The topologies and the differentiable structures of the images of special generic maps having simple structures

Abstract

Special generic maps are smooth maps at each singular point of which we can represent as (x 1 ,for suitable coordinates. Morse functions with exactly two singular points on homotopy spheres and canonical projections of unit spheres are special generic. They are known to restrict the topologies and the differentiable structures of the manifolds in various situations. On the other hands, various manifolds admit such maps.This article first presents a special generic map on a 7-dimensional manifold and the image. This results also seems to present a new example of 7-dimensional closed and simply-connected manifolds having non-vanishing triple Massey products and seems to be a new work related to similar works by Dranishnikov and Rudyak. We also review results on vanishing of products of cohomology classes, previously obtained by the author. The images of special generic maps are smoothly immersed manifolds whose dimensions are equal to the dimensions of the targets. The author studied the topologies of these images previously and studies on homology groups, cohomology rings and structures of them for special generic maps having simple structures will be presented as new results. Introduction.Throughout the present paper, manifolds and maps between manifolds are smooth or of class C ∞ . Diffeomorphisms on smooth manifolds are assumed to be smooth. We define the diffeomorphism group of a smooth manifold is the group of all the diffeomorphisms there. We assume that the structure groups of bundles whose fibers are smooth manifolds are subgroups of the diffeomorphism groups unless otherwise stated: in other words the bundles are smooth. The class of linear bundles is a subclass of the class of smooth bundles. A bundle is linear if the fiber is regarded as a unit sphere or a unit disc in a Euclidean space and the structure group acts linearly in a canonical way. In addition, (boundary) connected sums are discussed in the smooth category unless otherwise stated.A singular point p ∈ X of a smooth map c : X → Y is a point at which the rank of the differential dc p is smaller than both the dimensions dim X and dim Y . S(c) denotes the set of all the singular points of c (the singular set of c). We call c(S(c)) the singular value set of c. We call Y − c(S(c)) the regular value set of c.

A differential topological study of compact manifolds having simple structures

“…Notions and arguments in this section generalize several notions and arguments appearing first in [18]. We call equivalence classes under the natural equivalence relation in the PL and the smooth category PL types and diffeomorphism types, respectively.…”

“…The author previously studied the topologies and the differentiable structures of the images of in [18] and also in [16] and [17] before for example. [23] motivated the author to study this.…”

“…As a remark, in the introduction of [18], importance of compact (and smooth) manifolds with non-empty boundaries, in the knot theory, low-dimensional geometry, and several explicit studies of algebraic geometry such as topological properties of algebraic curves, complementary spaces of affine subspaces of real or complex vector spaces, and so on, is also explained shortly. It is also said that general algebraic topological or differential topological theory of compact manifolds possibly with non-empty boundaries is immature.…”

“…The third section is devoted to notions on compact manifolds with non-empty boundaries appearing as the domains of the immersions presented in the explanation on the images of special generic maps before and polyhedra they collapse to. They are essentially introduced first in [18]. We introduce these notions in a bit revised form.…”

“…It is also difficult to understand closed and simply-connected manifolds whose dimensions are greater than 4 in geometric and constructive ways due to the assumption that the dimensions are high and Nishioka and the author has obtained related results, mainly for 5 or 7dimensional closed and simply-connected manifolds, via higher dimensional variants of Morse functions with exactly two singular points on homotopy spheres and more general smooth maps: a homotopy sphere means a smooth manifold homeomorphic to a unit sphere in the present paper. Nishioka's work is [23] and related works of the author are some of [4]- [18]. Especially, Nishioka's result is on 5-dimensional Key words and phrases.…”