# Commentarii Mathematici Helvetici

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**Volume 77, Issue 1, 2002, pp. 78–124**

**DOI: 10.1007/s00014-002-8332-4**

Published online: 2002-03-31

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

A. Skopenkov^{[1]}(1) Moscow State University, Russian Federation

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}) $.

*Keywords: *Embedding, deleted product, engulfing, singular set, metastable case, isotopy, immersion, smoothing, knotted tori

Skopenkov A.: On the Haefliger-Hirsch-Wu invariants for embeddings and immersions. *Comment. Math. Helv.* 77 (2002), 78-124. doi: 10.1007/s00014-002-8332-4