Smooth maps like special generic maps

Abstract

In our paper, we introduce special-generic-like maps or SGL maps as smooth maps, present their fundamental algebraic topological and differential topological theory and give non-trivial examples.The new class generalize the class of so-called special generic maps. Special generic maps are smooth maps which are locally projections or the product maps of Morse functions and the identity maps on disks. Morse functions with exactly two singular points on spheres or Morse functions in Reeb's theorem are simplest examples. Special generic maps and the manifolds of their domains have been studied well. Their structures are simple and this help us to study explicitly. As important properties, they have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author. To cover wider classes of manifolds as the domains, the author previously introduced a class generalizing the class of special generic maps and smaller than our class: simply generalized special generic maps.

Restrictions on Manifolds admitting certain explicit special-generic-like maps and construction of maps with the manifolds

“…This is a new result on new explicit observations on maps in [22,23]. Main Theorem 1 constructs maps explicitly on manifolds satisfying some topological conditions.…”

“…Note that here we can consider natural identifications between the base spaces of the bundles and the fibers and we consider these identifications. This is a fundamental and natural method and related remarks are also in [23] for example. [23] also reviews shortly that [24] studies the structures and special properties of special generic maps and the manifolds obtained in such ways as most simplest examples.…”

“…This is a fundamental and natural method and related remarks are also in [23] for example. [23] also reviews shortly that [24] studies the structures and special properties of special generic maps and the manifolds obtained in such ways as most simplest examples. STEP 3 Construction of an FSGL map into R n like one in STEP 2 on some closed and simply-connected manifold M of dimension m = n + 2 whose 2nd integral homology group is free.…”

“…For related theory on symplectic toric manifolds and moment maps, see [3,5] for example. [22,23] say that our SGLs are introduced to cover classes of manifolds of the domains seeming to be difficult to obtain by special generic maps. symplectic toric manifolds admit no special generic maps in considerable cases.…”

“…Proof. General construction of an FSGL map on a closed and simply-connected manifold such that "m in Definition 2" is greater than or equal to 3 and that F f is a closed, connected and orientable surface of genus g ≥ 1 there is essentially same as one in the proof of Theorem 3 (Main Theorm 2) of [23] and our proof of main result of [13].…”