Prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem

Abstract

ABSTRACT. We find a connection between the Rokhlin theorem on the signature of a four-dimensional manifold and the notion of a prem-mapping that arises from the theory of embeddings of smooth manifolds.In the papers [1, 2], A. Szucs introduced the notion of a prem-mapping (an abbreviation of projected embedding, see Definition 1 below). This notion turned out to be very useful in the study of the cobordism group of embeddings of smooth manifolds in the case of nonstable codimension. Here and below a surface is the image of an arbitrary mapping s t in general position. We do not assume that a surface is embedded or immersed unless this is stated explicitly. Then for a mapping f: M 2 ~ R a of an oriented surface M 2 into three-dimensional Euclidean space R 3 there is a simple necessary and sufficient condition for .f to be a prem-mapping, see Criterion 1 below. For triple points of the surface f(M 2), there exists an additional structure (A, B-classification, see Definition 2); the criterion is related to this structure.A mapping f: M 2 ---, R 3 is not necessarily a prem-mapping. For example, consider a two-sheeted covering over the Boy surface (see [3,4]); put the covering space in general position by a small C ~176 deformation. Then this mapping is not a prem-mapping, see Proposition 2 below.Theorem 3 is the central result of the paper. This theorem is similar to the weLl-known Banchoff theorem, see [5]. It describes the relation between A-and B-structures of triple points of an oriented surface. More precisely, the condition M ~-~ S 2 implies parity restrictions for the number of triple points of type B. This theorem is a direct consequence of the Rokhlin theorem on the signature of a fourdimensional manifold in the form given in the paper [6]; also see the book [7]. On the other hand, it can be regarded as an elementary illustration of the Rokhlin theorem.Propositions 4, 5, and 6 make Theorem 3 more precise. In Proposition 4 we prove that for the torus (and thus for each surface of genus g) the statement of Theorem 3 does not hold in the general case. Proposition 5 shows that there is no direct generalization of Theorem 3 that gives integer-valued estimates. Proposition 6 detects a relationship between triple points of different types on an embedded surface. Propositions 4 and 6 are direct reformulations of the results of the paper [4]. Now let us define prem-mappings in the dimensions that we are interested in.

Lifting generic maps to embeddings. Triangulation and smoothing

“…Proof. Clearly (1) implies (2). The implication (2)⇒( 3) is proved in Lemma 3.1 in the case k = 1, and the general case is similar.…”

“…1 When the choice of a category is irrelevant, we will speak simply of "k-prems". The abbreviation "prem" was coined by A. Szűcz in the 90s (see [2], [55]), while the notion itself is older [14], [15], [19], [20], [26], [42], [43], [45], [52]. Other related work includes [6], [25], [46], [50], [54], [56], [58].…”

“…1 That is, a continuous/PL/smooth map which is a homeomorpism/PL homeomorphism/diffeomorphism onto its image. 2 A PL map is called non-degenerate if it has no point-inverses of dimension > 0.…”

Projected and near-projected embeddings

“…We call a map f : N → M a (PL/smooth) k-prem (k-codimensionally projected embedding) if it factors into the composition of some (PL/smooth) embedding N ֒→ M × R k and the projection M × R k → M. For example, a constant map f is a k-prem if and only if N embeds in R k . The abbreviation "prem" was coined by A. Szűcz in the 90s (see [2]), but is younger than many results about k-prems. Let us mention some of them (further references on the subject can be found in [4], [30], [31]).…”