Embedding obstructions in \(\mathbb{R}^d\) from the Goodwillie-Weiss calculus and Whitney disks (Q6078273)

This paper addresses the question of whether a given \(m\)-dimensional CW complex \(K\) can be embedded topologically in \({\mathbb R}^{d}\). This question was studied by \textit{E. R. Van Kampen} [Abh. Math. Semin. Univ. Hamb. 9, 72--78 (1932; JFM 58.0615.02)] (generalizing Kuratowski's approach to planar graphs): Let \(\operatorname{Emb}(K,X)\) denote the space of topological embeddings of \(K\) into a space \(X\), abbreviated to \(C(X,i)\) when \(K\) is a finite set with \(i\) points (that is, the configuration space of ordered \(i\)-tuples of distinct points in \(X\)). An embedding of \(K\) in \({\mathbb R}^{d}\) implies that there is a \(\Sigma_{2}\)-equivariant map \(C(K,2)\to C({\mathbb R}^{d},2)\), and van Kampen discovered a geometric obstruction to the existence of such a map. \textit{M. H. Freedman} et al. showed that the vanishing of this obstruction is not sufficient for embedability, even in the case \(m=2\) and \(d=4\) (see [Math. Res. Lett. 1, No. 2, 167--176 (1994; Zbl 0847.57005)]). In this paper, the authors present two sequences of higher obstructions to the existence of an embedding: the first starts with a generic map \(f:K\hookrightarrow {\mathbb R}^{4}\), where \(K\) is a finite \(2\)-dimensional simplicial complex, and looks at the Whitney disks for \(f\) -- that is, non-adjacent \(2\)-simplices \(\sigma_{i}\) and \(\sigma_{j}\) in \(K\) with \(f(\sigma_{i})\cdot f(\sigma_{j})=0\). This is generalized to a \textit{Whitney tower} encoding higher order intersections of surfaces in \(4\)-manifolds (see [\textit{R. Schneiderman} and \textit{P. Teichner}, Doc. Math. 19, 941--992 (2014; Zbl 1302.57057)]). Given such a tower, the authors produce cohomology classes \(\mathcal{W}_{n}(K)\) in \(H^{2n}_{\Sigma_{n}}\) of the simplicial configuration space \(C_{s}(K,n)\), serving as successive obstructions to embeddability. An alternative geometric description of the secondary obstruction \(\mathcal{W}_{3}(K)\) is given in Section 2, with a third triple colinearity interpretation in Section 5. In the second approach, the authors define \(T_{n}\operatorname{Emb}(K,{\mathbb R}^{d})\) to be the set of derived natural transformations \(C(K,-)\to C({\mathbb R}^{d},-)\), thought of as functors on the category \({\mathbb I}_{n}\) of finite sets of cardinality \(\leq n\) and injective maps. Here `derived' means that we use a cofibrant replacement for the source functor, and a fibrant replacement for the target. This yields a `Taylor tower' \[ \cdots \to T_{n}\operatorname{Emb}(K,{\mathbb R}^{d}) \to T_{n-1}\operatorname{Emb}(K,{\mathbb R}^{d})\to\cdots \to T_{2}\operatorname{Emb}(K,{\mathbb R}^{d}) \to\ast~, \] under \(\operatorname{Emb}(K,{\mathbb R}^{n})\), analogous to the Goodwillie-Weiss construction (with \({\mathbb I}_{n}\) replacing the category of manifolds diffeomorphic to a disjoint union of \(\leq n\) copies of \({\mathbb R}^{m}\) -- see [\textit{P. Boavida de Brito} and \textit{M. Weiss}, Homology Homotopy Appl. 15, No. 2, 361--383 (2013; Zbl 1291.18025)]). (Note the confusion of \(n\) and \(d\) in Definition 7.3). A pullback square relating successive layers in the tower yields a sequence of cohomology classes \(\mathcal{O}_{n}(K)\in H^{(d-2)(n-1)+2}_{\Sigma_{n}}(C(K,n),{\mathbb Z}^{(n-2)!})\) serving as successive obstruction to lifting a (derived) natural transformation \(C(K,-)\to C({\mathbb R}^{d},-)\) from \({\mathbb I}_{n-1}\) to \({\mathbb I}_{n}\). The authors discuss the relations between the various approaches, and show the connection in some cases.