalued map; Vietoris mapping

UR - http://eudml.org/doc/212413

ER -

## References

top- [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. Zbl0791.55001
- [3] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott-Foresman, Chicago, 1971. Zbl0216.19601
- [4] R. F. Brown and H. Schirmer, Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary, Topology Appl. 46 (1992), 65-79. Zbl0757.55002
- [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. Zbl0843.55003
- [6] V. R. Davidyan, Coincidence points of two maps, Russian Acad. Sci. Sb. Math. 40 (1980), 205-210. Zbl0465.55002
- [7] V. R. Davidyan, On coincidence of two maps for manifolds with boundary, Russian Math. Surveys 38 (1983), no. 2, 176. Zbl0548.55001
- [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. Zbl0135.23101
- [11] A. Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976), 215-244. Zbl0329.55007
- [12] A. Dold, A coincidence-fixed-point index, Enseign. Math. (2) 24 (1978), 41-53. Zbl0378.55003
- [13] A. N. Dranishnikov, Absolute extensors in dimension n and dimension-raising n-soft maps, Russian Math. Surveys 39 (1984), no. 5, 63-111. Zbl0572.54012
- [14] S. Eilenberg and D. Montgomery, Fixed point theorems for multi-valued transformations, Amer. J. Math. 68 (1946), 214-222. Zbl0060.40203
- [15] L. Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. (Rozprawy Mat.) 129 (1976). Zbl0324.55002
- [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. Zbl0409.55002
- [17] L. Górniewicz and A. Granas, Some general theorems in coincidence theory I, J. Math. Pures Appl. 60 (1981), 361-373. Zbl0482.55002
- [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. Zbl0760.54030
- [19] V. G. Gutev, A fixed-point theorem for $U{V}^{n}$ usco maps, Proc. Amer. Math. Soc. 124 (1996), 945-952. Zbl0861.54041
- [20] B. Halpern, A general coincidence theory, Pacific J. Math. 77 (1978), 451-471. Zbl0411.55001
- [21] D. S. Kahn, An example in Čech cohomology, Proc. Amer. Math. Soc. 16 (1969), 584. Zbl0141.40302
- [22] W. Kryszewski, Remarks on the Vietoris Theorem, Topol. Methods Nonlinear Anal. 8 (1996), 383-405. Zbl0891.55024
- [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. Zbl0757.55003
- [25] M. Nakaoka, Coincidence Lefschetz numbers for a pair of fibre preserving maps, J. Math. Soc. Japan 32 (1980), 751-779. Zbl0447.55001
- [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. Zbl0203.56001
- [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. Zbl0262.55005

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.