Branched Shadows and Complex Structures on 4-Manifolds

Abstract

Abstract. We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spin c -structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.

Stein domains and branched shadows of 4-manifolds

“…Lemma 2.24 [6] Let R i , R j , and R k be three regions of P adjacent along a common edge e ∈ Sing(P) so that R i is the preferred region of e. There exists an isotopy φ t : P → M, t ∈ [0, 1] whose support is contained in a small ball B around the center of e such that φ 1 (B ∩ P) contains three more complex points p i , p j , and p k , respectively, in R i , R j , and R k whose indices are, respectively, ν(p i ) = ±1, ν(p j ) = ∓1 and ν(p k ) = ∓1.…”

“…For a complete account on shadows (see [22,23]); for an introductory one, see [7]; for a detailed account of branched shadows we refer to [6]. From now on, all the manifolds and homeomorphisms will be smooth unless explicitly stated.…”

“…Lemma 2.18 [6] The gleam cochain gl(P) represents in H 2 (M; Z) the first Chern class of the field of oriented 2-planes V(P).…”

“…This causes that, roughly speaking, while annihilating hyperbolic and elliptic points, one has to count the number of times this annihilation "involves" the singular set. In [6], we proved a lemma which represents the translation in the world of shadows of the well known Harlamov-Eliashberg Annihilation Theorem for complex points over real surfaces: roughly speaking, it says that one can shift these points along a branched polyhedron, but while passing through the singular set, they "duplicate" so that the cochains I + and I − change by a coboundary. The resulting sufficient condition we get in Theorem 3.20, can be summarized, as follows: Theorem 1.…”

Stable maps and branched shadows of 3-manifolds

“…Shadows provide important topological properties for 4 and 3-manifolds as well as being an interesting notion for the study of quantum topology. We refer the reader to Costantino-Thurston [12] for triangulations of 4 and 3-manifolds, Costantino [9,10] for Stein structure, Spin c structure and complex structure of 4-manifolds and Martelli [26] for a classification of 4-manifolds admitting shadows without vertices. See also the survey paper [8] by Costantino.…”