Urtzi BUIJS, Antonio GARVIN, Aniceto MURILLO

Rational maps from Euclidean configuration spaces to spheres

In this note we give an algorithm to determine the rational homotopy type of the free and pointed mappingspaces $map(F(R^m,k),S^n)$ and $mapast(F(R^m,k),S^n)$ An explicit description of these spaces is given for k = 3. Thegeneral case for n odd is also presented as an immediate consequence of the rational version of a classical result of Thom.

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[1] Blagojević PVM, Ziegler G. Convex equipartitions via Equivariant Obstruction Theory. Israel Journal of Mathematics 2014; 200 (1): 49-77.
[2] Brown EH, Szczarba RH. On the rational homotopy type of function spaces. Transactions of the American Mathematical Society 1997; 349: 4931-4951.
[3] Buijs U, Murillo A. Basic constructions in rational homotopy theory of function spaces. Annales de l’institut Fourier 2006; 56 (3): 815-838.
[4] Buijs U, Murillo A. Rational Homotopy type of free and pointed mapping spaces between spheres. Proceedings of Ukrainian Mathematical Society 2013; 6 (6): 130-139.
[5] Cohen F, Lada T, May P. The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics vol 533. Berlin, Germany: Springer-Verlag, 1976.
[6] Fadell E, Neurwith L. Configuration spaces. Mathematica Scandinavica 1962; 10: 110-118.
[7] Félix Y, Halperin S, Thomas J. Rational homotopy theory. Springer GTM 2000; 205.
[8] Graham RL, Knuth DE, Patashnik O. Concrete Mathematics. Advanced Book Program, Reading, MA. UK: Addison–Wesley Publishing Company, 1989.
[9] Haefliger A. Rational homotopy of the space of sections of a nilpotent bundle. Transactions of the American Mathematical Society 1982; 273: 609-620.
[10] Hasegawa K. Minimal models of nilmanifolds. Proceedings of the American Mathematical Society 1989; 106 (1): 65-71.
[11] Hilton P, Mislin G, Roitberg J. Localization of nilpotent groups and spaces. North Holland: North Holland Mathematics Studies, 1975.
[12] Kontsevich M. Operads and motives in deformation quantization. Letters in Mathematical Physics 1999; 48 (1): 35-72.
[13] Nandakumar R, Ramana Rao N. Fair partitions of polygons: An elementary introduction. Proceedings of Indian Academy of Sciences (Mathematical Sciences) 2012; 122: 459-467.
[14] Thom R. L’homologie des espaces fonctionnels. Colloquium on Topological Algébrique Louvain 1956: 29-39 (in French).