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 F₀ and F₈ : T³ right arrow-hooked R⁵ of the 3-torus T³ with the following properties: (1) F₀#h is regularly homotopic to F₈ for some immersion h : S³ right arrow, looped R⁵, and (2) the immersion h as above cannot be chosen from a regular homotopy class containing an embedding.