×

On the homotopy equivalence of the spaces of proper and local maps. (English) Zbl 1297.55010

For any topological spaces \(X\) and \(Y\), let \(\mathrm{Map}(X,Y)\) denote the space of all continuous maps from \(X\) to \(Y\) with the compact open topology and let \({\mathcal M}(X,Y)\) be the space of all continuous maps \(f:D_f\to Y\) such that \(D_f\) is an open subset of \(X\). For a family \({\mathcal R}\) of subsets of \(Y\), we denote by \(\mathrm{Loc}(X,Y;{\mathcal R})\) the space of all maps \(f\in {\mathcal M}(X,Y)\) such that \(f^{-1}(R)\) is a compact subset of \(D_f\) for any \(R\in {\mathcal R}\). When \({\mathcal R}\) is a family of all compact subsets of \(Y\), we define the spaces of partial and proper maps \(\mathrm{Par}(X,Y)\) and \(\mathrm{Prop}(X,Y)\) by \(\mathrm{Par}(X,Y)=\mathrm{Loc}(X,Y;\emptyset )\) and \(\mathrm{Prop}(X,Y)=(X,Y;{\mathcal R})\), respectively. For \(n\geq 1\) and \(k\geq 0\), let \({\mathcal F}(n,k)\) and \({\mathcal P}(n,k)\) denote the spaces given by \({\mathcal F}(n,k)=\mathrm{Loc}(\mathbb{R}^{n+k},\mathbb{R}^k;\{\{{\mathbf 0}\}\})\) and \({\mathcal P}(n,k)=\mathrm{Prop}(\mathbb{R}^{n+k},\mathbb{R}^n)\), and let \({\mathcal F}_0(n,k)\) (resp. \({\mathcal P}_0(n,k)\)) be the component of \({\mathcal F}(n,k)\) (resp. \({\mathcal P}(n,k)\)) which contains the empty map.
Recently the authors showed that the inclusion map \({\mathcal P}(n,k)\to {\mathcal F}(n,k)\) is a weak homotopy equivalence for \(n\geq 1\) and \(k\geq 0\), and this paper is essentially intended as an essential complement of this result. In fact, in this paper the authors prove that two spaces \({\mathcal P}_0(n,k)\) and \({\mathcal F}_0(n,k)\) are not homotopy equivalent for \(n>1\) and \(k\geq 0\). Moreover, they also consider the case \(n=1\) and they prove that \({\mathcal P}_0(1,k)\) is contractible and that \({\mathcal F}_0(1,k)\) is weakly contractible.

MSC:

55P10 Homotopy equivalences in algebraic topology
55P15 Classification of homotopy type
54C35 Function spaces in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abd-Allah A.M., Brown R., A compact-open topology on partial maps with open domain, J. London Math. Soc., 1980, 21(3), 480-486 http://dx.doi.org/10.1112/jlms/s2-21.3.480; · Zbl 0436.54012
[2] Bartłomiejczyk P., Geba K., Izydorek M., Otopy classes of equivariant maps, J. Fixed Point Theory Appl., 2010, 7(1), 145-160 http://dx.doi.org/10.1007/s11784-010-0013-0; · Zbl 1205.55008
[3] Bartłomiejczyk P., Nowak-Przygodzki P., Gradient otopies of gradient local maps, Fund. Math., 2011, 214(1), 89-100 http://dx.doi.org/10.4064/fm214-1-6; · Zbl 1229.55003
[4] Bartłomiejczyk P., Nowak-Przygodzki P., Proper gradient otopies, Topology Appl., 2012, 159(10-11), 2570-2579 http://dx.doi.org/10.1016/j.topol.2012.04.014; · Zbl 1247.55002
[5] Bartłomiejczyk P., Nowak-Przygodzki P., The exponential law for partial, local and proper maps and its application to otopy theory, Commun. Contemp. Math. (in press), DOI: 10.1142/S0219199714500059; · Zbl 1303.55003
[6] Becker J.C., Gottlieb D.H., Vector fields and transfers, Manuscripta Math., 1991, 72(2), 111-130 http://dx.doi.org/10.1007/BF02568269; · Zbl 0736.55012
[7] Becker J.C., Gottlieb D.H., Spaces of local vector fields, In: Higher Homotopy Structures in Topology and Mathematical Physics, Contemp. Math., 227, American Mathematical Society, Providence, 1999, 21-28 http://dx.doi.org/10.1090/conm/227/03250;
[8] Cohen F.R., Fibration and product decompositions in nonstable homotopy theory, In: Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, 1175-1208 http://dx.doi.org/10.1016/B978-044481779-2/50025-0; · Zbl 0871.55008
[9] Dancer E.N., Geba K., Rybicki S.M., Classification of homotopy classes of equivariant gradient maps, Fund. Math., 2005, 185(1), 1-18 http://dx.doi.org/10.4064/fm185-1-1; · Zbl 1086.47031
[10] Serre J.-P., Homologie singulière des espaces fibrés, Applications, Ann. of Math., 1951, 54(3), 425-505 http://dx.doi.org/10.2307/1969485; · Zbl 0045.26003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.