Geodesic
On orthogonal projections of N\"{o}beling spaces
Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 627-658.

Voir la notice de l'article provenant de la source Math-Net.Ru

Suppose that $0\le k\infty$. We prove that there is a dense open subset of the Grassmann space $\operatorname{Gr}(2k+1,m)$ such that the orthogonal projection of the standard Nöbeling space $N^m_k$ (which lies in $\mathbb R^m$ for sufficiently large $m$) to every $(2k+1)$-dimensional plane in this subset is $k$-soft and possesses the strong $k$-universal property with respect to Polish spaces. Every such orthogonal projection is a natural counterpart of the standard Nöbeling space for the category of maps.
Keywords: Nöbeling space, strong fibrewise $k$-universal property, filtered finite-dimensional selection theorem, $\operatorname{AE}(k)$-space.
Mots-clés : Dranishnikov and Chigogidze resolutions 
@article{IM2_2020_84_4_a1,
     author = {S. M. Ageev},
     title = {On orthogonal projections of {N\"{o}beling} spaces},
     journal = {Izvestiya. Mathematics },
     pages = {627--658},
     publisher = {mathdoc},
     volume = {84},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a1/}
}
TY  - JOUR
AU  - S. M. Ageev
TI  - On orthogonal projections of N\"{o}beling spaces
JO  - Izvestiya. Mathematics 
PY  - 2020
SP  - 627
EP  - 658
VL  - 84
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a1/
LA  - en
ID  - IM2_2020_84_4_a1
ER  - 
%0 Journal Article
%A S. M. Ageev
%T On orthogonal projections of N\"{o}beling spaces
%J Izvestiya. Mathematics 
%D 2020
%P 627-658
%V 84
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a1/
%G en
%F IM2_2020_84_4_a1
S. M. Ageev. On orthogonal projections of N\"{o}beling spaces. Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 627-658. http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a1/

[1] H. Toruńczyk, “On CE-images of the Hilbert cube and characterization of $Q$-manifolds”, Fund. Math., 106:1 (1980), 31–40 | DOI | MR | Zbl

[2] H. Toruńczyk, “Characterizing Hilbert space topology”, Fund. Math., 111:3 (1981), 247–262 | DOI | MR | Zbl

[3] J. Mogilski, “Characterizing the topology of infinite-dimensional $\sigma$-compact manifolds”, Proc. Amer. Math. Soc., 92:1 (1984), 111–118 | DOI | MR | Zbl

[4] H. Toruńczyk, J. West, Fibrations and bundles with Hilbert cube manifold fibers, Mem. Amer. Math. Soc., 80, no. 406, Amer. Math. Soc., Providence, RI, 1989, iv+75 pp. | DOI | MR | Zbl

[5] M. Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc., 71, no. 380, Amer. Math. Soc., Providence, RI, 1988, vi+110 pp. | DOI | MR | Zbl

[6] S. Ageev, Axiomatic method of partitions in the theory of Menger and Nöbeling spaces, Topology Atlas preprint # 430, 2001, 98 pp. http://at.yorku.ca/v/a/a/a/87.htm

[7] S. M. Ageev, “Axiomatic method of partitions in the theory of Nöbeling spaces I. Improvement of partition connectivity”, Sb. Math., 198:3 (2007), 299–342 | DOI | DOI | MR | Zbl

[8] S. M. Ageev, “Axiomatic method of partitions in the theory of Nöbeling spaces. II. Unknotting theorem”, Sb. Math., 198:5 (2007), 597–625 | DOI | DOI | MR | Zbl

[9] S. M. Ageev, “Axiomatic method of partitions in the theory of Nöbeling spaces. III. Consistency of the axiom system”, Sb. Math., 198:7 (2007), 909–934 | DOI | DOI | MR | Zbl

[10] A. Nagórko, Characterization and topological rigidity of Nöbeling manifolds, Mem. Amer. Math. Soc., 223, no. 1048, Amer. Math. Soc., Providence, RI, 2013, viii+92 pp. | DOI | MR | Zbl

[11] M. Levin, Characterizing Nobeling spaces, 2006, arXiv: 0602361

[12] A. Chigogidze, M. M. Zarichnyi, “Universal Nöbeling spaces and pseudo-boundaries of Euclidean spaces”, Mat. Stud., 19:2 (2003), 193–200 | MR | Zbl

[13] A. N. Dranishnikov, “Absolute extensors in dimension $n$ and dimension-raising $n$-soft maps”, Russian Math. Surveys, 39:5 (1984), 63–111 | DOI | MR | Zbl

[14] A. Ch. Chigogidze, “$n$-soft mappings of $n$-dimensional spaces”, Math. Notes, 46:1 (1989), 558–562 | DOI | MR | Zbl

