On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

Abstract

We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g.¶ 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in \( \mathbb{R}^11 \).¶ 2) If \( p \leq q \) and \( m \geq \frac{3q}2 + p + 2 \) then the set of PL embeddings \( S^{p} \times S^{q} \to \mathbb{R}^m \) up to PL isotopy is in 1-1 correspondence with¶\( \pi_q(V_{m-q,p+1})\oplus\pi_p(V_{m-p,q+1}) \).