An Ekholm-Szűcs-type formula for codimension one immersions of 3-manifolds up to bordism

Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions

“…where Σ 2,0 (F ) is the (algebraic) number of umbilic points of F . This was essentially proved in the proof of [32,Lemma 3.2]. Here it should be mentioned that the sign convention in [32] differs from that used here; see Remark 2.6.…”

“…This was essentially proved in the proof of [32,Lemma 3.2]. Here it should be mentioned that the sign convention in [32] differs from that used here; see Remark 2.6. The proof of (2.5) is based on the Thom polynomial for umbilic singularities of a generic map G : M → R 4 of a closed oriented 4-manifold M : For the purposes of this paper, we weaken the condition for singular Seifert surfaces and use the following characterization.…”

“…as follows. If f : M → R 4 is an immersion of a closed 3-manifold representing a cobordism class of immersions then we map it to [32]). Note that μ(S 3 , s f ) is always equal to zero in Z 16 .…”

“…In [32], generic maps F : V → R 4 of compact oriented 4-manifolds were studied and the following result was derived: if ∂V = S 3 and the restriction f = F | ∂V : S 3 → R 4 is an immersion, and if F is non-singular in a neighbourhood of ∂V , then the Hirzebruch defect H (f ) satisfies…”

“…Theorem 1.1 is proved in Section 3; for other constructions of generators of the stable 3stem, see [4,10]. The proof uses singular Seifert surfaces, which were introduced in [6] and have since proved to be quite useful in the study of embeddings, immersions and other maps (see, for example, [1,6,7,12,27,32]). In fact, one of the main technical points of this paper is a concrete construction of singular Seifert surfaces for the immersions g n .…”