On the Haefliger-Hirsch-Wu invariants for embeddings and immersions (Q1601068)

This work is in the PL category and treats the isotopy classes of embeddings of closed PL manifolds into Euclidean spaces and regular concordance classes of immersions of polyhedra into Euclidean spaces. We shall present two sample results which in the reviewer's opinion give an idea about the content of this extensive study. Already at the beginning of the investigation of the possibility of embedding an \(n\)-dimensional polyhedron \(K\) into the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\), E. R. van Kampen has used the so-called simplicial deleted product \(\widetilde T \subset T \times T\) consisting of products \(\sigma \times \tau\) of disjoint simplices, \(\sigma \cap \tau = \emptyset\), of a triangulation \(T\) of \(K\). He constructed an obstruction to embeddability of an \(n\)-polyhedron into \(\mathbb R^{2n}\) which was developed further. From that time on, \(\widetilde T\) and the deleted product \(\widetilde K = K \times K \smallsetminus \Delta\) (where \(\Delta\) denotes the diagonal of \(K \times K\)) are significantly used in the theory of embeddings and immersions in the PL and the differentiable category. To each embedding \(f: K \rightarrow \mathbb R^m\) one assigns a map \(\widetilde f : \widetilde K \rightarrow S^{m-1}\) by \(\widetilde f(x,y) = (f(x) - f(y))/ |f(x) -f(y) |\). \(\mathbb Z_2\) acts on \(\widetilde K\) and \(S^{m-1}\) freely by permuting factors of \(\widetilde K\) and by antipodes, respectively. Since \(\widetilde f\) is a \(\mathbb Z_2\)-equivariant map it is a representative of the equivariant homotopy class, denoted \(\alpha (f)\), of equivariant maps \(\widetilde K: \rightarrow S^{m-1}\). Denote by \(\pi_{\text{eq}}^{m-1}(\widetilde K)\) the set of equivariant homotopy classes of \(\widetilde K\) into \(S^{m-1}\) . Since \(\alpha (f)\) is clearly an isotopy invariant of \(f\), one obtains a function \(\alpha^m(K) : \text{Emb}^m(K) \rightarrow \pi_{\text{eq}}^{m-1}(\widetilde K)\), which in this paper is called Haefliger-Wu function, where \(\text{Emb}^m(K)\) denotes the set of isotopy classes of embeddings of \(K\) into \(\mathbb R^m\) within a considered category TOP, PL or DIFF. Treating closed PL manifolds, the author transfers and generalizes \textit{A. Haefliger}'s [Comment. Math. Helv. 37, 155-176 (1962;l Zbl 0186.27302)] theorem. We quote the following result of the paper: Let \(M\) be a closed \(d\)-connected PL \(n\)-manifold, \(m \geqq n + 3\), then the Haefliger-Wu function \(\alpha_{\text{PL}}^m(M)\) is a bijection for \(2m \geqq 3n + 3 - d\) and a surjection for \(2m \geqq 3n + 2 - d\). In case \(d = 1\) the homological 1-connectedness suffices. The other result we wish to point out concerns immersions of an \(n\)-polyhedron \(K\) into \(\mathbb R^m\). In this case one considers the function \(\beta^m(K): \text{Imm}^m(K) \rightarrow \pi_{\text{eq}}^{m-1}(SK)\) that relates regular concordance classes of immersions \(\text{Imm}^m(K)\) of \(n\)-polyhedra \(K\) into \(\mathbb R^m\) and \(\pi_{\text{eq}}^{m-1}(SK)\), again within a considered category. If one takes a sufficiently small symmetric neighbourhood \(O\Delta\) of \(\Delta\) and considers the following subset \(SK = O\Delta \smallsetminus \Delta \subset K\times K\), then by a similar reasoning there is a function \(\beta_{\text{eq}}: \text{Imm}^m(K) \rightarrow\pi_{\text{eq}}^{m-1}(SK)\) between the set of regular concordance classes \(\text{Imm}^m(K)\) of immersions of \(K\) into \(\mathbb R^m\) and the set of equivariant homotopy classes \(\pi_{\text{eq}}^{m-1}(SK)\) of maps of \(SK\) into \(S^{m-1}\). In this paper \(\beta^m\) is called Haefliger-Hirsch function [\textit{A. Haefliger} and \textit{M. W. Hirsch}, Ann. Math. (2) 75, 231-241 (1962; Zbl 0186.27301)]. Let us quote the following result: If \(K\) is an \(n\)-dimensional polyhedron, \(m \geqq n+3\), then \(\beta_{\text{PL}}^m(K)\) is a bijection for \(2m\geqq 3n + 3\) and a surjection for \(2m \geqq 3n + 2\). The above quoted statements are samples of results that one can find in the paper. There are other results about epimorphisms and bijections of \(\alpha^m\) and \(\beta^m\). Furthermore the paper contains examples which show that particular inequalities are sharp.