Cent. Eur. J. Math. • 12(9) • 2014 • 1330-1336 DOI: 10.2478/s11533-014-0410-5 Central European Journal of Mathematics On the homotopy equivalence of the spaces of proper and local maps Research Article Piotr Bartłomiejczyk 1∗ , Piotr Nowak-Przygodzki 2† 1 Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland 2 Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland Received 22 January 2013; accepted 28 November 2013 Abstract: We prove that for n> 1 the space of proper maps P 0 (n, k ) and the space of local maps F 0 (n, k ) are not homotopy equivalent. MSC: 55P10, 54C35 Keywords: Proper map • Local map • Homotopy equivalence © Versita Sp. z o.o. Introduction This paper is intended as an essential complement to our previous work [5], in which it is shown that the inclusion of the space of proper local maps P(n, k ) into the space of local maps F(n, k ) is a weak homotopy equivalence. Namely, in this article we prove that the above spaces are not homotopy equivalent for n> 1. Unfortunately, the problem in the case n = 1 remains unsolved, but we give some observations that may be useful for further studies. It is worth pointing out that the idea of studying spaces of partial, local and proper maps comes from [1–4, 6, 7, 9]. To be more precise, the space of partial (resp. proper) maps appears first in [1] (resp. [7]). The notion of local maps is introduced in [6] and, independently, in [9]. The relation between gradient and usual local maps (also in the equivariant case) is studied in [2–4]. Finally, in [5] authors introduce the topology on the set of local maps and prove that the inclusion of the space of proper maps into the space of local maps is a weak homotopy equivalence if we restrict ourselves to local maps with domains in R n+k and ranges in R n . ∗ E-mail: pb@mat.ug.edu.pl † E-mail: piotrnp@wp.pl 1330 Author copy