[15] V. V. Fedorchuk, A. Ch. Chigogidze, Absolyutnye retrakty i beskonechnye mnogoobraziya, Nauka, M., 1992, 232 pp. | MR

[16] S. M. Ageev, G. N. Gruzdev, Z. N. Silaeva, “Kharakterizatsiya $0$-mernoi rezolventy Chigogidze”, Vestn. Belorus. gos. un-ta. Ser. 1. Fiz. Matem. Inform., 2006, no. 2, 100–103 | MR | Zbl

[17] E. V. Shchepin, N. B. Brodskii, “Selections of filtered multivalued mappings”, Proc. Steklov Inst. Math., 212 (1996), 209–229 | MR | Zbl

[18] R. Engelking, General topology, Monogr. Mat., 60, PWN–Polish Scientific Publishers, Warsaw, 1977, 626 pp. | MR | MR | Zbl

[19] K. Borsuk, Theory of retracts, Monogr. Mat., 44, PWN–Polish Scientific Publishers, Warsaw, 1967, 251 pp. | MR | MR | Zbl

[20] Sze-tsen Hu, Theory of retracts, Wayne State Univ. Press, Detroit, 1965, 234 pp. | MR | Zbl

[21] P. L. Bowers, “General position properties satisfied by finite products of dendrites”, Trans. Amer. Math. Soc., 288:2 (1985), 739–753 | DOI | MR | Zbl

[22] M. Bestvina, J. Mogilski, “Characterizing certain incomplete infinite-dimensional absolute retracts”, Michigan Math. J., 33:3 (1986), 291–313 | DOI | MR | Zbl

[23] P. L. Bowers, “Limitation topologies on function spaces”, Trans. Amer. Math. Soc., 314:1 (1989), 421–431 | DOI | MR | Zbl

[24] J. Nagata, Modern dimension theory, Sigma Ser. Pure Math., 2, rev. ed., Heldermann Verlag, Berlin, 1983, ix+284 pp. | MR | Zbl

[25] P. S. Aleksandrov, B. A. Pasynkov, Vvedenie v teoriyu razmernosti, Nauka, M., 1973, 575 pp. | MR | Zbl

[26] J. van Mill, Infinite-dimensional topology. Prerequisites and introduction, North-Holland Math. Library, 43, North-Holland Publishing Co., Amsterdam, 1989, xii+401 pp. | MR | Zbl

[27] S. M. Ageev, D. Repovš, “A new construction of semi-free actions on Menger manifolds”, Proc. Amer. Math. Soc., 129:5 (2001), 1551–1562 | DOI | MR | Zbl

[28] E. Michael, “Continuous selections. II”, Ann. of Math. (2), 64:3 (1956), 562–580 | DOI | MR | Zbl

[29] W. Hurewicz, H. Wallman, Dimension theory, Princeton Math. Ser., 4, Princeton Univ. Press, Princeton, NJ, 1941, vii+165 pp. | MR | Zbl

[30] P. L. Bowers, “Dense embeddings of nowhere locally compact separable metric spaces”, Topology Appl., 26:1 (1987), 1–12 | DOI | MR | Zbl

[31] S. M. Ageev, S. A. Bogatyj, “Sewing in some classes of spaces”, Moscow Univ. Math. Bull., 49:6 (1994), 18–21 | MR | Zbl

[32] D. Repovš, P. V. Semenov, Continuous selections of multivalued mappings, Math. Appl., 455, Kluwer Acad. Publ., Dordrecht, 1998, viii+356 pp. | DOI | MR | Zbl

[33] N. B. Brodskii, “Extension of maps to the hyperspace of $UV^n$-compacta”, Russian Math. Surveys, 54:6 (1999), 1236–1237 | DOI | DOI | MR | Zbl

[34] N. B. Brodskii, “Extension of $UV^n$-valued mappings”, Math. Notes, 66:3 (1999), 283–291 | DOI | DOI | MR | Zbl

[35] N. B. Brodsky, “Sections of maps with fibers homeomorphic to a two-dimensional manifold”, Topology Appl., 120:1-2 (2002), 77–83 | DOI | MR | Zbl

[36] N. Brodsky, A. Chigogidze, E. V. Ščepin, “Sections of Serre fibrations with 2-manifold fibers”, Topology Appl., 155:8 (2008), 773–782 | DOI | MR | Zbl

[37] S. M. Ageev, M. Cencelj, D. Repovš, “Preserving $Z$-sets by Dranishnikov's resolution”, Topology Appl., 156:13 (2009), 2175–2188 | DOI | MR | Zbl

[38] E. Michael, “Continuous selections avoiding a set”, Topology Appl., 28:3 (1988), 195–213 | DOI | MR | Zbl

[39] P. L. Bowers, “Dense embeddings of sigma-compact, nowhere locally compact metric spaces”, Proc. Amer. Math. Soc., 95:1 (1985), 123–130 | DOI | MR | Zbl