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

TL;DR

This paper explores the topologies and differentiable structures of images of special generic maps, presenting new examples in 7-dimensional manifolds and analyzing their cohomological properties.

Contribution

It introduces a new example of a 7-dimensional closed, simply-connected manifold with special generic maps and studies the cohomology structures of their images.

Findings

New example of 7-dimensional manifold with special generic map

Analysis of cohomology rings and structures of images

Results on non-vanishing triple Massey products

Abstract

Special generic maps are smooth maps at each singular point of which we can represent as $(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ 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 are presented as new results.