Geodesic
Configuration-like spaces and coincidences of maps on orbits
Karasev, Roman1 ; Volovikov, Alexey2
1Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700
2Department of Mathematics, University of Texas at Brownsville, 80 Fort Brown, Brownsville TX 78520, USA, Department of Higher Mathematics, Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University), Pr. Vernadskogo 78, Moscow 117454, Russia
Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 1033-1052
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

In this paper we study the spaces of q–tuples of points in a Euclidean space, without k–wise coincidences (configuration-like spaces). A transitive group action by permuting these points is considered, and some new upper bounds on the genus (in the sense of Krasnosel’skii–Schwarz and Clapp–Puppe) for this action are given. Some theorems of Cohen–Lusk type for coincidence points of continuous maps to Euclidean spaces are deduced.

DOI : 10.2140/agt.2011.11.1033
Keywords: configuration space, coincidence, equivariant topology, Krasnosel'skii–Schwarz genus

Karasev, Roman  1   ; Volovikov, Alexey  2

1 Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700
2 Department of Mathematics, University of Texas at Brownsville, 80 Fort Brown, Brownsville TX 78520, USA, Department of Higher Mathematics, Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University), Pr. Vernadskogo 78, Moscow 117454, Russia 
@article{10_2140_agt_2011_11_1033,
     author = {Karasev, Roman and Volovikov, Alexey},
     title = {Configuration-like spaces and coincidences of maps on orbits},
     journal = {Algebraic and Geometric Topology},
     pages = {1033--1052},
     year = {2011},
     volume = {11},
     number = {2},
     doi = {10.2140/agt.2011.11.1033},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1033/}
}
TY  - JOUR
AU  - Karasev, Roman
AU  - Volovikov, Alexey
TI  - Configuration-like spaces and coincidences of maps on orbits
JO  - Algebraic and Geometric Topology
PY  - 2011
SP  - 1033
EP  - 1052
VL  - 11
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1033/
DO  - 10.2140/agt.2011.11.1033
ID  - 10_2140_agt_2011_11_1033
ER  - 
%0 Journal Article
%A Karasev, Roman
%A Volovikov, Alexey
%T Configuration-like spaces and coincidences of maps on orbits
%J Algebraic and Geometric Topology
%D 2011
%P 1033-1052
%V 11
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2011.11.1033/
%R 10.2140/agt.2011.11.1033
%F 10_2140_agt_2011_11_1033
Karasev, Roman; Volovikov, Alexey. Configuration-like spaces and coincidences of maps on orbits. Algebraic and Geometric Topology, Tome 11 (2011) no. 2, pp. 1033-1052. doi: 10.2140/agt.2011.11.1033

[1] T Bartsch, On the genus of representation spheres, Comment. Math. Helv. 65 (1990) 85

[2] T Bartsch, Topological methods for variational problems with symmetries, 1560, Springer (1993)

[3] I Basabe, J Gonzalez, Y B Rudyak, D Tamaki, Higher topological complexity and homotopy dimension of configuration spaces on spheres

[4] D V Bolotov, The Cohen–Lusk conjecture, Mat. Zametki 70 (2001) 22

[5] K Borsuk, Drei Sätze über die n–dimensionale euklidische Sphäre, Fund. Math. 20 (1933) 177

[6] M Clapp, D Puppe, Invariants of the Lusternik–Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986) 603

[7] M Clapp, D Puppe, Critical point theory with symmetries, J. Reine Angew. Math. 418 (1991) 1

[8] F Cohen, E L Lusk, Configuration-like spaces and the Borsuk–Ulam theorem, Proc. Amer. Math. Soc. 56 (1976) 313

[9] P E Conner, E E Floyd, On the construction of periodic maps without fixed points, Proc. Amer. Math. Soc. 10 (1959) 354

[10] P E Conner, E E Floyd, Fixed point free involutions and equivariant maps, Bull. Amer. Math. Soc. 66 (1960) 416

[11] R N Karasev, The genus and the category of configuration spaces, Topology Appl. 156 (2009) 2406

[12] R N Karasev, Periodic billiard trajectories in smooth convex bodies, Geom. Funct. Anal. 19 (2009) 423

[13] R N Karasev, A Y Volovikov, Knaster’s problem for almost (Zp)k–orbits, Topology Appl. 157 (2010) 941

[14] B Knaster, Problem 4, Colloq. Math. 30 (1947) 30

[15] M A Krasnosel’Skiĭ, On the estimation of the number of critical points of functionals, Uspehi Matem. Nauk (N.S.) 7 (1952) 157

[16] M A Krasnosel’Skiĭ, On special coverings of a finite-dimensional sphere, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955) 961

[17] M A Krasnosel’Skiĭ, Stability of the critical values of even functionals on the sphere, Mat. Sb. N.S. 37(79) (1955) 301

[18] V V Makeev, The Knaster problem and almost spherical sections, Mat. Sb. 180 (1989) 424

[19] F Roth, On the category of Euclidean configuration spaces and associated fibrations, from: "Groups, homotopy and configuration spaces", Geom. Topol. Monogr. 13 (2008) 447

[20] A S Schwarz, Some estimates of the genus of a topological space in the sense of Krasnosel’skiĭ, Uspehi Mat. Nauk (N.S.) 12 (1957) 209

[21] A S Schwarz, The genus of a fibre space, Trudy Moskov Mat. Obšč. 11 (1962) 99

[22] S Smale, On the topology of algorithms I, J. Complexity 3 (1987) 81

[23] V A Vasil’Ev, Cohomology of braid groups and the complexity of algorithms, Funktsional. Anal. i Prilozhen. 22 (1988) 15, 96

[24] A Y Volovikov, A Bourgin–Yang-type theorem for Zpn–action, Mat. Sb. 183 (1992) 115

[25] A Y Volovikov, On the index of G–spaces, Mat. Sb. 191 (2000) 3

[26] A Y Volovikov, Coincidence points of mappings of Zpn–spaces, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005) 53

[27] A Y Volovikov, On the Cohen–Lusk theorem, Fundam. Prikl. Mat. 13 (2007) 61

[28] C T Yang, Continuous functions from spheres to euclidean spaces, Ann of Math. (2) 62 (1955) 284

[29] C T Yang, On theorems of Borsuk–Ulam, Kakutani–Yamabe–Yujobô and Dyson II, Ann. of Math. (2) 62 (1955) 271

Cité par Sources :