Regular homotopy classes of immersions of 3-manifolds into 5-space

 We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and of the 3-torus T ³ with the following properties: (1) is regularly homotopic to F ₈ for some immersion , and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding. Received: 29 March 2001