The following characterization of trivial \(\ell_2\)-bundles is well-known: a map \(f: X \to Y\) is homeomorphic to the trivial \(\ell_2\)-bundle \(\pi_Y: Y\times \ell_2 \to Y\) for a Polish space (a separable completely metrizable space) \(Y\) that is an ANE if and only if for \(k=\infty\), \(f\) is a surjective \(k\)-soft map of a Polish \(k\)-dimensional space \(X\) possessing the fibrewise \(k\)-universal property with respect to Polish spaces. This paper is considered as a preparation for a future solution to the existence problem of a finite-dimensional counterpart of this \(\pi_Y\): does there exist a map which is uniquely characterized by the above condition with \(k\) finite?
The main theorem states: Let \(\sigma = \{T_\beta \mid \beta\in B\}\), where \(|B| \geq k+1\), be a finite family of pairwise intersecting planes in \({\mathbb R}^m\), where \(m > (2k+1) + (k+1)^2\), of codimension \(k+1\) which are in general position. If a countable family \(\Sigma^\prime\) of planes of codimension \(k+1\) in \({\mathbb R}^m\) contains the translation \(\Sigma=\sigma+{\mathbb Q}^m\), then there exists a dense \(G_\delta\)-subset \({\mathcal L}\) of the Grassmann manifold \({\mathrm {Gr}} (2k+1, m)\) (the set of all \((2k+1)\)-dimensional planes in \({\mathbb R}^m\)) such that for every \(L \in {\mathcal L}\), the restriction \(p|N_{\Sigma^\prime}: N_{\Sigma^\prime} = {\mathbb R}^m\setminus \cup\Sigma^\prime \to L\) of the orthogonal projection \(p: {\mathbb R}^m \to L\) is \(k\)-soft and possesses the strong fibrewise \(k\)-universal property with respect to Polish spaces.
If \({\mathcal N} \subset L\) is a standard \(k\)-dimensional Nöbeling space and \(\dim N_{\Sigma^\prime} = k\) (this condition is satisfied if \(\sigma\) contains all the coordinate planes of codimension \(k+1\)), then \({\mathcal N}^\bullet = p^{-1}({\mathcal N}) \cap N_{\Sigma^\prime} \subset {\mathbb R}^m\) is a standard \(k\)-dimensional Nöbeling space, and the main theorem implies that the map \(p|{\mathcal N}^\bullet: {\mathcal N}^\bullet \to {\mathcal N}\), which is called a geometric Chigogidze resolution in this paper, is \(k\)-soft and possesses the strong fibrewise \(k\)-universal property with respect to Polish spaces.